Decimal, Binary, Hex

Decimal versus binary

When we refer to a number such as 532 in base 10, what we actually mean is the number:

(5 * 10^2) + (3 * 10^1) + (2 * 10^0) = 532

Note that in each pair of parentheses, the first number of the product is a single-digit number in the range [0, 9], and the second number is a unique power of 10 (in sorted order from greatest to least).

We can do a similar calculation for numbers in other bases such as binary. For example, 0100 in binary (base 2) is equivalent to:

(0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (0 * 2^0) = 0 + 4 + 0 + 0 = 4 (in decimal)

Since we are working in base 2 now instead of base 10, all of the exponent bases are 2s instead of 10s, and all of the first numbers in each parenthesized product is an integer in the range [0, 1].

However, a number like 0100 might be interpreted as one-hundred in decimal. Thus, we sometimes prefix binary numbers with "0b" (zero b) to differentiate binary numbers from numbers in base 10 representation (so instead of using 0100, we would say that 4 in decimal is equivalent to 0b0100 in binary).

Now let's try a more complicated example: convert 0b10101001 to decimal. We will assume this is unsigned for now (more on representing positive versus negative numbers later).

0b10101001

            = 1 * 2^7 + 0 * 2^6 + 1 * 2^5 + 0 * 2^4 + 1 * 2^3 + 0 * 2^2  + 0 * 2^1 + 1 * 2^0

            = 2^7 + 2^5 + 2^3 + 2^0

            = 128 + 32 + 8 + 1

            = 169

What the hex is going on here?

Binary isn't the only base that is useful. Hexadecimal is the base-16 number system, as opposed to the base 2 system (binary) and base 10 system (decimal). The prefix we use for hexadecimal is "0x". To represent the numbers 0-9, we simply use those digits. To represent 10-15, we use the letters A-F.

A	B	C	D	E	F
10	11	12	13	14	15

Note: Hexadecimal is oftentimes referred to as "hex" for short. Do not confuse this with the rarely-used base 6 representation heximal. That is, if you hear someone talking about a "hex" number, they are almost certainly referring to a number in base 16 rather than a number in base 6.

Calculating the decimal value of a hexadecimal number follows much the same pattern as binary. For an example, we will convert 0xCA1 to decimal:

0xCA1

            = 12 * 16^2 + 10 * 16^1 + 1 * 16^0

            = 3072 + 160 + 1

            = 3233

As you can see, with only 3 digits in hex, we were able to represent a 4-digit number in decimal, which is also a 12-bit number in binary. In fact, hexadecimal is a pretty convenient way of shortening a binary string to make it more readable because each 4-bit nibble of a binary string corresponds to a single digit in hex.

Let's take the binary representation 0b10101001. Suppose we break this into 4-bit chunks, 0b1010 and 0b1001. We can convert each 4-bit chunk into a hex value.

0b1010 = 10 = 0xA

            0b1001 = 9 = 0x9

Thus, we can compress the binary representation 0b10101001 into the hex representation 0xA9. In much the same vein, you can "decompress" from hex to binary by converting each hex digit into a 4-bit binary representation.

Self-tests: Decimal, Binary, Hex

Convert the following to decimal:

`0b1111`	Toggle Solution `15`
`0xCA`	Toggle Solution `202`

Convert the following to 8-bit binary numbers:

`67`	Toggle Solution `0b01000011`
`0xB1`	Toggle Solution `0b10110001`

Convert the following to hex:

`0b01001011`	Toggle Solution `0x4B`
`255`	Toggle Solution `0xFF`

What would be the length of the corresponding binary representation? 0xAF02E9D3

Toggle Solution

32-bits