Tutorial: Working with Big Integers in Julia

Julia provides excellent support for working with arbitrarily large integers, often called “big integers” or “bignums.” This is incredibly useful when dealing with numbers that exceed the capacity of standard integer types (like Int64). This tutorial covers the basics of using big integers in Julia.

1. Creating Big Integers:

Julia’s BigInt type represents big integers. You can create BigInt values in several ways:

• Using the big() function: This is the most common way to create a big integer from a regular integer or a string.

x = big(12345678901234567890)  # Creates a BigInt from an integer literal
y = big("98765432109876543210") # Creates a BigInt from a string
z = big(Int64(9223372036854775807)) #Creates a BigInt from a Int64 (Max Int64 value)

println(x)
println(y)
println(z)

• From arithmetic operations: If an arithmetic operation on integers would result in a number too large to fit in a standard integer type, Julia will automatically promote the result to a BigInt.

a = 2^63  # This will be an Int64
b = 2^64  # This will automatically be a BigInt because it's too large for Int64

println(typeof(a)) # Output: Int64
println(typeof(b)) # Output: BigInt

2. Arithmetic Operations:

You can perform all the standard arithmetic operations (+, -, *, /, ^, %, div, mod) on BigInt values, just like with regular integers. Julia overloads these operators to work seamlessly with big integers.

x = big(12345)
y = big(67890)

sum = x + y
difference = x - y
product = x * y
quotient = x ÷ y  # Integer division (div)
remainder = x % y  # Modulo operation
power = x^y

println("Sum: ", sum)
println("Difference: ", difference)
println("Product: ", product)
println("Quotient: ", quotient)
println("Remainder: ", remainder)
println("Power: ", power)

3. Comparisons:

You can compare BigInt values using the standard comparison operators (==, !=, <, >, <=, >=).

x = big(100)
y = big(200)

println(x == y)  # Output: false
println(x < y)   # Output: true
println(x >= y)  # Output: false

4. Functions and Big Integers:

Many built-in Julia functions work correctly with BigInt arguments. For example:

x = big(12345678901234567890)
println(sqrt(x)) # Calculate the square root (returns a Float64 by default)
println(factorial(big(100))) # Calculate 100 factorial (a very large number)

Important Note: While BigInts offer arbitrary precision, they are generally slower than standard integer types. Use them when you truly need to work with numbers exceeding the limits of Int64. For most everyday calculations, Int64 is sufficient and more efficient.

5. Converting to other types:

You can convert a BigInt to other number types, but be careful about potential loss of information if the BigInt is too large.

x = big(123)
y = Int64(x) # Convert to Int64

z = big(2^100)
#w = Int64(z) # This will throw an InexactError because z is too large

f = Float64(z)  # Convert to Float64 (might lose some precision)
println(f)

Example: Calculating a large Fibonacci number:

function fibonacci(n::Integer)
    if n <= 1
        return big(n)  # Important: Return BigInt for large n
    else
        return fibonacci(n-1) + fibonacci(n-2)
    end
end

println(fibonacci(100)) # Calculate the 100th Fibonacci number (a large BigInt)

This example demonstrates the power of BigInt for calculations involving very large numbers, where standard integer types would overflow. Remember to use big() when returning values from recursive functions like this to ensure that you’re working with BigInts throughout the calculation.