## Fundamental Basics: Numerical Systems

There’s an old joke: *“There are only 10 types of people in the world: those who understand binary, and those who don’t.”* If you don’t get it now, hopefully you will after this post.

For most everyday tasks, we’ve been accustomed to using base-10, or “decimal” numerical system. Its what we’re taught from grade school and makes the most logical sense for humans. A quick recap on integer base-10 goes back to “ones”, “tens”, “hundreds”, and we say digits go from 0-9.

Let’s take a sample number 325. We say 325 is comprised of (3*100) + (2*10) + (5*1):

|--100's ||---10's |||----1's 325

Or, an equivalent way of notation is:

|--10^2 ||---10^1 |||----10^0 325

*..in case you’ve forgotten algebra, I’ve linked Google Calculator above.* For each number column, the digit represents that many 10^x. Like, 3 in the “hundreds column” is so because 10^2 is 100, and there’s 3 of them.

It’s called base-10, because the base number is 10.

In Binary, the base number is 2, a.k.a. base-2. In base-2, the digits go from 0-1, and each column is 2^x. So, if we take binary “1011”:

|---2^3 ||----2^2 |||-----2^1 ||||------2^0 1011

Reading the binary digits left to right, there’s 1 of (2^3) + 0 of (2^2) +1 of (2^1) + 1 of (2^0), OR (1*8) + (0*4) + (1*2) + (1*1) = 11 decimal.

While human brains process and store numbers in base-10, computer systems store data in binary, hence our discussion as a fundamental concept for software and embedded systems.

In nomenclature, each binary digit is called a “bit”. (binary 1011 is a 4 bit number). Data variables and CPU architectures are often referred to as number of bits. A 8-bit processor indicates that it natively processes data in 8-bit wide registers. A 16 bit data variable means it can hold up to 16 bits for that variable.

|-2^15 ||--2^14 || ... || |--2^1 || ||---2^0 1010 1100 0011 1011

Writing out just 16 bits already starts to look like writing out credit card numbers. Imagine how long and tedious it gets to write out/read 32 bits, 64…

Luckily, there’s another numerical format: Hexadecimal, or “Hex”. Hex is base-16. There’s 16 digits. The first ten digits are 0-9. To avoid ambiguity, the eleventh through sixteenth digits are denoted A-F, so while binary digits go [0, 1] and decimal digits go [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], hex digits go [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F].

Perhaps a table will help:

|decimal | binary | hex | |--------|--------|-----| | 0 | 0000 | 0 | | 1 | 0001 | 1 | | 2 | 0010 | 2 | | 3 | 0011 | 3 | | 4 | 0100 | 4 | | 5 | 0101 | 5 | | 6 | 0110 | 6 | | 7 | 0111 | 7 | | 8 | 1000 | 8 | | 9 | 1001 | 9 | | 10 | 1010 | A | | 11 | 1011 | B | | 12 | 1100 | C | | 13 | 1101 | D | | 14 | 1110 | E | | 15 | 1111 | F | |--------|--------|-----|

From the table, each hex digit can be represented by a 4-bit binary value. That’s convenient in that you can group a binary representation into 4-bits and convert it to hex:

binary: 1010 1100 0011 1011 hex: A C 3 B

The binary value 1010110000111011 (commonly written as **0b**1010110000111011 or 1010110000111011**b**) is equivalent to hex value AC3B (commonly written as **0x**AC3B, or AC3B**h)**

To convert to decimal, the value 0xAC3B can be viewed as:

|---16^3 ||----16^2 |||-----16^1 ||||------16^0 AC3B

where the number has

("A" *16^3) + ("C" *16^2) + (3 *16^1) + ("B" *16^0), or (10 *4096) + (12 * 256) + (3 * 16) + (11 * 1) == 44091 dec.

As with base-10 math, there are calculators available that aid in converting numerical representation.

On Windows, the “calc.exe” in programmer mode (View->Programmer) is handy:

You can also try the Google:

As a building block in software and embedded systems development, having a working knowledge of these three numerical systems are a prerequisite. There is another format, Octal, that’s base-8, that is usually also introduced in Computer Science. I think after going through binary and hex it isn’t to hard to figure out.