Modulo who?

When programmer and mathematician are talking about modulus or modulo, there is often a confusion what this term means. For programmer modulo means an operator that finds the remainder of division of one number by another, e.g. 5 mod 2 = 1. For mathematician modulo is a congruence relation between two numbers: a and b are said to be congruent modulo n, written a ≡ b (mod n), if their difference a − b is an integer multiple of n.

These two definitions are not equivalent. The former is a special case of the latter: if b mod n = a then a ≡ b (mod n). The inverse is not true in general case. 5 mod 2 = 1, and 1 ≡ 5 (mod 2) because 1 - 5 = -4 is integer multiple of 2. Now 5 ≡ 1 (mod 2) because 5 - 1=4 is evenly divisible by 2, but 1 mod 2 = 1, not 5.

The biggest confusion happens when programmer and mathematician start arguing about Gauss' famous golden theorem where both definitions of modulus can be used.

Tony Hoare: 50 Years with Legacy Code

Q: With your time in the computer science field, I'm guessing that you've seen some common trends, things that have remained consistent throughout time since 1960. What are those trends that have remained consistent and how do you think it will continue to the future?

A: I'm sorry to say it's the mistakes that remained consistent, not the virtues. The problems that afflicted us in the 1960s were the difficulty of predicting performance of large systems, the difficulty of discovering requirements, the difficulty of implementing code that was coherent across large-scale module boundaries. All of these things are still with us. I suppose I could say that even in 1960 living with legacy code was there. Dykstra once said that every day when he comes into work he thanks providence that he is not charged with the responsibility of living with legacy code - that's certainly still with us.

watch the interview