What does mathematics have to do with programming languages?

Answer 1

Logic is a major aspect of mathematics, particularly boolean algebra. While it is possible to write programs using nothing more than elementary decimal arithmetic, you cannot always write efficient algorithms using nothing more than high-level abstractions like decimal; you need to talk to the machine in its own native language.

For instance, to determine if a positive whole number is odd or even, a human will simply look at the least significant digit. If it is 0, 2, 4, 6 or 8 then the value is clearly even otherwise it is odd. But that's a lot of processing for a computer because all values are encoded in binary, not decimal. To do it the human way you would need to divide the binary value by decimal 10 and then examine the remainder to determine the final digit. Then you have to compare that digit with 0, 2, 4, 6 and 8 to decide if it is even or not. An alternative approach that is more efficient is to divide the binary value by 2 and then check the remainder. If it is 0 then the number is even, otherwise it is odd. But there is an even better method. If the least-significant bit of the binary value is 0 then the number must be even, otherwise it must be odd. That's just simple binary logic but unless you are familiar with binary mathematics it may not be entirely obvious.