Fractions Introduction - Python Programming

Recipes, measurements, and mathematical calculations often involve exact ratios that lose precision when converted to decimals. The Fraction module represents rational numbers as numerator/denominator pairs, automatically simplifying results and maintaining perfect accuracy.

Fraction A numeric type representing exact rational numbers as a ratio of two integers, with automatic simplification to lowest terms.

Creating Fractions

Multiple ways to create fraction objects:

                            create.py
                        
Replay: real traced execution (multi-file project)

                            # Create Fraction

from fractions import Fraction
import math

# Create Fraction
print("Create Fraction:")

# From integers (numerator, denominator)
f1 = Fraction(3, 4)
print(f"From ints (3, 4): {f1}")

# From float
f2 = Fraction(0.75)
print(f"From float (0.75): {f2}")

# From string
f3 = Fraction('3/4')
print(f"From string ('3/4'): {f3}")

f4 = Fraction('0.75')
print(f"From string ('0.75'): {f4}")

# From Decimal
from decimal import Decimal
d = Decimal('0.75')
f5 = Fraction(d)
print(f"From Decimal: {f5}")

# Whole numbers
f6 = Fraction(5)
print(f"From int (5): {f6}")

# Negative fractions
f7 = Fraction(-3, 4)
print(f"Negative (-3, 4): {f7}")

f8 = Fraction(3, -4)
print(f"Negative (3, -4): {f8}")

# Automatic simplification
print("\nAutomatic simplification:")
f9 = Fraction(6, 8)
print(f"6/8 simplified: {f9}")

f10 = Fraction(12, 16)
print(f"12/16 simplified: {f10}")

f11 = Fraction(100, 200)
print(f"100/200 simplified: {f11}")

# Numerator and denominator
print("\nNumerator and denominator:")
f = Fraction(3, 4)
print(f"Fraction: {f}")
print(f"Numerator: {f.numerator}")
print(f"Denominator: {f.denominator}")

# Zero
print("\nZero:")
zero = Fraction(0)
print(f"Zero: {zero}")
print(f"Numerator: {zero.numerator}")
print(f"Denominator: {zero.denominator}")

# From float (precision note)
print("\nFrom float (precision):")
f12 = Fraction(0.1)
print(f"Fraction(0.1): {f12}")  # Shows float precision issues

f13 = Fraction('0.1')
print(f"Fraction('0.1'): {f13}")  # Exact

# limit_denominator
print("\nlimit_denominator:")
f14 = Fraction(0.1)
print(f"Original: {f14}")
print(f"Limited to 10: {f14.limit_denominator(10)}")
print(f"Limited to 100: {f14.limit_denominator(100)}")

# Approximate pi
print("\nApproximate pi:")
pi_frac = Fraction(str(math.pi))
print(f"Pi as fraction: {pi_frac}")
print(f"Pi limited to 1000: {pi_frac.limit_denominator(1000)}")
print(f"Pi limited to 100: {pi_frac.limit_denominator(100)}")

# from_float and from_decimal
print("\nfrom_float and from_decimal:")
f15 = Fraction.from_float(0.75)
print(f"from_float(0.75): {f15}")

f16 = Fraction.from_decimal(Decimal('0.75'))
print(f"from_decimal('0.75'): {f16}")

# Convert to other types
print("\nConvert to other types:")
f = Fraction(3, 4)
print(f"Fraction: {f}")
print(f"float: {float(f)}")
print(f"int (truncate): {int(f)}")
print(f"Decimal: {Decimal(f.numerator) / Decimal(f.denominator)}")

# Common fractions
print("\nCommon fractions:")
fractions = [
    (1, 2), (1, 3), (1, 4), (1, 5),
    (2, 3), (3, 4), (3, 5), (4, 5)
]
for num, den in fractions:
    f = Fraction(num, den)
    print(f"{num}/{den} = {f} = {float(f):.4f}")


                        
1f1 ← 3/4, f2 ← 3/4, f3 ← 3/4, f4 ← 3/4, d ← 0.75, f5 ← 3/4, f6 ← 5⋯6# Create Fraction7print("Create Fraction:")89# From integers (numerator, denominator)10f1→ 3/4 = Fraction(3, 4)11print(f"From ints (3, 4): {f13/4}")1213# From float14f2→ 3/4 = Fraction(0.75)15print(f"From float (0.75): {f23/4}")1617# From string18f3→ 3/4 = Fraction('3/4')19print(f"From string ('3/4'): {f33/4}")2021f4→ 3/4 = Fraction('0.75')22print(f"From string ('0.75'): {f43/4}")2324# From Decimal25from decimal import Decimal26d→ 0.75 = Decimal('0.75')27f5→ 3/4 = Fraction(d0.75)28print(f"From Decimal: {f53/4}")2930# Whole numbers31f6→ 5 = Fraction(5)32print(f"From int (5): {f65}")3334# Negative fractions35f7→ -3/4 = Fraction(-3, 4)36print(f"Negative (-3, 4): {f7-3/4}")3738f8→ -3/4 = Fraction(3, -4)39print(f"Negative (3, -4): {f8-3/4}")4041# Automatic simplification42print("\nAutomatic simplification:")43f9→ 3/4 = Fraction(6, 8)44print(f"6/8 simplified: {f93/4}")4546f10→ 3/4 = Fraction(12, 16)47print(f"12/16 simplified: {f103/4}")4849f11→ 1/2 = Fraction(100, 200)50print(f"100/200 simplified: {f111/2}")5152# Numerator and denominator53print("\nNumerator and denominator:")54f→ 3/4 = Fraction(3, 4)55print(f"Fraction: {f3/4}")56print(f"Numerator: {f.numerator3}")57print(f"Denominator: {f.denominator4}")5859# Zero60print("\nZero:")61zero→ 0 = Fraction(0)62print(f"Zero: {zero0}")63print(f"Numerator: {zero.numerator0}")64print(f"Denominator: {zero.denominator1}")6566# From float (precision note)67print("\nFrom float (precision):")68f12→ 3602879701896397/36028797018963968 = Fraction(0.1)69print(f"Fraction(0.1): {f123602879701896397/36028797018963968}")  # Shows float precision issues7071f13→ 1/10 = Fraction('0.1')72print(f"Fraction('0.1'): {f131/10}")  # Exact7374# limit_denominator75print("\nlimit_denominator:")76f14→ 3602879701896397/36028797018963968 = Fraction(0.1)77print(f"Original: {f143602879701896397/36028797018963968}")78print(f"Limited to 10: {f143602879701896397/36028797018963968.limit_denominator(10)}")79print(f"Limited to 100: {f143602879701896397/36028797018963968.limit_denominator(100)}")8081# Approximate pi82print("\nApproximate pi:")83pi_frac→ 3141592653589793/1000000000000000 = Fraction(str(math.pi3.141592653589793))84print(f"Pi as fraction: {pi_frac3141592653589793/1000000000000000}")85print(f"Pi limited to 1000: {pi_frac3141592653589793/1000000000000000.limit_denominator(1000)}")86print(f"Pi limited to 100: {pi_frac3141592653589793/1000000000000000.limit_denominator(100)}")8788# from_float and from_decimal89print("\nfrom_float and from_decimal:")90f15→ 3/4 = Fraction<class 'fractions.Fraction'>.from_float(0.75)91print(f"from_float(0.75): {f153/4}")9293f16→ 3/4 = Fraction<class 'fractions.Fraction'>.from_decimal(Decimal('0.75'))94print(f"from_decimal('0.75'): {f163/4}")9596# Convert to other types97print("\nConvert to other types:")98f→ 3/4 = Fraction(3, 4)99print(f"Fraction: {f3/4}")100print(f"float: {float(f3/4)}")101print(f"int (truncate): {int(f3/4)}")102print(f"Decimal: {Decimal(f.numerator3) / Decimal(f.denominator4)}")103104# Common fractions105print("\nCommon fractions:")106fractions→ [(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (3, 4), (3, 5), (4, 5)] = [107    (1, 2), (1, 3), (1, 4), (1, 5),108    (2, 3), (3, 4), (3, 5), (4, 5)109]110for num, den in fractions:⋯outputCreate Fraction:
From ints (3, 4): 3/4
From float (0.75): 3/4
From string ('3/4'): 3/4
From string ('0.75'): 3/4
From Decimal: 3/4
From int (5): 5
Negative (-3, 4): -3/4
Negative (3, -4): -3/4

Automatic simplification:
6/8 simplified: 3/4
12/16 simplified: 3/4
100/200 simplified: 1/2

Numerator and denominator:
Fraction: 3/4
Numerator: 3
Denominator: 4

Zero:
Zero: 0
Numerator: 0
Denominator: 1

From float (precision):
Fraction(0.1): 3602879701896397/36028797018963968
Fraction('0.1'): 1/10

limit_denominator:
Original: 3602879701896397/36028797018963968
Limited to 10: 1/10
Limited to 100: 1/10

Approximate pi:
Pi as fraction: 3141592653589793/1000000000000000
Pi limited to 1000: 355/113
Pi limited to 100: 311/99

from_float and from_decimal:
from_float(0.75): 3/4
from_decimal('0.75'): 3/4

Convert to other types:
Fraction: 3/4
float: 0.75
int (truncate): 0
Decimal: 0.75

Common fractions:
2f ← 1/2pass 1 of 8⋯109]110for num1, den2 in fractions[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (3, 4), (3, 5), (4, 5)]:111    f→ 1/2 = Fraction(num1, den2)112    print(f"{num1}/{den2} = {f1/2} = {float(f):.4f}")⋯output1/2 = 1/2 = 0.5000All 8 passes — pass 1 is the card abovepassnumdenf
1121/2
2131/3
3141/4
4151/5
5232/3
6343/4
7353/5
8454/5

All 8 passes — pass 1 is the card above
pass	`num`	`den`	`f`
1	1	2	1/2
2	1	3	1/3
3	1	4	1/4
4	1	5	1/5
5	2	3	2/3
6	3	4	3/4
7	3	5	3/5
8	4	5	4/5

automatic simplification Fractions automatically reduce to lowest terms - Fraction(6, 8) becomes Fraction(3, 4).

Arithmetic Operations

Fractions support all standard math operations:

                            people
                            people = 8 (default)people = 4people = 12
                        

                            arithmetic.py
                        
Replay: real traced execution (multi-file project)

                            # Arithmetic operations

from fractions import Fraction

# Arithmetic
a = Fraction(3, 4)
b = Fraction(2, 5)

print(f"a = {a}")
print(f"b = {b}")
print()

# Addition
print("Addition:")
print(f"a + b = {a + b}")
print(f"  = {a.numerator * b.denominator + b.numerator * a.denominator}/{a.denominator * b.denominator}")

# Subtraction
print("\nSubtraction:")
print(f"a - b = {a - b}")

# Multiplication
print("\nMultiplication:")
print(f"a * b = {a * b}")
print(f"  = {a.numerator * b.numerator}/{a.denominator * b.denominator}")

# Division
print("\nDivision:")
print(f"a / b = {a / b}")
print(f"  = {a.numerator * b.denominator}/{a.denominator * b.numerator}")

# Floor division
print("\nFloor division:")
print(f"a // b = {a // b}")

# Modulo
print("\nModulo:")
print(f"a % b = {a % b}")

# Power
print("\nPower:")
f = Fraction(2, 3)
print(f"{f}^2 = {f ** 2}")
print(f"{f}^3 = {f ** 3}")
print(f"{f}^-1 = {f ** -1}")  # Reciprocal

# Negation
print("\nNegation:")
print(f"-a = {-a}")

# Absolute value
print("\nAbsolute value:")
neg = Fraction(-3, 4)
print(f"abs({neg}) = {abs(neg)}")

# Reciprocal
print("\nReciprocal:")
f = Fraction(3, 4)
print(f"Reciprocal of {f} = {1 / f}")
print(f"  = {Fraction(f.denominator, f.numerator)}")

# Mixed operations
print("\nMixed operations:")
print(f"Fraction + int: {a + 2}")
print(f"int + Fraction: {2 + a}")
print(f"Fraction * int: {a * 3}")
print(f"Fraction / int: {a / 2}")

# Chaining
print("\nChaining:")
result = Fraction(1, 2) + Fraction(1, 3) + Fraction(1, 6)
print(f"1/2 + 1/3 + 1/6 = {result}")

# Complex expression
print("\nComplex expression:")
expr = (Fraction(3, 4) * Fraction(2, 5) + Fraction(1, 2)) / Fraction(3, 8)
print(f"(3/4 * 2/5 + 1/2) / 3/8 = {expr}")

# Sum of series
print("\nSum of series:")
# 1/1 + 1/2 + 1/3 + 1/4 + 1/5
series_sum = sum(Fraction(1, i) for i in range(1, 6))
print(f"1/1 + 1/2 + 1/3 + 1/4 + 1/5 = {series_sum}")
print(f"  = {float(series_sum):.6f}")

# Harmonic series
print("\nHarmonic series (first 10 terms):")
harmonic = sum(Fraction(1, i) for i in range(1, 11))
print(f"H_10 = {harmonic}")
print(f"  ≈ {float(harmonic):.10f}")

# Factorial fractions
print("\nFactorial fractions:")
# 1/1! + 1/2! + 1/3! + 1/4!
import math
factorial_sum = sum(Fraction(1, math.factorial(i)) for i in range(1, 5))
print(f"1/1! + 1/2! + 1/3! + 1/4! = {factorial_sum}")
print(f"  = {float(factorial_sum):.10f}")
print(f"  (approaching e-1 = {math.e - 1:.10f})")

# Dividing whole cake
print("\nDividing cake:")
cake = Fraction(1)  # Whole cake
people = 8
slice_size = cake / people
print(f"Cake for {people} people: {slice_size} each")
print(f"3 people get: {slice_size * 3}")

# Recipe scaling
print("\nRecipe scaling:")
original = Fraction(2, 3)  # 2/3 cup
scale_factor = Fraction(3, 2)  # 1.5x recipe
scaled = original * scale_factor
print(f"Original: {original} cup")
print(f"Scaled by {scale_factor}: {scaled} cup")
print(f"  = {float(scaled)} cup")

# Arithmetic operations

from fractions import Fraction

# Arithmetic
a = Fraction(3, 4)
b = Fraction(2, 5)

print(f"a = {a}")
print(f"b = {b}")
print()

# Addition
print("Addition:")
print(f"a + b = {a + b}")
print(f"  = {a.numerator * b.denominator + b.numerator * a.denominator}/{a.denominator * b.denominator}")

# Subtraction
print("\nSubtraction:")
print(f"a - b = {a - b}")

# Multiplication
print("\nMultiplication:")
print(f"a * b = {a * b}")
print(f"  = {a.numerator * b.numerator}/{a.denominator * b.denominator}")

# Division
print("\nDivision:")
print(f"a / b = {a / b}")
print(f"  = {a.numerator * b.denominator}/{a.denominator * b.numerator}")

# Floor division
print("\nFloor division:")
print(f"a // b = {a // b}")

# Modulo
print("\nModulo:")
print(f"a % b = {a % b}")

# Power
print("\nPower:")
f = Fraction(2, 3)
print(f"{f}^2 = {f ** 2}")
print(f"{f}^3 = {f ** 3}")
print(f"{f}^-1 = {f ** -1}")  # Reciprocal

# Negation
print("\nNegation:")
print(f"-a = {-a}")

# Absolute value
print("\nAbsolute value:")
neg = Fraction(-3, 4)
print(f"abs({neg}) = {abs(neg)}")

# Reciprocal
print("\nReciprocal:")
f = Fraction(3, 4)
print(f"Reciprocal of {f} = {1 / f}")
print(f"  = {Fraction(f.denominator, f.numerator)}")

# Mixed operations
print("\nMixed operations:")
print(f"Fraction + int: {a + 2}")
print(f"int + Fraction: {2 + a}")
print(f"Fraction * int: {a * 3}")
print(f"Fraction / int: {a / 2}")

# Chaining
print("\nChaining:")
result = Fraction(1, 2) + Fraction(1, 3) + Fraction(1, 6)
print(f"1/2 + 1/3 + 1/6 = {result}")

# Complex expression
print("\nComplex expression:")
expr = (Fraction(3, 4) * Fraction(2, 5) + Fraction(1, 2)) / Fraction(3, 8)
print(f"(3/4 * 2/5 + 1/2) / 3/8 = {expr}")

# Sum of series
print("\nSum of series:")
# 1/1 + 1/2 + 1/3 + 1/4 + 1/5
series_sum = sum(Fraction(1, i) for i in range(1, 6))
print(f"1/1 + 1/2 + 1/3 + 1/4 + 1/5 = {series_sum}")
print(f"  = {float(series_sum):.6f}")

# Harmonic series
print("\nHarmonic series (first 10 terms):")
harmonic = sum(Fraction(1, i) for i in range(1, 11))
print(f"H_10 = {harmonic}")
print(f"  ≈ {float(harmonic):.10f}")

# Factorial fractions
print("\nFactorial fractions:")
# 1/1! + 1/2! + 1/3! + 1/4!
import math
factorial_sum = sum(Fraction(1, math.factorial(i)) for i in range(1, 5))
print(f"1/1! + 1/2! + 1/3! + 1/4! = {factorial_sum}")
print(f"  = {float(factorial_sum):.10f}")
print(f"  (approaching e-1 = {math.e - 1:.10f})")

# Dividing whole cake
print("\nDividing cake:")
cake = Fraction(1)  # Whole cake
people = 4
slice_size = cake / people
print(f"Cake for {people} people: {slice_size} each")
print(f"3 people get: {slice_size * 3}")

# Recipe scaling
print("\nRecipe scaling:")
original = Fraction(2, 3)  # 2/3 cup
scale_factor = Fraction(3, 2)  # 1.5x recipe
scaled = original * scale_factor
print(f"Original: {original} cup")
print(f"Scaled by {scale_factor}: {scaled} cup")
print(f"  = {float(scaled)} cup")

# Arithmetic operations

from fractions import Fraction

# Arithmetic
a = Fraction(3, 4)
b = Fraction(2, 5)

print(f"a = {a}")
print(f"b = {b}")
print()

# Addition
print("Addition:")
print(f"a + b = {a + b}")
print(f"  = {a.numerator * b.denominator + b.numerator * a.denominator}/{a.denominator * b.denominator}")

# Subtraction
print("\nSubtraction:")
print(f"a - b = {a - b}")

# Multiplication
print("\nMultiplication:")
print(f"a * b = {a * b}")
print(f"  = {a.numerator * b.numerator}/{a.denominator * b.denominator}")

# Division
print("\nDivision:")
print(f"a / b = {a / b}")
print(f"  = {a.numerator * b.denominator}/{a.denominator * b.numerator}")

# Floor division
print("\nFloor division:")
print(f"a // b = {a // b}")

# Modulo
print("\nModulo:")
print(f"a % b = {a % b}")

# Power
print("\nPower:")
f = Fraction(2, 3)
print(f"{f}^2 = {f ** 2}")
print(f"{f}^3 = {f ** 3}")
print(f"{f}^-1 = {f ** -1}")  # Reciprocal

# Negation
print("\nNegation:")
print(f"-a = {-a}")

# Absolute value
print("\nAbsolute value:")
neg = Fraction(-3, 4)
print(f"abs({neg}) = {abs(neg)}")

# Reciprocal
print("\nReciprocal:")
f = Fraction(3, 4)
print(f"Reciprocal of {f} = {1 / f}")
print(f"  = {Fraction(f.denominator, f.numerator)}")

# Mixed operations
print("\nMixed operations:")
print(f"Fraction + int: {a + 2}")
print(f"int + Fraction: {2 + a}")
print(f"Fraction * int: {a * 3}")
print(f"Fraction / int: {a / 2}")

# Chaining
print("\nChaining:")
result = Fraction(1, 2) + Fraction(1, 3) + Fraction(1, 6)
print(f"1/2 + 1/3 + 1/6 = {result}")

# Complex expression
print("\nComplex expression:")
expr = (Fraction(3, 4) * Fraction(2, 5) + Fraction(1, 2)) / Fraction(3, 8)
print(f"(3/4 * 2/5 + 1/2) / 3/8 = {expr}")

# Sum of series
print("\nSum of series:")
# 1/1 + 1/2 + 1/3 + 1/4 + 1/5
series_sum = sum(Fraction(1, i) for i in range(1, 6))
print(f"1/1 + 1/2 + 1/3 + 1/4 + 1/5 = {series_sum}")
print(f"  = {float(series_sum):.6f}")

# Harmonic series
print("\nHarmonic series (first 10 terms):")
harmonic = sum(Fraction(1, i) for i in range(1, 11))
print(f"H_10 = {harmonic}")
print(f"  ≈ {float(harmonic):.10f}")

# Factorial fractions
print("\nFactorial fractions:")
# 1/1! + 1/2! + 1/3! + 1/4!
import math
factorial_sum = sum(Fraction(1, math.factorial(i)) for i in range(1, 5))
print(f"1/1! + 1/2! + 1/3! + 1/4! = {factorial_sum}")
print(f"  = {float(factorial_sum):.10f}")
print(f"  (approaching e-1 = {math.e - 1:.10f})")

# Dividing whole cake
print("\nDividing cake:")
cake = Fraction(1)  # Whole cake
people = 12
slice_size = cake / people
print(f"Cake for {people} people: {slice_size} each")
print(f"3 people get: {slice_size * 3}")

# Recipe scaling
print("\nRecipe scaling:")
original = Fraction(2, 3)  # 2/3 cup
scale_factor = Fraction(3, 2)  # 1.5x recipe
scaled = original * scale_factor
print(f"Original: {original} cup")
print(f"Scaled by {scale_factor}: {scaled} cup")
print(f"  = {float(scaled)} cup")


                        
1a ← 3/4, b ← 2/5, f ← 2/3, neg ← -3/4, result ← 1, expr ← 32/15⋯5# Arithmetic6a→ 3/4 = Fraction(3, 4)7b→ 2/5 = Fraction(2, 5)89print(f"a = {a3/4}")10print(f"b = {b2/5}")11print()1213# Addition14print("Addition:")15print(f"a + b = {a3/4 + b2/5}")16print(f"  = {a.numerator3 * b.denominator5 + b.numerator2 * a.denominator4}/{a.denominator * b.denominator}")1718# Subtraction19print("\nSubtraction:")20print(f"a - b = {a3/4 - b2/5}")2122# Multiplication23print("\nMultiplication:")24print(f"a * b = {a3/4 * b2/5}")25print(f"  = {a.numerator3 * b.numerator2}/{a.denominator4 * b.denominator5}")2627# Division28print("\nDivision:")29print(f"a / b = {a3/4 / b2/5}")30print(f"  = {a.numerator3 * b.denominator5}/{a.denominator4 * b.numerator2}")3132# Floor division33print("\nFloor division:")34print(f"a // b = {a3/4 // b2/5}")3536# Modulo37print("\nModulo:")38print(f"a % b = {a3/4 % b2/5}")3940# Power41print("\nPower:")42f→ 2/3 = Fraction(2, 3)43print(f"{f2/3}^2 = {f ** 2}")44print(f"{f2/3}^3 = {f ** 3}")45print(f"{f2/3}^-1 = {f ** -1}")  # Reciprocal4647# Negation48print("\nNegation:")49print(f"-a = {-a3/4}")5051# Absolute value52print("\nAbsolute value:")53neg→ -3/4 = Fraction(-3, 4)54print(f"abs({neg-3/4}) = {abs(neg)}")5556# Reciprocal57print("\nReciprocal:")58f→ 3/4 = Fraction(3, 4)59print(f"Reciprocal of {f3/4} = {1 / f}")60print(f"  = {Fraction(f.denominator4, f.numerator3)}")6162# Mixed operations63print("\nMixed operations:")64print(f"Fraction + int: {a3/4 + 2}")65print(f"int + Fraction: {2 + a3/4}")66print(f"Fraction * int: {a3/4 * 3}")67print(f"Fraction / int: {a3/4 / 2}")6869# Chaining70print("\nChaining:")71result→ 1 = Fraction(1, 2) + Fraction(1, 3) + Fraction(1, 6)72print(f"1/2 + 1/3 + 1/6 = {result1}")7374# Complex expression75print("\nComplex expression:")76expr→ 32/15 = (Fraction(3, 4) * Fraction(2, 5) + Fraction(1, 2)) / Fraction(3, 8)77print(f"(3/4 * 2/5 + 1/2) / 3/8 = {expr32/15}")7879# Sum of series80print("\nSum of series:")81# 1/1 + 1/2 + 1/3 + 1/4 + 1/582series_sum→ 137/60 = sum(Fraction(1, i) for i in range(1, 6))83print(f"1/1 + 1/2 + 1/3 + 1/4 + 1/5 = {series_sum137/60}")84print(f"  = {float(series_sum137/60):.6f}")8586# Harmonic series87print("\nHarmonic series (first 10 terms):")88harmonic→ 7381/2520 = sum(Fraction(1, i) for i in range(1, 11))89print(f"H_10 = {harmonic7381/2520}")90print(f"  ≈ {float(harmonic7381/2520):.10f}")9192# Factorial fractions93print("\nFactorial fractions:")94# 1/1! + 1/2! + 1/3! + 1/4!95import math96factorial_sum→ 41/24 = sum(Fraction(1, math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.factorial(i)) for i in range(1, 5))97print(f"1/1! + 1/2! + 1/3! + 1/4! = {factorial_sum41/24}")98print(f"  = {float(factorial_sum41/24):.10f}")99print(f"  (approaching e-1 = {math.e2.718281828459045 - 1:.10f})")100101# Dividing whole cake102print("\nDividing cake:")103cake→ 1 = Fraction(1)  # Whole cake104people→ 8 = 8  #@people=4, 12105slice_size→ 1/8 = cake1 / people8106print(f"Cake for {people8} people: {slice_size1/8} each")107print(f"3 people get: {slice_size1/8 * 3}")108109# Recipe scaling110print("\nRecipe scaling:")111original→ 2/3 = Fraction(2, 3)  # 2/3 cup112scale_factor→ 3/2 = Fraction(3, 2)  # 1.5x recipe113scaled→ 1 = original2/3 * scale_factor3/2114print(f"Original: {original2/3} cup")115print(f"Scaled by {scale_factor3/2}: {scaled1} cup")116print(f"  = {float(scaled1)} cup")⋯outputa = 3/4
b = 2/5
Addition:
a + b = 23/20
  = 23/20

Subtraction:
a - b = 7/20

Multiplication:
a * b = 3/10
  = 6/20

Division:
a / b = 15/8
  = 15/8

Floor division:
a // b = 1

Modulo:
a % b = 7/20

Power:
2/3^2 = 4/9
2/3^3 = 8/27
2/3^-1 = 3/2

Negation:
-a = -3/4

Absolute value:
abs(-3/4) = 3/4

Reciprocal:
Reciprocal of 3/4 = 4/3
  = 4/3

Mixed operations:
Fraction + int: 11/4
int + Fraction: 11/4
Fraction * int: 9/4
Fraction / int: 3/8

Chaining:
1/2 + 1/3 + 1/6 = 1

Complex expression:
(3/4 * 2/5 + 1/2) / 3/8 = 32/15

Sum of series:
1/1 + 1/2 + 1/3 + 1/4 + 1/5 = 137/60
  = 2.283333

Harmonic series (first 10 terms):
H_10 = 7381/2520
  ≈ 2.9289682540

Factorial fractions:
1/1! + 1/2! + 1/3! + 1/4! = 41/24
  = 1.7083333333
  (approaching e-1 = 1.7182818285)

Dividing cake:
Cake for 8 people: 1/8 each
3 people get: 3/8

Recipe scaling:
Original: 2/3 cup
Scaled by 3/2: 1 cup
  = 1.0 cup
1a ← 3/4, b ← 2/5, f ← 2/3, neg ← -3/4, result ← 1, expr ← 32/15⋯5# Arithmetic6a→ 3/4 = Fraction(3, 4)7b→ 2/5 = Fraction(2, 5)89print(f"a = {a3/4}")10print(f"b = {b2/5}")11print()1213# Addition14print("Addition:")15print(f"a + b = {a3/4 + b2/5}")16print(f"  = {a.numerator3 * b.denominator5 + b.numerator2 * a.denominator4}/{a.denominator * b.denominator}")1718# Subtraction19print("\nSubtraction:")20print(f"a - b = {a3/4 - b2/5}")2122# Multiplication23print("\nMultiplication:")24print(f"a * b = {a3/4 * b2/5}")25print(f"  = {a.numerator3 * b.numerator2}/{a.denominator4 * b.denominator5}")2627# Division28print("\nDivision:")29print(f"a / b = {a3/4 / b2/5}")30print(f"  = {a.numerator3 * b.denominator5}/{a.denominator4 * b.numerator2}")3132# Floor division33print("\nFloor division:")34print(f"a // b = {a3/4 // b2/5}")3536# Modulo37print("\nModulo:")38print(f"a % b = {a3/4 % b2/5}")3940# Power41print("\nPower:")42f→ 2/3 = Fraction(2, 3)43print(f"{f2/3}^2 = {f ** 2}")44print(f"{f2/3}^3 = {f ** 3}")45print(f"{f2/3}^-1 = {f ** -1}")  # Reciprocal4647# Negation48print("\nNegation:")49print(f"-a = {-a3/4}")5051# Absolute value52print("\nAbsolute value:")53neg→ -3/4 = Fraction(-3, 4)54print(f"abs({neg-3/4}) = {abs(neg)}")5556# Reciprocal57print("\nReciprocal:")58f→ 3/4 = Fraction(3, 4)59print(f"Reciprocal of {f3/4} = {1 / f}")60print(f"  = {Fraction(f.denominator4, f.numerator3)}")6162# Mixed operations63print("\nMixed operations:")64print(f"Fraction + int: {a3/4 + 2}")65print(f"int + Fraction: {2 + a3/4}")66print(f"Fraction * int: {a3/4 * 3}")67print(f"Fraction / int: {a3/4 / 2}")6869# Chaining70print("\nChaining:")71result→ 1 = Fraction(1, 2) + Fraction(1, 3) + Fraction(1, 6)72print(f"1/2 + 1/3 + 1/6 = {result1}")7374# Complex expression75print("\nComplex expression:")76expr→ 32/15 = (Fraction(3, 4) * Fraction(2, 5) + Fraction(1, 2)) / Fraction(3, 8)77print(f"(3/4 * 2/5 + 1/2) / 3/8 = {expr32/15}")7879# Sum of series80print("\nSum of series:")81# 1/1 + 1/2 + 1/3 + 1/4 + 1/582series_sum→ 137/60 = sum(Fraction(1, i) for i in range(1, 6))83print(f"1/1 + 1/2 + 1/3 + 1/4 + 1/5 = {series_sum137/60}")84print(f"  = {float(series_sum137/60):.6f}")8586# Harmonic series87print("\nHarmonic series (first 10 terms):")88harmonic→ 7381/2520 = sum(Fraction(1, i) for i in range(1, 11))89print(f"H_10 = {harmonic7381/2520}")90print(f"  ≈ {float(harmonic7381/2520):.10f}")9192# Factorial fractions93print("\nFactorial fractions:")94# 1/1! + 1/2! + 1/3! + 1/4!95import math96factorial_sum→ 41/24 = sum(Fraction(1, math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.factorial(i)) for i in range(1, 5))97print(f"1/1! + 1/2! + 1/3! + 1/4! = {factorial_sum41/24}")98print(f"  = {float(factorial_sum41/24):.10f}")99print(f"  (approaching e-1 = {math.e2.718281828459045 - 1:.10f})")100101# Dividing whole cake102print("\nDividing cake:")103cake→ 1 = Fraction(1)  # Whole cake104people→ 4 = 4105slice_size→ 1/4 = cake1 / people4106print(f"Cake for {people4} people: {slice_size1/4} each")107print(f"3 people get: {slice_size1/4 * 3}")108109# Recipe scaling110print("\nRecipe scaling:")111original→ 2/3 = Fraction(2, 3)  # 2/3 cup112scale_factor→ 3/2 = Fraction(3, 2)  # 1.5x recipe113scaled→ 1 = original2/3 * scale_factor3/2114print(f"Original: {original2/3} cup")115print(f"Scaled by {scale_factor3/2}: {scaled1} cup")116print(f"  = {float(scaled1)} cup")⋯outputa = 3/4
b = 2/5
Addition:
a + b = 23/20
  = 23/20

Subtraction:
a - b = 7/20

Multiplication:
a * b = 3/10
  = 6/20

Division:
a / b = 15/8
  = 15/8

Floor division:
a // b = 1

Modulo:
a % b = 7/20

Power:
2/3^2 = 4/9
2/3^3 = 8/27
2/3^-1 = 3/2

Negation:
-a = -3/4

Absolute value:
abs(-3/4) = 3/4

Reciprocal:
Reciprocal of 3/4 = 4/3
  = 4/3

Mixed operations:
Fraction + int: 11/4
int + Fraction: 11/4
Fraction * int: 9/4
Fraction / int: 3/8

Chaining:
1/2 + 1/3 + 1/6 = 1

Complex expression:
(3/4 * 2/5 + 1/2) / 3/8 = 32/15

Sum of series:
1/1 + 1/2 + 1/3 + 1/4 + 1/5 = 137/60
  = 2.283333

Harmonic series (first 10 terms):
H_10 = 7381/2520
  ≈ 2.9289682540

Factorial fractions:
1/1! + 1/2! + 1/3! + 1/4! = 41/24
  = 1.7083333333
  (approaching e-1 = 1.7182818285)

Dividing cake:
Cake for 4 people: 1/4 each
3 people get: 3/4

Recipe scaling:
Original: 2/3 cup
Scaled by 3/2: 1 cup
  = 1.0 cup
1a ← 3/4, b ← 2/5, f ← 2/3, neg ← -3/4, result ← 1, expr ← 32/15⋯5# Arithmetic6a→ 3/4 = Fraction(3, 4)7b→ 2/5 = Fraction(2, 5)89print(f"a = {a3/4}")10print(f"b = {b2/5}")11print()1213# Addition14print("Addition:")15print(f"a + b = {a3/4 + b2/5}")16print(f"  = {a.numerator3 * b.denominator5 + b.numerator2 * a.denominator4}/{a.denominator * b.denominator}")1718# Subtraction19print("\nSubtraction:")20print(f"a - b = {a3/4 - b2/5}")2122# Multiplication23print("\nMultiplication:")24print(f"a * b = {a3/4 * b2/5}")25print(f"  = {a.numerator3 * b.numerator2}/{a.denominator4 * b.denominator5}")2627# Division28print("\nDivision:")29print(f"a / b = {a3/4 / b2/5}")30print(f"  = {a.numerator3 * b.denominator5}/{a.denominator4 * b.numerator2}")3132# Floor division33print("\nFloor division:")34print(f"a // b = {a3/4 // b2/5}")3536# Modulo37print("\nModulo:")38print(f"a % b = {a3/4 % b2/5}")3940# Power41print("\nPower:")42f→ 2/3 = Fraction(2, 3)43print(f"{f2/3}^2 = {f ** 2}")44print(f"{f2/3}^3 = {f ** 3}")45print(f"{f2/3}^-1 = {f ** -1}")  # Reciprocal4647# Negation48print("\nNegation:")49print(f"-a = {-a3/4}")5051# Absolute value52print("\nAbsolute value:")53neg→ -3/4 = Fraction(-3, 4)54print(f"abs({neg-3/4}) = {abs(neg)}")5556# Reciprocal57print("\nReciprocal:")58f→ 3/4 = Fraction(3, 4)59print(f"Reciprocal of {f3/4} = {1 / f}")60print(f"  = {Fraction(f.denominator4, f.numerator3)}")6162# Mixed operations63print("\nMixed operations:")64print(f"Fraction + int: {a3/4 + 2}")65print(f"int + Fraction: {2 + a3/4}")66print(f"Fraction * int: {a3/4 * 3}")67print(f"Fraction / int: {a3/4 / 2}")6869# Chaining70print("\nChaining:")71result→ 1 = Fraction(1, 2) + Fraction(1, 3) + Fraction(1, 6)72print(f"1/2 + 1/3 + 1/6 = {result1}")7374# Complex expression75print("\nComplex expression:")76expr→ 32/15 = (Fraction(3, 4) * Fraction(2, 5) + Fraction(1, 2)) / Fraction(3, 8)77print(f"(3/4 * 2/5 + 1/2) / 3/8 = {expr32/15}")7879# Sum of series80print("\nSum of series:")81# 1/1 + 1/2 + 1/3 + 1/4 + 1/582series_sum→ 137/60 = sum(Fraction(1, i) for i in range(1, 6))83print(f"1/1 + 1/2 + 1/3 + 1/4 + 1/5 = {series_sum137/60}")84print(f"  = {float(series_sum137/60):.6f}")8586# Harmonic series87print("\nHarmonic series (first 10 terms):")88harmonic→ 7381/2520 = sum(Fraction(1, i) for i in range(1, 11))89print(f"H_10 = {harmonic7381/2520}")90print(f"  ≈ {float(harmonic7381/2520):.10f}")9192# Factorial fractions93print("\nFactorial fractions:")94# 1/1! + 1/2! + 1/3! + 1/4!95import math96factorial_sum→ 41/24 = sum(Fraction(1, math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.factorial(i)) for i in range(1, 5))97print(f"1/1! + 1/2! + 1/3! + 1/4! = {factorial_sum41/24}")98print(f"  = {float(factorial_sum41/24):.10f}")99print(f"  (approaching e-1 = {math.e2.718281828459045 - 1:.10f})")100101# Dividing whole cake102print("\nDividing cake:")103cake→ 1 = Fraction(1)  # Whole cake104people→ 12 = 12105slice_size→ 1/12 = cake1 / people12106print(f"Cake for {people12} people: {slice_size1/12} each")107print(f"3 people get: {slice_size1/12 * 3}")108109# Recipe scaling110print("\nRecipe scaling:")111original→ 2/3 = Fraction(2, 3)  # 2/3 cup112scale_factor→ 3/2 = Fraction(3, 2)  # 1.5x recipe113scaled→ 1 = original2/3 * scale_factor3/2114print(f"Original: {original2/3} cup")115print(f"Scaled by {scale_factor3/2}: {scaled1} cup")116print(f"  = {float(scaled1)} cup")⋯outputa = 3/4
b = 2/5
Addition:
a + b = 23/20
  = 23/20

Subtraction:
a - b = 7/20

Multiplication:
a * b = 3/10
  = 6/20

Division:
a / b = 15/8
  = 15/8

Floor division:
a // b = 1

Modulo:
a % b = 7/20

Power:
2/3^2 = 4/9
2/3^3 = 8/27
2/3^-1 = 3/2

Negation:
-a = -3/4

Absolute value:
abs(-3/4) = 3/4

Reciprocal:
Reciprocal of 3/4 = 4/3
  = 4/3

Mixed operations:
Fraction + int: 11/4
int + Fraction: 11/4
Fraction * int: 9/4
Fraction / int: 3/8

Chaining:
1/2 + 1/3 + 1/6 = 1

Complex expression:
(3/4 * 2/5 + 1/2) / 3/8 = 32/15

Sum of series:
1/1 + 1/2 + 1/3 + 1/4 + 1/5 = 137/60
  = 2.283333

Harmonic series (first 10 terms):
H_10 = 7381/2520
  ≈ 2.9289682540

Factorial fractions:
1/1! + 1/2! + 1/3! + 1/4! = 41/24
  = 1.7083333333
  (approaching e-1 = 1.7182818285)

Dividing cake:
Cake for 12 people: 1/12 each
3 people get: 1/4

Recipe scaling:
Original: 2/3 cup
Scaled by 3/2: 1 cup
  = 1.0 cup

Conversions

Converting between fractions and other types:

                            convert.py
                        
Replay: real traced execution (multi-file project)

                            # Converting and formatting

from fractions import Fraction
from decimal import Decimal
import math

# Convert to other types
print("Convert to other types:")
f = Fraction(3, 4)

print(f"Fraction: {f}")
print(f"float: {float(f)}")
print(f"int (truncate): {int(f)}")
print(f"str: {str(f)}")
print()

# To Decimal
print("To Decimal:")
dec = Decimal(f.numerator) / Decimal(f.denominator)
print(f"{f} as Decimal: {dec}")

# Different fractions
print("\nDifferent fractions to float:")
fractions = [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(2, 3)]
for frac in fractions:
    print(f"{frac} = {float(frac):.10f}")

# Formatting
print("\nFormatting:")
f = Fraction(22, 7)  # Approximation of pi
print(f"Default: {f}")
print(f"Float: {float(f):.6f}")
print(f"Percentage: {float(f) * 100:.2f}%")

# Mixed number representation
print("\nMixed number representation:")
def to_mixed_number(frac):
    """Convert improper fraction to mixed number."""
    whole = frac.numerator // frac.denominator
    remainder = frac.numerator % frac.denominator
    if remainder == 0:
        return f"{whole}"
    elif whole == 0:
        return f"{remainder}/{frac.denominator}"
    else:
        return f"{whole} {remainder}/{frac.denominator}"

improper_fracs = [Fraction(7, 4), Fraction(11, 3), Fraction(5, 2), Fraction(8, 4)]
for f in improper_fracs:
    print(f"{f} = {to_mixed_number(f)}")

# Decimal to Fraction
print("\nDecimal to Fraction:")
decimals = [0.5, 0.25, 0.75, 0.125, 0.333]
for d in decimals:
    f = Fraction(d).limit_denominator(1000)
    print(f"{d} ≈ {f}")

# String to Fraction
print("\nString to Fraction:")
strings = ['1/2', '3/4', '0.5', '0.75', '22/7']
for s in strings:
    f = Fraction(s)
    print(f"'{s}' = {f} = {float(f):.6f}")

# Pi approximations
print("\nPi approximations:")
pi_approx = [
    Fraction(22, 7),
    Fraction(355, 113),
    Fraction(103993, 33102),
]
for f in pi_approx:
    error = abs(float(f) - math.pi)
    print(f"{f} = {float(f):.10f} (error: {error:.2e})")
print(f"Actual π = {math.pi:.10f}")

# Common percentages as fractions
print("\nCommon percentages:")
percentages = [10, 20, 25, 33.33, 50, 75, 100]
for pct in percentages:
    f = Fraction(str(pct / 100)).limit_denominator(100)
    print(f"{pct}% = {f} = {float(f):.4f}")

# Continued fractions (representation)
print("\nContinued fraction representation:")
def continued_fraction(frac, max_terms=10):
    """Get continued fraction representation."""
    terms = []
    n, d = frac.numerator, frac.denominator
    for _ in range(max_terms):
        if d == 0:
            break
        q, r = divmod(n, d)
        terms.append(q)
        n, d = d, r
        if r == 0:
            break
    return terms

f = Fraction(22, 7)
cf = continued_fraction(f)
print(f"{f}: {cf}")

f = Fraction(355, 113)
cf = continued_fraction(f)
print(f"{f}: {cf}")

# Reconstruct from numerator/denominator
print("\nReconstruct fraction:")
num, den = 3, 4
reconstructed = Fraction(num, den)
print(f"From {num}/{den}: {reconstructed}")

# Using with f-strings
print("\nF-string formatting:")
f = Fraction(3, 4)
print(f"Fraction: {f}")
print(f"As float: {float(f):.2f}")
print(f"As percentage: {float(f):.1%}")


                        
1f ← 3/4, dec ← 0.75, fractions ← [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(2, 3)]⋯7# Convert to other types8print("Convert to other types:")9f→ 3/4 = Fraction(3, 4)1011print(f"Fraction: {f3/4}")12print(f"float: {float(f3/4)}")13print(f"int (truncate): {int(f3/4)}")14print(f"str: {str(f3/4)}")15print()1617# To Decimal18print("To Decimal:")19dec→ 0.75 = Decimal(f.numerator3) / Decimal(f.denominator4)20print(f"{f3/4} as Decimal: {dec0.75}")2122# Different fractions23print("\nDifferent fractions to float:")24fractions→ [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(2, 3)] = [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(2, 3)]25for frac in fractions:⋯outputConvert to other types:
Fraction: 3/4
float: 0.75
int (truncate): 0
str: 3/4
To Decimal:
3/4 as Decimal: 0.75

Different fractions to float:
2for frac in fractions:pass 1 of 4⋯24fractions = [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(2, 3)]25for frac1/2 in fractions[Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(2, 3)]:26    print(f"{frac1/2} = {float(frac):.10f}")⋯output1/2 = 0.5000000000All 4 passes — pass 1 is the card abovepassfrac
11/2
21/3
31/4
42/3
3f ← 22/7, improper_fracs ← [Fraction(7, 4), Fraction(11, 3), Fraction(5, 2), Fraction(2, 1)]⋯28# Formatting29print("\nFormatting:")30f→ 22/7 = Fraction(22, 7)  # Approximation of pi31print(f"Default: {f22/7}")32print(f"Float: {float(f22/7):.6f}")33print(f"Percentage: {float(f22/7) * 100:.2f}%")3435# Mixed number representation36print("\nMixed number representation:")37def to_mixed_number(frac):38    """Convert improper fraction to mixed number."""39    whole = frac.numerator // frac.denominator40    remainder = frac.numerator % frac.denominator41    if remainder == 0:42        return f"{whole}"43    elif whole == 0:44        return f"{remainder}/{frac.denominator}"45    else:46        return f"{whole} {remainder}/{frac.denominator}"4748improper_fracs→ [Fraction(7, 4), Fraction(11, 3), Fraction(5, 2), Fraction(2, 1)] = [Fraction(7, 4), Fraction(11, 3), Fraction(5, 2), Fraction(8, 4)]49for f in improper_fracs:⋯output
Formatting:
Default: 22/7
Float: 3.142857
Percentage: 314.29%

Mixed number representation:
4for f in improper_fracs:pass 1 of 4⋯48improper_fracs = [Fraction(7, 4), Fraction(11, 3), Fraction(5, 2), Fraction(8, 4)]49for f7/4 in improper_fracs[Fraction(7, 4), Fraction(11, 3), Fraction(5, 2), Fraction(2, 1)]:50    print(f"{f7/4} = {to_mixed_number(f)}")⋯All 4 passes — pass 1 is the card abovepassfremainderwhole
17/4——
211/3——
35/2——
4202
5whole ← 1, remainder ← 3pass 1 of 4⋯36print("\nMixed number representation:")37def to_mixed_number(frac7/4):38    """Convert improper fraction to mixed number."""39    whole→ 1 = frac.numerator7 // frac.denominator440    remainder→ 3 = frac.numerator7 % frac.denominator441    if remainder == 0:⋯All 4 passes — pass 1 is the card abovepassfracfrac.numeratorfrac.denominatorwholeremainder
17/47413
211/311332
35/25221
422120
6else:pass 1 of 3⋯43elif whole == 0:44    return f"{remainder}/{frac.denominator}"45else:46    return f"{whole1} {remainder3}/{frac.denominator4}"⋯All 3 passes — pass 1 is the card abovepasswholeremainderfrac.denominator
1134
2323
3212
7print(f"{f} = {to_mixed_number(f)}")⋯49for f in improper_fracs:50    print(f"{f7/4} = {to_mixed_number(f)}")⋯output7/4 = 1 3/4
8print(f"{f} = {to_mixed_number(f)}")⋯49for f in improper_fracs:50    print(f"{f11/3} = {to_mixed_number(f)}")⋯output11/3 = 3 2/3
9print(f"{f} = {to_mixed_number(f)}")⋯49for f in improper_fracs:50    print(f"{f5/2} = {to_mixed_number(f)}")⋯output5/2 = 2 1/2
10if remainder == 0:⋯40remainder = frac.numerator % frac.denominator41if remainder0 == 0:42    return f"{whole2}"43elif whole == 0:⋯
11print(f"{f} = {to_mixed_number(f)}")⋯49for f in improper_fracs:50    print(f"{f2} = {to_mixed_number(f)}")⋯output2 = 2
12decimals ← [0.5, 0.25, 0.75, 0.125, 0.333]⋯52# Decimal to Fraction53print("\nDecimal to Fraction:")54decimals→ [0.5, 0.25, 0.75, 0.125, 0.333] = [0.5, 0.25, 0.75, 0.125, 0.333]55for d in decimals:⋯output
Decimal to Fraction:
13f ← 1/2pass 1 of 5⋯54decimals = [0.5, 0.25, 0.75, 0.125, 0.333]55for d0.5 in decimals[0.5, 0.25, 0.75, 0.125, 0.333]:56    f→ 1/2 = Fraction(d0.5).limit_denominator(1000)57    print(f"{d0.5} ≈ {f1/2}")⋯output0.5 ≈ 1/2All 5 passes — pass 1 is the card abovepassdf
10.51/2
20.251/4
30.753/4
40.1251/8
50.333333/1000
14strings ← ['1/2', '3/4', '0.5', '0.75', '22/7']⋯59# String to Fraction60print("\nString to Fraction:")61strings→ ['1/2', '3/4', '0.5', '0.75', '22/7'] = ['1/2', '3/4', '0.5', '0.75', '22/7']62for s in strings:⋯output
String to Fraction:
15f ← 1/2pass 1 of 5⋯61strings = ['1/2', '3/4', '0.5', '0.75', '22/7']62for s1/2 in strings['1/2', '3/4', '0.5', '0.75', '22/7']:63    f→ 1/2 = Fraction(s1/2)64    print(f"'{s1/2}' = {f1/2} = {float(f):.6f}")⋯output'1/2' = 1/2 = 0.500000All 5 passes — pass 1 is the card abovepasssf
11/21/2
23/43/4
30.51/2
40.753/4
522/722/7
16pi_approx ← [Fraction(22, 7), Fraction(355, 113), Fraction(103993, 33102)]⋯66# Pi approximations67print("\nPi approximations:")68pi_approx→ [Fraction(22, 7), Fraction(355, 113), Fraction(103993, 33102)] = [69    Fraction(22, 7),70    Fraction(355, 113),71    Fraction(103993, 33102),72]73for f in pi_approx:⋯output
Pi approximations:
17error ← 0.0012644892673496777pass 1 of 3⋯72]73for f22/7 in pi_approx[Fraction(22, 7), Fraction(355, 113), Fraction(103993, 33102)]:74    error→ 0.0012644892673496777 = abs(float(f22/7) - math.pi3.141592653589793)75    print(f"{f22/7} = {float(f):.10f} (error: {error0.0012644892673496777:.2e})")76print(f"Actual π = {math.pi:.10f}")⋯output22/7 = 3.1428571429 (error: 1.26e-03)All 3 passes — pass 1 is the card abovepassferror
122/70.0012644892673496777
2355/1132.667641894049666e-07
3103993/331025.778906242426274e-10
18percentages ← [10, 20, 25, 33.33, 50, 75, 100]⋯75    print(f"{f} = {float(f):.10f} (error: {error:.2e})")76print(f"Actual π = {math.pi3.141592653589793:.10f}")7778# Common percentages as fractions79print("\nCommon percentages:")80percentages→ [10, 20, 25, 33.33, 50, 75, 100] = [10, 20, 25, 33.33, 50, 75, 100]81for pct in percentages:⋯outputActual π = 3.1415926536

Common percentages:
19f ← 1/10pass 1 of 7⋯80percentages = [10, 20, 25, 33.33, 50, 75, 100]81for pct10 in percentages[10, 20, 25, 33.33, 50, 75, 100]:82    f→ 1/10 = Fraction(str(pct10 / 100)).limit_denominator(100)83    print(f"{pct10}% = {f1/10} = {float(f):.4f}")⋯output10% = 1/10 = 0.1000All 7 passes — pass 1 is the card abovepasspctf
1101/10
2201/5
3251/4
433.331/3
5501/2
6753/4
71001
20f ← 22/7⋯85# Continued fractions (representation)86print("\nContinued fraction representation:")87def continued_fraction(frac, max_terms=10):88    """Get continued fraction representation."""89    terms = []90    n, d = frac.numerator, frac.denominator91    for _ in range(max_terms):92        if d == 0:93            break94        q, r = divmod(n, d)95        terms.append(q)96        n, d = d, r97        if r == 0:98            break99    return terms100101f→ 22/7 = Fraction(22, 7)102cf = continued_fraction(f22/7)103print(f"{f}: {cf}")⋯output
Continued fraction representation:
21terms ← [], n ← 22, d ← 7pass 1 of 2⋯86print("\nContinued fraction representation:")87def continued_fraction(frac22/7, max_terms10=10):88    """Get continued fraction representation."""89    terms→ [] = []90    n→ 22, d→ 7 = frac.numerator22, frac.denominator791    for _ in range(max_terms):⋯
22q ← 3, r ← 1, terms ← [3], n ← 7, d ← 1pass 1 of 5⋯90n, d = frac.numerator, frac.denominator91for _0 in range(max_terms10):92    if d == 0:93        break94    q→ 3, r→ 1 = divmod(n22, d7)95    terms→ [3].append(q3)96    n→ 7, d→ 1 = d, r197    if r == 0:⋯All 5 passes — pass 1 is the card abovepass_qrtermsnd
1031[] → [3]22 → 77 → 1
2170[3] → [3, 7]7 → 11 → 0
30316[] → [3]355 → 113113 → 16
4171[3] → [3, 7]113 → 1616 → 1
52160[3, 7] → [3, 7, 16]16 → 11 → 0
23if r == 0:pass 1 of 2⋯96    n, d = d, r97    if r0 == 0:98        break99return terms⋯
24return terms⋯98        break99return terms[3, 7]⋯
25cf ← [3, 7], f ← 355/113⋯101f = Fraction(22, 7)102cf→ [3, 7] = continued_fraction(f22/7)103print(f"{f22/7}: {cf[3, 7]}")104105f→ 355/113 = Fraction(355, 113)106cf = continued_fraction(f355/113)107print(f"{f}: {cf}")⋯output22/7: [3, 7]
26terms ← [], n ← 355, d ← 113pass 2 of 2⋯86print("\nContinued fraction representation:")87def continued_fraction(frac355/113, max_terms10=10):88    """Get continued fraction representation."""89    terms→ [] = []90    n→ 355, d→ 113 = frac.numerator355, frac.denominator11391    for _ in range(max_terms):⋯
27if r == 0:pass 2 of 2⋯96    n, d = d, r97    if r0 == 0:98        break99return terms⋯
28return terms⋯98        break99return terms[3, 7, 16]⋯
29cf ← [3, 7, 16], num ← 3, den ← 4, reconstructed ← 3/4, f ← 3/4⋯105f = Fraction(355, 113)106cf→ [3, 7, 16] = continued_fraction(f355/113)107print(f"{f355/113}: {cf[3, 7, 16]}")108109# Reconstruct from numerator/denominator110print("\nReconstruct fraction:")111num→ 3, den→ 4 = 3, 4112reconstructed→ 3/4 = Fraction(num3, den4)113print(f"From {num3}/{den4}: {reconstructed3/4}")114115# Using with f-strings116print("\nF-string formatting:")117f→ 3/4 = Fraction(3, 4)118print(f"Fraction: {f3/4}")119print(f"As float: {float(f3/4):.2f}")120print(f"As percentage: {float(f3/4):.1%}")⋯output355/113: [3, 7, 16]

Reconstruct fraction:
From 3/4: 3/4

F-string formatting:
Fraction: 3/4
As float: 0.75
As percentage: 75.0%

All 4 passes — pass 1 is the card above
pass	`frac`
1	1/2
2	1/3
3	1/4
4	2/3

All 4 passes — pass 1 is the card above
pass	`f`	`remainder`	`whole`
1	7/4	—	—
2	11/3	—	—
3	5/2	—	—
4	2	0	2

All 4 passes — pass 1 is the card above
pass	`frac`	`frac.numerator`	`frac.denominator`	`whole`	`remainder`
1	7/4	7	4	1	3
2	11/3	11	3	3	2
3	5/2	5	2	2	1
4	2	2	1	2	0

All 3 passes — pass 1 is the card above
pass	`whole`	`remainder`	`frac.denominator`
1	1	3	4
2	3	2	3
3	2	1	2

All 5 passes — pass 1 is the card above
pass	`d`	`f`
1	0.5	1/2
2	0.25	1/4
3	0.75	3/4
4	0.125	1/8
5	0.333	333/1000

All 5 passes — pass 1 is the card above
pass	`s`	`f`
1	1/2	1/2
2	3/4	3/4
3	0.5	1/2
4	0.75	3/4
5	22/7	22/7

All 3 passes — pass 1 is the card above
pass	`f`	`error`
1	22/7	0.0012644892673496777
2	355/113	2.667641894049666e-07
3	103993/33102	5.778906242426274e-10

All 7 passes — pass 1 is the card above
pass	`pct`	`f`
1	10	1/10
2	20	1/5
3	25	1/4
4	33.33	1/3
5	50	1/2
6	75	3/4
7	100	1

All 5 passes — pass 1 is the card above
pass	`_`	`q`	`r`	`terms`	`n`	`d`
1	0	3	1	[] → [3]	22 → 7	7 → 1
2	1	7	0	[3] → [3, 7]	7 → 1	1 → 0
3	0	3	16	[] → [3]	355 → 113	113 → 16
4	1	7	1	[3] → [3, 7]	113 → 16	16 → 1
5	2	16	0	[3, 7] → [3, 7, 16]	16 → 1	1 → 0

limit_denominator() Finds the closest fraction with a denominator at or below a specified limit - useful for approximating floats as simple fractions.

Comparisons

Comparing fractions:

                            comparison.py
                        
Replay: real traced execution (multi-file project)

                            # Comparison operations

from fractions import Fraction

# Comparison
a = Fraction(3, 4)
b = Fraction(2, 3)
c = Fraction(6, 8)  # Same as 3/4

print(f"a = {a}")
print(f"b = {b}")
print(f"c = {c}")
print()

# Equality
print("Equality:")
print(f"a == b: {a == b}")
print(f"a == c: {a == c}")
print(f"a != b: {a != b}")

# Ordering
print("\nOrdering:")
print(f"a < b: {a < b}")
print(f"a <= b: {a <= b}")
print(f"a > b: {a > b}")
print(f"a >= b: {a >= b}")

# Compare with numbers
print("\nCompare with numbers:")
f = Fraction(1, 2)
print(f"{f} == 0.5: {f == 0.5}")
print(f"{f} > 0.4: {f > 0.4}")
print(f"{f} < 1: {f < 1}")

# max and min
print("\nmax and min:")
fractions = [Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)]
print(f"Fractions: {fractions}")
print(f"Max: {max(fractions)}")
print(f"Min: {min(fractions)}")

# Sort
print("\nSort:")
unsorted = [Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)]
sorted_fracs = sorted(unsorted)
print(f"Unsorted: {unsorted}")
print(f"Sorted: {sorted_fracs}")

# Sort descending
print("\nSort descending:")
desc = sorted(unsorted, reverse=True)
print(f"Descending: {desc}")

# Find max in different forms
print("\nCompare different representations:")
vals = [Fraction(3, 4), 0.75, Fraction('0.75')]
print(f"Values: {vals}")
print(f"All equal: {vals[0] == vals[1] == vals[2]}")

# Range check
print("\nRange check:")
value = Fraction(1, 2)
lower = Fraction(1, 4)
upper = Fraction(3, 4)
in_range = lower <= value <= upper
print(f"{value} in [{lower}, {upper}]: {in_range}")

# Zero comparison
print("\nZero comparison:")
zero = Fraction(0)
pos = Fraction(1, 2)
neg = Fraction(-1, 2)

print(f"{zero} == 0: {zero == 0}")
print(f"{pos} > 0: {pos > 0}")
print(f"{neg} < 0: {neg < 0}")

# Proper vs improper fractions
print("\nProper vs improper fractions:")
proper = Fraction(3, 4)  # < 1
improper = Fraction(5, 4)  # > 1

print(f"{proper} is proper (< 1): {proper < 1}")
print(f"{improper} is improper (> 1): {improper > 1}")

# Compare fractions visually
print("\nCompare fractions:")
fracs = [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5)]
for f in fracs:
    bar = '█' * int(f * 20)
    print(f"{str(f):>4} = {float(f):.4f} {bar}")

# Unit fractions (1/n)
print("\nUnit fractions (sorted):")
units = [Fraction(1, i) for i in range(2, 11)]
for f in units:
    print(f"1/{f.denominator} = {float(f):.6f}")

# Egyptian fractions comparison
print("\nEgyptian fractions (distinct unit fractions):")
# 5/6 = 1/2 + 1/3
frac = Fraction(5, 6)
decomp = Fraction(1, 2) + Fraction(1, 3)
print(f"{frac} == 1/2 + 1/3: {frac == decomp}")

# Tolerance comparison (not usually needed)
print("\nTolerance comparison (for demonstration):")
f1 = Fraction(1, 3)
f2 = Fraction(333, 1000)  # Approximate 1/3
diff = abs(f1 - f2)
tolerance = Fraction(1, 100)
approx_equal = diff <= tolerance
print(f"{f1} ≈ {f2} (tolerance {tolerance}): {approx_equal}")
print(f"Difference: {diff}")


                        
1a ← 3/4, b ← 2/3, c ← 3/4, f ← 1/2, fractions ← [Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)]⋯5# Comparison6a→ 3/4 = Fraction(3, 4)7b→ 2/3 = Fraction(2, 3)8c→ 3/4 = Fraction(6, 8)  # Same as 3/4910print(f"a = {a3/4}")11print(f"b = {b2/3}")12print(f"c = {c3/4}")13print()1415# Equality16print("Equality:")17print(f"a == b: {a3/4 == b2/3}")18print(f"a == c: {a3/4 == c3/4}")19print(f"a != b: {a3/4 != b2/3}")2021# Ordering22print("\nOrdering:")23print(f"a < b: {a3/4 < b2/3}")24print(f"a <= b: {a3/4 <= b2/3}")25print(f"a > b: {a3/4 > b2/3}")26print(f"a >= b: {a3/4 >= b2/3}")2728# Compare with numbers29print("\nCompare with numbers:")30f→ 1/2 = Fraction(1, 2)31print(f"{f1/2} == 0.5: {f == 0.5}")32print(f"{f1/2} > 0.4: {f > 0.4}")33print(f"{f1/2} < 1: {f < 1}")3435# max and min36print("\nmax and min:")37fractions→ [Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)] = [Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)]38print(f"Fractions: {fractions[Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)]}")39print(f"Max: {max(fractions[Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)])}")40print(f"Min: {min(fractions[Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)])}")4142# Sort43print("\nSort:")44unsorted→ [Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)] = [Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)]45sorted_fracs→ [Fraction(1, 4), Fraction(1, 2), Fraction(2, 3), Fraction(3, 4)] = sorted(unsorted[Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)])46print(f"Unsorted: {unsorted[Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)]}")47print(f"Sorted: {sorted_fracs[Fraction(1, 4), Fraction(1, 2), Fraction(2, 3), Fraction(3, 4)]}")4849# Sort descending50print("\nSort descending:")51desc→ [Fraction(3, 4), Fraction(2, 3), Fraction(1, 2), Fraction(1, 4)] = sorted(unsorted[Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)], reverse=True)52print(f"Descending: {desc[Fraction(3, 4), Fraction(2, 3), Fraction(1, 2), Fraction(1, 4)]}")5354# Find max in different forms55print("\nCompare different representations:")56vals→ [Fraction(3, 4), 0.75, Fraction(3, 4)] = [Fraction(3, 4), 0.75, Fraction('0.75')]57print(f"Values: {vals[Fraction(3, 4), 0.75, Fraction(3, 4)]}")58print(f"All equal: {vals[0]3/4 == vals[1]0.75 == vals[2]3/4}")5960# Range check61print("\nRange check:")62value→ 1/2 = Fraction(1, 2)63lower→ 1/4 = Fraction(1, 4)64upper→ 3/4 = Fraction(3, 4)65in_range→ True = lower1/4 <= value1/2 <= upper3/466print(f"{value1/2} in [{lower1/4}, {upper3/4}]: {in_rangeTrue}")6768# Zero comparison69print("\nZero comparison:")70zero→ 0 = Fraction(0)71pos→ 1/2 = Fraction(1, 2)72neg→ -1/2 = Fraction(-1, 2)7374print(f"{zero0} == 0: {zero == 0}")75print(f"{pos1/2} > 0: {pos > 0}")76print(f"{neg-1/2} < 0: {neg < 0}")7778# Proper vs improper fractions79print("\nProper vs improper fractions:")80proper→ 3/4 = Fraction(3, 4)  # < 181improper→ 5/4 = Fraction(5, 4)  # > 18283print(f"{proper3/4} is proper (< 1): {proper < 1}")84print(f"{improper5/4} is improper (> 1): {improper > 1}")8586# Compare fractions visually87print("\nCompare fractions:")88fracs→ [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5)] = [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5)]89for f in fracs:⋯outputa = 3/4
b = 2/3
c = 3/4
Equality:
a == b: False
a == c: True
a != b: True

Ordering:
a < b: False
a <= b: False
a > b: True
a >= b: True

Compare with numbers:
1/2 == 0.5: True
1/2 > 0.4: True
1/2 < 1: True

max and min:
Fractions: [Fraction(1, 2), Fraction(1, 3), Fraction(2, 3), Fraction(1, 4)]
Max: 2/3
Min: 1/4

Sort:
Unsorted: [Fraction(2, 3), Fraction(1, 4), Fraction(3, 4), Fraction(1, 2)]
Sorted: [Fraction(1, 4), Fraction(1, 2), Fraction(2, 3), Fraction(3, 4)]

Sort descending:
Descending: [Fraction(3, 4), Fraction(2, 3), Fraction(1, 2), Fraction(1, 4)]

Compare different representations:
Values: [Fraction(3, 4), 0.75, Fraction(3, 4)]
All equal: True

Range check:
1/2 in [1/4, 3/4]: True

Zero comparison:
0 == 0: True
1/2 > 0: True
-1/2 < 0: True

Proper vs improper fractions:
3/4 is proper (< 1): True
5/4 is improper (> 1): True

Compare fractions:
2bar ← ██████████pass 1 of 4⋯88fracs = [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5)]89for f1/2 in fracs[Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5)]:90    bar→ ██████████ = '█' * int(f1/2 * 20)91    print(f"{str(f1/2):>4} = {float(f):.4f} {bar██████████}")⋯output 1/2 = 0.5000 ██████████All 4 passes — pass 1 is the card abovepassfbar
11/2██████████
21/3██████
31/4█████
41/5████
3units ← [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5), Fraction(1, 6), Fraction(1, 7), Fraction(1, 8), Fraction(1, 9), Fraction(1, 10)]⋯93# Unit fractions (1/n)94print("\nUnit fractions (sorted):")95units→ [Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5), Fraction(1, 6), Fraction(1, 7), Fraction(1, 8), Fraction(1, 9), Fraction(1, 10)] = [Fraction(1, i) for i in range(2, 11)]96for f in units:⋯output
Unit fractions (sorted):
4for f in units:pass 1 of 9⋯95units = [Fraction(1, i) for i in range(2, 11)]96for f1/2 in units[Fraction(1, 2), Fraction(1, 3), Fraction(1, 4), Fraction(1, 5), Fraction(1, 6), Fraction(1, 7), Fraction(1, 8), Fraction(1, 9), Fraction(1, 10)]:97    print(f"1/{f.denominator2} = {float(f1/2):.6f}")⋯output1/2 = 0.500000All 9 passes — pass 1 is the card abovepassff.denominator
11/22
21/33
31/44
41/55
51/66
61/77
71/88
81/99
91/1010
5frac ← 5/6, decomp ← 5/6, f1 ← 1/3, f2 ← 333/1000, diff ← 1/3000⋯99# Egyptian fractions comparison100print("\nEgyptian fractions (distinct unit fractions):")101# 5/6 = 1/2 + 1/3102frac→ 5/6 = Fraction(5, 6)103decomp→ 5/6 = Fraction(1, 2) + Fraction(1, 3)104print(f"{frac5/6} == 1/2 + 1/3: {frac == decomp5/6}")105106# Tolerance comparison (not usually needed)107print("\nTolerance comparison (for demonstration):")108f1→ 1/3 = Fraction(1, 3)109f2→ 333/1000 = Fraction(333, 1000)  # Approximate 1/3110diff→ 1/3000 = abs(f11/3 - f2333/1000)111tolerance→ 1/100 = Fraction(1, 100)112approx_equal→ True = diff1/3000 <= tolerance1/100113print(f"{f11/3} ≈ {f2333/1000} (tolerance {tolerance1/100}): {approx_equalTrue}")114print(f"Difference: {diff1/3000}")⋯output
Egyptian fractions (distinct unit fractions):
5/6 == 1/2 + 1/3: True

Tolerance comparison (for demonstration):
1/3 ≈ 333/1000 (tolerance 1/100): True
Difference: 1/3000

All 4 passes — pass 1 is the card above
pass	`f`	`bar`
1	1/2	██████████
2	1/3	██████
3	1/4	█████
4	1/5	████

All 9 passes — pass 1 is the card above
pass	`f`	`f.denominator`
1	1/2	2
2	1/3	3
3	1/4	4
4	1/5	5
5	1/6	6
6	1/7	7
7	1/8	8
8	1/9	9
9	1/10	10

Math Operations

Using fractions with math functions:

                            math.py
                        
Replay: real traced execution (multi-file project)

                            # Mathematical operations

from fractions import Fraction
import math

# GCD and simplification
print("GCD and simplification:")

def show_gcd(a, b):
    """Show GCD and simplification."""
    g = math.gcd(a, b)
    print(f"gcd({a}, {b}) = {g}")
    print(f"  {a}/{b} = {a//g}/{b//g}")

show_gcd(6, 8)
show_gcd(12, 16)
show_gcd(100, 150)
print()

# LCM (least common multiple)
print("LCM for adding fractions:")
def show_lcm_addition(f1, f2):
    """Show LCM method for adding fractions."""
    lcm = (f1.denominator * f2.denominator) // math.gcd(f1.denominator, f2.denominator)
    print(f"{f1} + {f2}:")
    print(f"  LCM of {f1.denominator} and {f2.denominator} = {lcm}")
    print(f"  = {f1.numerator * (lcm // f1.denominator)}/{lcm} + {f2.numerator * (lcm // f2.denominator)}/{lcm}")
    print(f"  = {f1 + f2}")

show_lcm_addition(Fraction(1, 3), Fraction(1, 4))
show_lcm_addition(Fraction(2, 5), Fraction(3, 7))
print()

# Powers and roots
print("Powers:")
f = Fraction(2, 3)
for exp in range(1, 5):
    result = f ** exp
    print(f"({f})^{exp} = {result} = {float(result):.6f}")

print("\nNegative powers (reciprocals):")
for exp in range(1, 4):
    result = f ** -exp
    print(f"({f})^-{exp} = {result} = {float(result):.6f}")

print("\nSquare root (as float):")
f = Fraction(9, 16)
sqrt = math.sqrt(float(f))
print(f"√{f} = {sqrt}")
sqrt_frac = Fraction(3, 4)  # Manual
print(f"  = {sqrt_frac} (exact)")

# Series calculations
print("\nGeometric series:")
# Sum: a + ar + ar² + ar³ + ... (n terms)
def geometric_sum(a, r, n):
    """Sum of geometric series."""
    if r == 1:
        return a * n
    return a * (1 - r**n) / (1 - r)

a = Fraction(1, 2)
r = Fraction(1, 2)
for n in [3, 5, 10]:
    s = geometric_sum(a, r, n)
    print(f"Sum of {n} terms (a={a}, r={r}): {s} = {float(s):.10f}")

# Approaching limit
print("\nApproaching limit (1 - 1/2^n):")
for n in range(1, 11):
    limit = 1 - Fraction(1, 2**n)
    print(f"n={n:2}: {limit} = {float(limit):.10f}")

# Fibonacci with fractions
print("\nFibonacci ratios:")
def fibonacci_ratio(n):
    """Get ratio of consecutive Fibonacci numbers."""
    if n <= 1:
        return None
    a, b = Fraction(0), Fraction(1)
    for _ in range(n):
        a, b = b, a + b
    return b / a if a != 0 else None

golden = (1 + math.sqrt(5)) / 2
print("Ratio of consecutive Fibonacci numbers (approaching φ):")
for n in range(5, 16):
    ratio = fibonacci_ratio(n)
    if ratio:
        error = abs(float(ratio) - golden)
        print(f"F({n})/F({n-1}) = {ratio.limit_denominator(10000)} ≈ {float(ratio):.10f} (error: {error:.2e})")
print(f"Golden ratio φ = {golden:.10f}")

# Harmonic mean
print("\nHarmonic mean:")
def harmonic_mean(nums):
    """Calculate harmonic mean."""
    return len(nums) / sum(Fraction(1, x) for x in nums)

values = [Fraction(2), Fraction(3), Fraction(4)]
hm = harmonic_mean(values)
print(f"Harmonic mean of {[float(v) for v in values]}: {hm} = {float(hm):.6f}")

# Egyptian fractions
print("\nEgyptian fraction decomposition:")
def egyptian_fractions(frac, max_terms=10):
    """Decompose fraction into sum of unit fractions."""
    terms = []
    remaining = frac
    n = 2
    while remaining > 0 and len(terms) < max_terms:
        unit = Fraction(1, n)
        if unit <= remaining:
            terms.append(n)
            remaining -= unit
        n += 1
        if remaining == 0:
            break
    return terms

f = Fraction(5, 6)
decomp = egyptian_fractions(f)
print(f"{f} = {' + '.join(f'1/{d}' for d in decomp)}")
# Verify
verify = sum(Fraction(1, d) for d in decomp)
print(f"Verify: {verify}")

# Mediant
print("\nMediant (between two fractions):")
def mediant(f1, f2):
    """Calculate mediant of two fractions."""
    return Fraction(f1.numerator + f2.numerator, f1.denominator + f2.denominator)

a, b = Fraction(1, 3), Fraction(1, 2)
m = mediant(a, b)
print(f"Mediant of {a} and {b}: {m} = {float(m):.6f}")
print(f"Check: {float(a):.6f} < {float(m):.6f} < {float(b):.6f}")


                        
1print("GCD and simplification:")⋯6# GCD and simplification7print("GCD and simplification:")89def show_gcd(a, b):10    """Show GCD and simplification."""11    g = math.gcd(a, b)12    print(f"gcd({a}, {b}) = {g}")13    print(f"  {a}/{b} = {a//g}/{b//g}")1415show_gcd(6, 8)16show_gcd(12, 16)⋯outputGCD and simplification:
2g ← 2pass 1 of 3⋯9def show_gcd(a6, b8):10    """Show GCD and simplification."""11    g→ 2 = math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.gcd(a6, b8)12    print(f"gcd({a6}, {b8}) = {g2}")13    print(f"  {a6}/{b8} = {a//g2}/{b//g}")⋯outputgcd(6, 8) = 2
  6/8 = 3/4All 3 passes — pass 1 is the card abovepassabg
1682
212164
310015050
3show_gcd(6, 8)⋯15show_gcd(6, 8)16show_gcd(12, 16)17show_gcd(100, 150)⋯
4show_gcd(12, 16)⋯15show_gcd(6, 8)16show_gcd(12, 16)17show_gcd(100, 150)18print()⋯
5show_gcd(100, 150)⋯16show_gcd(12, 16)17show_gcd(100, 150)18print()1920# LCM (least common multiple)21print("LCM for adding fractions:")22def show_lcm_addition(f1, f2):23    """Show LCM method for adding fractions."""24    lcm = (f1.denominator * f2.denominator) // math.gcd(f1.denominator, f2.denominator)25    print(f"{f1} + {f2}:")26    print(f"  LCM of {f1.denominator} and {f2.denominator} = {lcm}")27    print(f"  = {f1.numerator * (lcm // f1.denominator)}/{lcm} + {f2.numerator * (lcm // f2.denominator)}/{lcm}")28    print(f"  = {f1 + f2}")2930show_lcm_addition(Fraction(1, 3), Fraction(1, 4))31show_lcm_addition(Fraction(2, 5), Fraction(3, 7))⋯outputLCM for adding fractions:
6lcm ← 12pass 1 of 2⋯21print("LCM for adding fractions:")22def show_lcm_addition(f11/3, f21/4):23    """Show LCM method for adding fractions."""24    lcm→ 12 = (f1.denominator3 * f2.denominator4) // math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.gcd(f1.denominator, f2.denominator)25    print(f"{f11/3} + {f21/4}:")26    print(f"  LCM of {f1.denominator3} and {f2.denominator4} = {lcm12}")27    print(f"  = {f1.numerator1 * (lcm12 // f1.denominator3)}/{lcm} + {f2.numerator1 * (lcm // f2.denominator4)}/{lcm}")28    print(f"  = {f11/3 + f21/4}")⋯output1/3 + 1/4:
  LCM of 3 and 4 = 12
  = 4/12 + 3/12
  = 7/12
7show_lcm_addition(Fraction(1, 3), Fraction(1, 4))⋯30show_lcm_addition(Fraction(1, 3), Fraction(1, 4))31show_lcm_addition(Fraction(2, 5), Fraction(3, 7))32print()⋯
8lcm ← 35pass 2 of 2⋯21print("LCM for adding fractions:")22def show_lcm_addition(f12/5, f23/7):23    """Show LCM method for adding fractions."""24    lcm→ 35 = (f1.denominator5 * f2.denominator7) // math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.gcd(f1.denominator, f2.denominator)25    print(f"{f12/5} + {f23/7}:")26    print(f"  LCM of {f1.denominator5} and {f2.denominator7} = {lcm35}")27    print(f"  = {f1.numerator2 * (lcm35 // f1.denominator5)}/{lcm} + {f2.numerator3 * (lcm // f2.denominator7)}/{lcm}")28    print(f"  = {f12/5 + f23/7}")⋯output2/5 + 3/7:
  LCM of 5 and 7 = 35
  = 14/35 + 15/35
  = 29/35
9f ← 2/3⋯30show_lcm_addition(Fraction(1, 3), Fraction(1, 4))31show_lcm_addition(Fraction(2, 5), Fraction(3, 7))32print()3334# Powers and roots35print("Powers:")36f→ 2/3 = Fraction(2, 3)37for exp in range(1, 5):⋯outputPowers:
10result ← 2/3pass 1 of 4⋯36f = Fraction(2, 3)37for exp1 in range(1, 5):38    result→ 2/3 = f2/3 ** exp139    print(f"({f2/3})^{exp1} = {result2/3} = {float(result):.6f}")⋯output(2/3)^1 = 2/3 = 0.666667All 4 passes — pass 1 is the card abovepassexpresult
112/3
224/9
338/27
4416/81
11print(" Negative powers (reciprocals):")⋯41print("\nNegative powers (reciprocals):")42for exp in range(1, 4):⋯output
Negative powers (reciprocals):
12result ← 3/2pass 1 of 3⋯41print("\nNegative powers (reciprocals):")42for exp1 in range(1, 4):43    result→ 3/2 = f2/3 ** -exp144    print(f"({f2/3})^-{exp1} = {result3/2} = {float(result):.6f}")⋯output(2/3)^-1 = 3/2 = 1.500000All 3 passes — pass 1 is the card abovepassexpresult
113/2
229/4
3327/8
13f ← 9/16, sqrt ← 0.75, sqrt_frac ← 3/4, a ← 1/2, r ← 1/2⋯46print("\nSquare root (as float):")47f→ 9/16 = Fraction(9, 16)48sqrt→ 0.75 = math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.sqrt(float(f9/16))49print(f"√{f9/16} = {sqrt0.75}")50sqrt_frac→ 3/4 = Fraction(3, 4)  # Manual51print(f"  = {sqrt_frac3/4} (exact)")5253# Series calculations54print("\nGeometric series:")55# Sum: a + ar + ar² + ar³ + ... (n terms)56def geometric_sum(a, r, n):57    """Sum of geometric series."""58    if r == 1:59        return a * n60    return a * (1 - r**n) / (1 - r)6162a→ 1/2 = Fraction(1, 2)63r→ 1/2 = Fraction(1, 2)64for n in [3, 5, 10]:⋯output
Square root (as float):
√9/16 = 0.75
  = 3/4 (exact)

Geometric series:
14for n in [3, 5, 10]:pass 1 of 3⋯63r = Fraction(1, 2)64for n3 in [3, 5, 10]:65    s = geometric_sum(a1/2, r1/2, n3)66    print(f"Sum of {n} terms (a={a}, r={r}): {s} = {float(s):.10f}")⋯All 3 passes — pass 1 is the card abovepassn
13
25
310
15def geometric_sum(a, r, n):pass 1 of 3⋯55# Sum: a + ar + ar² + ar³ + ... (n terms)56def geometric_sum(a1/2, r1/2, n3):57    """Sum of geometric series."""58    if r == 1:59        return a * n60    return a1/2 * (1 - r1/2**n3) / (1 - r)⋯All 3 passes — pass 1 is the card abovepassn
13
25
310
16s ← 7/8⋯64for n in [3, 5, 10]:65    s→ 7/8 = geometric_sum(a1/2, r1/2, n3)66    print(f"Sum of {n3} terms (a={a1/2}, r={r1/2}): {s7/8} = {float(s):.10f}")⋯outputSum of 3 terms (a=1/2, r=1/2): 7/8 = 0.8750000000
17s ← 31/32⋯64for n in [3, 5, 10]:65    s→ 31/32 = geometric_sum(a1/2, r1/2, n5)66    print(f"Sum of {n5} terms (a={a1/2}, r={r1/2}): {s31/32} = {float(s):.10f}")⋯outputSum of 5 terms (a=1/2, r=1/2): 31/32 = 0.9687500000
18s ← 1023/1024⋯64for n in [3, 5, 10]:65    s→ 1023/1024 = geometric_sum(a1/2, r1/2, n10)66    print(f"Sum of {n10} terms (a={a1/2}, r={r1/2}): {s1023/1024} = {float(s):.10f}")⋯outputSum of 10 terms (a=1/2, r=1/2): 1023/1024 = 0.9990234375
19print(" Approaching limit (1 - 1/2^n):")⋯68# Approaching limit69print("\nApproaching limit (1 - 1/2^n):")70for n in range(1, 11):⋯output
Approaching limit (1 - 1/2^n):
20limit ← 1/2pass 1 of 10⋯69print("\nApproaching limit (1 - 1/2^n):")70for n1 in range(1, 11):71    limit→ 1/2 = 1 - Fraction(1, 2**n1)72    print(f"n={n1:2}: {limit1/2} = {float(limit):.10f}")⋯outputn= 1: 1/2 = 0.5000000000All 10 passes — pass 1 is the card abovepassnlimit
111/2
223/4
337/8
4415/16
5531/32
6663/64
77127/128
88255/256
99511/512
10101023/1024
21golden ← 1.618033988749895⋯74# Fibonacci with fractions75print("\nFibonacci ratios:")76def fibonacci_ratio(n):77    """Get ratio of consecutive Fibonacci numbers."""78    if n <= 1:79        return None80    a, b = Fraction(0), Fraction(1)81    for _ in range(n):82        a, b = b, a + b83    return b / a if a != 0 else None8485golden→ 1.618033988749895 = (1 + math<module 'math' from '/usr/local/lib/python3.12/lib-dynload/math.cpython-312-x86_64-linux-gnu.so'>.sqrt(5)) / 286print("Ratio of consecutive Fibonacci numbers (approaching φ):")87for n in range(5, 16):⋯output
Fibonacci ratios:
Ratio of consecutive Fibonacci numbers (approaching φ):
22for n in range(5, 16):pass 1 of 11⋯86print("Ratio of consecutive Fibonacci numbers (approaching φ):")87for n5 in range(5, 16):88    ratio = fibonacci_ratio(n5)89    if ratio:⋯All 11 passes — pass 1 is the card abovepassn
15
26
37
48
59
610
711
812
913
1014
1115
23a ← 0, b ← 1pass 1 of 11⋯75print("\nFibonacci ratios:")76def fibonacci_ratio(n5):77    """Get ratio of consecutive Fibonacci numbers."""78    if n <= 1:79        return None80    a→ 0, b→ 1 = Fraction(0), Fraction(1)81    for _ in range(n):⋯All 11 passes — pass 1 is the card abovepassnab
1501
2601
3701
4801
5901
61001
71101
81201
91301
101401
111501
24a ← 1, b ← 1pass 1 of 110⋯80a, b = Fraction(0), Fraction(1)81for _0 in range(n5):82    a→ 1, b→ 1 = b, a + b83return b / a if a != 0 else None⋯110 passes — pass 1 is the card abovepass_nab
1050 → 11
21511 → 2
3251 → 22 → 3
4352 → 33 → 5
5453 → 55 → 8
6060 → 11
71611 → 2
8261 → 22 → 3
9362 → 33 → 5
⋯ 99 more passes ⋯
1091315233 → 377377 → 610
1101415377 → 610610 → 987
25return b / a if a != 0 else None⋯82    a, b = b, a + b83return b8 / a5 if a != 0 else None⋯
26ratio ← 8/5⋯87for n in range(5, 16):88    ratio→ 8/5 = fibonacci_ratio(n5)89    if ratio:⋯
27error ← 0.018033988749894814pass 1 of 11⋯88    ratio = fibonacci_ratio(n)89    if ratio8/5:90        error→ 0.018033988749894814 = abs(float(ratio8/5) - golden1.618033988749895)91        print(f"F({n5})/F({n-1}) = {ratio8/5.limit_denominator(10000)} ≈ {float(ratio):.10f} (error: {error0.018033988749894814:.2e})")92print(f"Golden ratio φ = {golden:.10f}")⋯outputF(5)/F(4) = 8/5 ≈ 1.6000000000 (error: 1.80e-02)All 11 passes — pass 1 is the card abovepassrationerror
18/550.018033988749894814
213/860.0069660112501050975
321/1370.0026493733652794837
434/2180.0010136302977241662
555/3490.00038692992636546464
689/55100.00014782943192326314
7144/89115.6460660007306984e-05
8233/144122.15668056606777e-05
9377/233138.237676933475768e-06
10610/377143.1465286196574738e-06
11987/610151.2018646489142526e-06
28return b / a if a != 0 else None⋯82    a, b = b, a + b83return b13 / a8 if a != 0 else None⋯
29ratio ← 13/8⋯87for n in range(5, 16):88    ratio→ 13/8 = fibonacci_ratio(n6)89    if ratio:⋯
30return b / a if a != 0 else None⋯82    a, b = b, a + b83return b21 / a13 if a != 0 else None⋯
31ratio ← 21/13⋯87for n in range(5, 16):88    ratio→ 21/13 = fibonacci_ratio(n7)89    if ratio:⋯
32return b / a if a != 0 else None⋯82    a, b = b, a + b83return b34 / a21 if a != 0 else None⋯
33ratio ← 34/21⋯87for n in range(5, 16):88    ratio→ 34/21 = fibonacci_ratio(n8)89    if ratio:⋯
34return b / a if a != 0 else None⋯82    a, b = b, a + b83return b55 / a34 if a != 0 else None⋯
35ratio ← 55/34⋯87for n in range(5, 16):88    ratio→ 55/34 = fibonacci_ratio(n9)89    if ratio:⋯
36return b / a if a != 0 else None⋯82    a, b = b, a + b83return b89 / a55 if a != 0 else None⋯
37ratio ← 89/55⋯87for n in range(5, 16):88    ratio→ 89/55 = fibonacci_ratio(n10)89    if ratio:⋯
38return b / a if a != 0 else None⋯82    a, b = b, a + b83return b144 / a89 if a != 0 else None⋯
39ratio ← 144/89⋯87for n in range(5, 16):88    ratio→ 144/89 = fibonacci_ratio(n11)89    if ratio:⋯
40return b / a if a != 0 else None⋯82    a, b = b, a + b83return b233 / a144 if a != 0 else None⋯
41ratio ← 233/144⋯87for n in range(5, 16):88    ratio→ 233/144 = fibonacci_ratio(n12)89    if ratio:⋯
42return b / a if a != 0 else None⋯82    a, b = b, a + b83return b377 / a233 if a != 0 else None⋯
43ratio ← 377/233⋯87for n in range(5, 16):88    ratio→ 377/233 = fibonacci_ratio(n13)89    if ratio:⋯
44return b / a if a != 0 else None⋯82    a, b = b, a + b83return b610 / a377 if a != 0 else None⋯
45ratio ← 610/377⋯87for n in range(5, 16):88    ratio→ 610/377 = fibonacci_ratio(n14)89    if ratio:⋯
46return b / a if a != 0 else None⋯82    a, b = b, a + b83return b987 / a610 if a != 0 else None⋯
47ratio ← 987/610⋯87for n in range(5, 16):88    ratio→ 987/610 = fibonacci_ratio(n15)89    if ratio:⋯
48values ← [Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)]⋯91        print(f"F({n})/F({n-1}) = {ratio.limit_denominator(10000)} ≈ {float(ratio):.10f} (error: {error:.2e})")92print(f"Golden ratio φ = {golden1.618033988749895:.10f}")9394# Harmonic mean95print("\nHarmonic mean:")96def harmonic_mean(nums):97    """Calculate harmonic mean."""98    return len(nums) / sum(Fraction(1, x) for x in nums)99100values→ [Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)] = [Fraction(2), Fraction(3), Fraction(4)]101hm = harmonic_mean(values[Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)])102print(f"Harmonic mean of {[float(v) for v in values]}: {hm} = {float(hm):.6f}")⋯outputGolden ratio φ = 1.6180339887

Harmonic mean:
49def harmonic_mean(nums):⋯95print("\nHarmonic mean:")96def harmonic_mean(nums[Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)]):97    """Calculate harmonic mean."""98    return len(nums[Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)]) / sum(Fraction(1, x) for x in nums)⋯
50hm ← 36/13, f ← 5/6⋯100values = [Fraction(2), Fraction(3), Fraction(4)]101hm→ 36/13 = harmonic_mean(values[Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)])102print(f"Harmonic mean of {[float(v) for v in values[Fraction(2, 1), Fraction(3, 1), Fraction(4, 1)]]}: {hm36/13} = {float(hm):.6f}")103104# Egyptian fractions105print("\nEgyptian fraction decomposition:")106def egyptian_fractions(frac, max_terms=10):107    """Decompose fraction into sum of unit fractions."""108    terms = []109    remaining = frac110    n = 2111    while remaining > 0 and len(terms) < max_terms:112        unit = Fraction(1, n)113        if unit <= remaining:114            terms.append(n)115            remaining -= unit116        n += 1117        if remaining == 0:118            break119    return terms120121f→ 5/6 = Fraction(5, 6)122decomp = egyptian_fractions(f5/6)123print(f"{f} = {' + '.join(f'1/{d}' for d in decomp)}")⋯outputHarmonic mean of [2.0, 3.0, 4.0]: 36/13 = 2.769231

Egyptian fraction decomposition:
51terms ← [], remaining ← 5/6, n ← 2⋯105print("\nEgyptian fraction decomposition:")106def egyptian_fractions(frac5/6, max_terms10=10):107    """Decompose fraction into sum of unit fractions."""108    terms→ [] = []109    remaining→ 5/6 = frac5/6110    n→ 2 = 2111    while remaining > 0 and len(terms) < max_terms:⋯
52unit ← 1/2pass 1 of 2⋯110n = 2111while remaining5/6 > 0 and len(terms[]) < max_terms10:112    unit→ 1/2 = Fraction(1, n2)113    if unit <= remaining:⋯
53terms ← [2], remaining ← 1/3pass 1 of 2⋯112unit = Fraction(1, n)113if unit1/2 <= remaining5/6:114    terms→ [2].append(n2)115    remaining→ 1/3 -= unit1/2116n += 1⋯
54n ← 3⋯115    remaining -= unit116n→ 3 += 1117if remaining == 0:⋯
55unit ← 1/3pass 2 of 2⋯110n = 2111while remaining1/3 > 0 and len(terms[2]) < max_terms10:112    unit→ 1/3 = Fraction(1, n3)113    if unit <= remaining:⋯
56terms ← [2, 3], remaining ← 0pass 2 of 2⋯112unit = Fraction(1, n)113if unit1/3 <= remaining1/3:114    terms→ [2, 3].append(n3)115    remaining→ 0 -= unit1/3116n += 1⋯
57n ← 4⋯115    remaining -= unit116n→ 4 += 1117if remaining == 0:⋯
58if remaining == 0:⋯116    n += 1117    if remaining0 == 0:118        break119return terms⋯
59return terms⋯118        break119return terms[2, 3]⋯
60decomp ← [2, 3], verify ← 5/6, a ← 1/3, b ← 1/2⋯121f = Fraction(5, 6)122decomp→ [2, 3] = egyptian_fractions(f5/6)123print(f"{f5/6} = {' + '.join(f'1/{d}' for d in decomp[2, 3])}")124# Verify125verify→ 5/6 = sum(Fraction(1, d) for d in decomp[2, 3])126print(f"Verify: {verify5/6}")127128# Mediant129print("\nMediant (between two fractions):")130def mediant(f1, f2):131    """Calculate mediant of two fractions."""132    return Fraction(f1.numerator + f2.numerator, f1.denominator + f2.denominator)133134a→ 1/3, b→ 1/2 = Fraction(1, 3), Fraction(1, 2)135m = mediant(a1/3, b1/2)136print(f"Mediant of {a} and {b}: {m} = {float(m):.6f}")⋯output5/6 = 1/2 + 1/3
Verify: 5/6

Mediant (between two fractions):
61def mediant(f1, f2):⋯129print("\nMediant (between two fractions):")130def mediant(f11/3, f21/2):131    """Calculate mediant of two fractions."""132    return Fraction(f1.numerator1 + f2.numerator1, f1.denominator3 + f2.denominator2)⋯
62m ← 2/5⋯134a, b = Fraction(1, 3), Fraction(1, 2)135m→ 2/5 = mediant(a1/3, b1/2)136print(f"Mediant of {a1/3} and {b1/2}: {m2/5} = {float(m):.6f}")137print(f"Check: {float(a1/3):.6f} < {float(m2/5):.6f} < {float(b1/2):.6f}")⋯outputMediant of 1/3 and 1/2: 2/5 = 0.400000
Check: 0.333333 < 0.400000 < 0.500000

All 3 passes — pass 1 is the card above
pass	`a`	`b`	`g`
1	6	8	2
2	12	16	4
3	100	150	50

All 4 passes — pass 1 is the card above
pass	`exp`	`result`
1	1	2/3
2	2	4/9
3	3	8/27
4	4	16/81

All 3 passes — pass 1 is the card above
pass	`exp`	`result`
1	1	3/2
2	2	9/4
3	3	27/8

All 3 passes — pass 1 is the card above
pass	`n`
1	3
2	5
3	10

All 3 passes — pass 1 is the card above
pass	`n`
1	3
2	5
3	10

All 10 passes — pass 1 is the card above
pass	`n`	`limit`
1	1	1/2
2	2	3/4
3	3	7/8
4	4	15/16
5	5	31/32
6	6	63/64
7	7	127/128
8	8	255/256
9	9	511/512
10	10	1023/1024

All 11 passes — pass 1 is the card above
pass	`n`
1	5
2	6
3	7
4	8
5	9
6	10
7	11
8	12
9	13
10	14
11	15

All 11 passes — pass 1 is the card above
pass	`n`	`b`
1	5	1
2	6	1
3	7	1
4	8	1
5	9	1
6	10	1
7	11	1
8	12	1
9	13	1
10	14	1
11	15	1

110 passes — pass 1 is the card above
pass	`_`	`n`	`a`	`b`
1	0	5	0 → 1	1
2	1	5	1	1 → 2
3	2	5	1 → 2	2 → 3
4	3	5	2 → 3	3 → 5
5	4	5	3 → 5	5 → 8
6	0	6	0 → 1	1
7	1	6	1	1 → 2
8	2	6	1 → 2	2 → 3
9	3	6	2 → 3	3 → 5
⋯ 99 more passes ⋯
109	13	15	233 → 377	377 → 610
110	14	15	377 → 610	610 → 987

All 11 passes — pass 1 is the card above
pass	`ratio`	`n`	`error`
1	8/5	5	0.018033988749894814
2	13/8	6	0.0069660112501050975
3	21/13	7	0.0026493733652794837
4	34/21	8	0.0010136302977241662
5	55/34	9	0.00038692992636546464
6	89/55	10	0.00014782943192326314
7	144/89	11	5.6460660007306984e-05
8	233/144	12	2.15668056606777e-05
9	377/233	13	8.237676933475768e-06
10	610/377	14	3.1465286196574738e-06
11	987/610	15	1.2018646489142526e-06

Exercise: practical.py

Scale a recipe by a fraction and calculate total ingredient amounts