We’ve talked about:
- How to represent numbers in binary.
- How to convert to and from arbitrary bases, most relevantly hexadecimal to compactly display binary data.
That’s neat, but it raises a more concrete question: How do computers really store numbers?
Bytes and Words
The smallest unit that modern computers typically store is a byte, or a block of 8 bits.
If we want to store a non-negative integer, one byte lets us store a value between 0 and (2^8)-1 = 255.
Internally, one of the most common things computers use numbers for is as memory addresses. Going back to our diagram of a computer, we’ve got a CPU and RAM. We store data in RAM, and can do operations on the CPU.
So if we want to add two numbers:
- There’s an adder circuit in the CPU.
- But first the two numbers need to be loaded from RAM into registers in the CPU.
- To load the numbers from memory we need addresses, which need to fit in registers too.
If we have 8-bit registers (a “word” is 8 bits), then we can store 8-bit memory addresses. Typically each address in memory stores one byte of data, so an 8 bit computer can have 256 bytes of RAM.
You can’t do much with only a couple hundred bytes, so the first broadly adopted computers were 16-bit machines. A register or memory address (“word”) was two bytes, and the maximum RAM they could handle was 2^16 = 65536 bytes. This is the 80’s.
In the 90’s, new home computers had 32-bit words and could address 2^32 = 4GB of RAM (although actual computers from 1995 shipped with maybe 8MB).
And in the 00’s, home computers moved to 64-bit words. That’s 2^64 = 16 EB of addressable RAM. The 2010’s and 2020’s so far haven’t maxed that out, even for heavy duty servers.
But even on a 64-bit system, you sometimes don’t want to spend 8 bytes for every integer. So we commonly see 8, 16, 32, and 64-bit integers.
Negative Numbers
Plan A: Sign bit.
Problems:
- There’s two zeros.
- You need more complicated adders.
Crazy trick that solves both problems: Two’s complement.
- To make a number negative: Flip all bits, add 1.
- Show this for some 4-bit numbers.
- Add -1 + 5.
Weird thing: Extra negative range.
Show negative numbers in hex.
Floating Point Numbers
binary32: 24 bit significand, 7 bit exponent, 1 sign bit
https://en.wikipedia.org/wiki/Minifloat
Let’s look at the 8-bit 1.4.3 table.
