Some of my readers who have been teaching Advanced Placement Computer Science (APCS) will remember the BigInt case study. It was a case study involving mathematics using large (very large) integers. As released by the College Board it supported adding, subtracting, and multiplying large integers. You will notice that division was not included. In fact, asking students to implement division was part of the exam.
BigInt introduced the idea that multiplication was actually multiple addition. By extension, students were to figure out that division is multiple subtraction.
Computer science really requires understanding how mathematics works at a deep level. It becomes obvious (one would hope) when trying to understand how Binary, Octal, and Hexadecimal work. We don’t often spend much if any time trying to understand subtraction though.
Recently, on BlueSky I can across a message by Andrew Virnuls linking to a blog post titled Two's Complement and Negative Binary Numbers that explains subtracting by adding negative numbers.
Let me draw the two previous notes in this post together with some history of mine. Back in my university days I worked on a course connecting some test hardware to a computer. The computer was a Digital Equipment PDP-8. Now the 8 was an interesting machine. It didn’t have a hard drive and it was programmed in assembly language entered in Binary. Where as most computers we use today use hexadecimal representation (base 16) the PDP-8 used Octal (base 8). The word size was 12 bits. Not 64, 32, or even 16 – 12.
This word size places some limits and one of those limits was the number of machine language/ Assembly language instructions. There was no multiply, divide or even subtraction instruction. We had to write code to do those things similar to how code was written in BigInt for those operations. We also had to write code to do subtraction. There was an instruction to create the two’s compliment of a number though. That was handy. So we wrote code to find and use the two’s compliment of a number in order to do subtraction.
We used the subtraction routine to implement division. Though to be honest, we tried to avoid having to do multiplication or division in our project to keep performance reasonable.
I think we’re all glad that today’s computers have a lot more layers of abstraction than the PDP-8 had! Of course, and a lot of students do not realize this, most powerful assembly language instructions are actually the result of what is called microcode that works transparently behind the scenes.
We keep moving up the path of abstraction. Hal Berenson addressed this recently in a post called 98% of Developers can’t program a computer which is actually a bit of a success story including how artificial intelligence is helping with higher levels of abstraction.
