Calculating the Values of Numbers in Non-Base10 Number Systems¶
Here's a table representing how the different number systems
increment:
| Binary₂ | Octal₈ | Decimal₁₀ | Hexadecimal₁₆ |
|---|---|---|---|
0000 |
0 |
0 |
0 |
0001 |
1 |
1 |
1 |
0010 |
2 |
2 |
2 |
0011 |
3 |
3 |
3 |
0100 |
4 |
4 |
4 |
0101 |
5 |
5 |
5 |
0110 |
6 |
6 |
6 |
0111 |
7 |
7 |
7 |
1000 |
10 |
8 |
8 |
1001 |
11 |
9 |
9 |
1010 |
12 |
10 |
A |
1011 |
13 |
11 |
B |
1100 |
14 |
12 |
C |
1101 |
15 |
13 |
D |
1110 |
16 |
14 |
E |
1111 |
17 |
15 |
F |
Calculating in Any Number System¶
When we want to calculate a binary (base 2 ₂), octal (base 8 ₈),
or hexadecimal (base 16 ₁₆) number to a human-readable format,
we calculate it into a decimal number (base 10 ₁₀).
This is the number system we know, understand, and use in our everyday lives.
Formula¶
The numbers' positions are indexed starting at 0 from right to left.
For example, in the number 1234, 4 would have position 0, and 1 would have position 3
Any non-zero number raised to the power of 0 is equal to 1.
The formula is as follows:
# Binary
base = 2
value = 10101
( (value * base^4) + (value * base^3) + (value * base^2) + (value * base^1) + (value * base^0) )
# 10101₂ = 21₁₀
Binary (Base 2)¶
Binary is base 2.
This is because it only has 2 numbers, 0 and 1.
To calculate the value of the binary number 10101₂ (₂ means "base 2"):
# Number: 10101₂
( (1 * 2^4) + (0 * 2^3) + (1 * 2^2) + (0 * 2^1) + (1 * 2^0))₁₀
( (16) + (0) + (4) + (0) + (1) )₁₀
10101₂ = 21₁₀
Hexadecimal¶
Hexadecimal is base 16 (it has 16 "states of nature").
It has 16 numbers, 0 through 9 and A through F.
| Hexadecimal | Binary |
|---|---|
0 |
0000 |
1 |
0001 |
2 |
0010 |
3 |
0011 |
4 |
0100 |
5 |
0101 |
6 |
0110 |
7 |
0111 |
8 |
1000 |
9 |
1001 |
A |
1010 |
B |
1011 |
C |
1100 |
D |
1101 |
E |
1110 |
F |
1111 |
Converting values to hexadecimal is the same as converting them to binary, except you'll be using base16 instead of base2.
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