Binary Multiplier - Multiplication Basics

Multiplication Basics

The method taught in school for multiplying decimal numbers is based on calculating partial products, shifting them to the left and then adding them together. The most difficult part is to obtain the partial products, as that involves multiplying a long number by one digit (from 0 to 9):

123 x 456 ===== 738 (this is 123 x 6) 615 (this is 123 x 5, shifted one position to the left) + 492 (this is 123 x 4, shifted two positions to the left) ===== 56088

A binary computer does exactly the same, but with binary numbers. In binary encoding each long number is multiplied by one digit (either 0 or 1), and that is much easier than in decimal, as the product by 0 or 1 is just 0 or the same number. Therefore, the multiplication of two binary numbers comes down to calculating partial products (which are 0 or the first number), shifting them left, and then adding them together (a binary addition, of course):

1011 (this is 11 in decimal) x 1110 (this is 14 in decimal) ====== 0000 (this is 1011 x 0) 1011 (this is 1011 x 1, shifted one position to the left) 1011 (this is 1011 x 1, shifted two positions to the left) + 1011 (this is 1011 x 1, shifted three positions to the left) ========= 10011010 (this is 154 in binary)

This is much simpler than in the decimal system, as there is no table of multiplication to remember: just shifts and adds.

This method is mathematically correct and has the advantage that a small CPU may perform the multiplication by using the shift and add features of its arithmetic logic unit rather than a specialized circuit. The method is slow, however, as it involves many intermediate additions. These additions take a lot of time. Faster multipliers may be engineered in order to do fewer additions; a modern processor can multiply two 64-bit numbers with 16 additions (rather than 64), and can do several steps in parallel.

The second problem is that the basic school method handles the sign with a separate rule ("+ with + yields +", "+ with - yields -", etc.). Modern computers embed the sign of the number in the number itself, usually in the two's complement representation. That forces the multiplication process to be adapted to handle two's complement numbers, and that complicates the process a bit more. Similarly, processors that use ones' complement, sign-and-magnitude, IEEE-754 or other binary representations require specific adjustments to the multiplication process.