Number Systems: Introductions

Number systems allow us to count various things with a lot more ease than before. The most common one is Decimal or Base 10, this is most likely because humans generally have 10 fingers, 5 on both hands.

Base 10

Base 10 uses 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

This combined with places allows us to use large numbers such as 231.

If we split 231 down we get:

• 2 x 100s
• 3 x 10s
• 1 x 1s

It is interesting to note that each place increases a multiple of 10, but also can be explained exponentially so:

• 1 = x10^0
• 10 = x10^1
• 100 = x10^2
• 1000 = x10^3

Base 2 - Binary

Base 2 uses 2 symbols: 0, 1

These usually represents on and off or true and false.

Instead of each place being represented by x10^n, base 2 places are represented by x2^n.

So the first 8 places of binary are:

• 1 = x2^0
• 2 = x2^1
• 4 = x2^2
• 8 = x2^3
• 16 = x2^4
• 32 = x2^5
• 64 = x2^6
• 128 = x2^7

We can now convert 231 base 10 into base 2

128 64 32 16 8 4 2 1

---------------------------

    1   1   1   0 0 1 1 1

So in binary 231 is represented as 11100111 and we can check this by adding each place that has a value of 1: 128 + 64 + 32 + 4 + 2 + 1 = 231

Base 16 - Hexadecimal

Base 16 uses 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A (10), B(11), C(12), D(13), E(14), F (15)

As with base 2, base 16 places are represented by something different to x10^n. The places are represented by x16^n. So the first 4 places of base 16 are:

• 1 = x16^0
• 16 = x16^1
• 256 = x16^2
• 4096 = x16^3

We can now figure out how to write 231 in base 16

4096 256 16 1

-------------------

                  E 7

So in base 16, 231 can be represented as E7 and we can check this by adding the each places total together.

E or 14 x 16 = 224

7 x 1 = 7. 

224 + 7 = 231