Multiplication

Multiplication is a fundamental mathematical operation that combines two numbers with a third, called the product.

For natural numbers, it can be understood as a repetition of addition: to multiply is to add the same number several times.

Any multiplication of big/large/long numbers exceeding one million or one billion often generates errors on standard calculators.

The main problem stems from how calculators and computers represent numbers in memory.

— Representation limit: Common integers have a fixed size in bits. On a 32-bit architecture, an unsigned integer is bounded by $ 2^{32}-1 = 4294967295 $, while a signed integer is bounded by $ 2147483647 $. Beyond this limit, there is an overflow, which can produce an incorrect result.

— Display limit: Even if the calculation was correct, some calculators or software truncate the display beyond a certain number of digits.

— Calculation cost: The larger the numbers, the more time and memory are required to multiply them.

To multiply numbers with thousands (or even millions) of digits, the user must employ arbitrary-precision arithmetic tools.

These tools do not store numbers in a fixed format (32 or 64 bits), but rather as lists of digits, and apply specialized algorithms (classical multiplication, Karatsuba, or methods based on the Fourier transform).

This allows for an exact result, without rounding and without resorting to scientific notation, even for very large integers.

$ 5000000 \times 1000000000 = 5000000000000000 $ (5 million multiplied by 1 billion)

Multiplication tables are an organized chart that summarizes the products of common whole numbers, usually from $ 1 $ to $ 10 $.

Traditionally multiplication tables refers to this table:

The Karatsuba algorithm is a fast multiplication technique for large numbers. In order to improve calculation time the multiplication is accelerated by decomposing it:

ab * cd = (a * 10^k + b) * (c * 10^k + d) = ac * 10^2k + (ad + bc) * 10^k + bd

This multiplication needs 4 values ac, ad, bc and bd. More:

(a * 10^k + b) * (c * 10^k + d) = ac * 10^2k + (ac + bd - (a - b)(c - d)) * 10^k + bd

The same multiplication needs 3 values: ac, bd and (a - b)(c - d).