The everyday number system most people use is the decimal number system.
Decimal is a base-10 system.
That means it uses ten digits:
0 1 2 3 4 5 6 7 8 9
After the digit 9, the system needs another position.
That is why counting continues:
8
9
10
11
12
The value of a digit depends not only on the digit itself but also on its position.
Consider:
4,372
The digits have different place values.
| Position | Place value | Contribution |
|---|---|---|
| thousands | 1,000 | 4 × 1,000 = 4,000 |
| hundreds | 100 | 3 × 100 = 300 |
| tens | 10 | 7 × 10 = 70 |
| ones | 1 | 2 × 1 = 2 |
Add the contributions:
4,000 + 300 + 70 + 2 = 4,372
The position gives each digit its weight.
Starting at the ones position:
1
10
100
1,000
10,000
Each place to the left is worth ten times the place immediately to its right.
These values are powers of 10:
10^0 = 1
10^1 = 10
10^2 = 100
10^3 = 1,000
You do not need advanced exponent work to use place value.
The important pattern is that each position has a defined weight.
The digit 5 does not always mean the quantity five in a larger number.
In:
5
it means five ones.
In:
50
the 5 represents five tens, or fifty.
In:
500
the 5 represents five hundreds.
The symbol is the same.
Its contribution changes because its position changes.
Consider:
507
The 0 in the tens position tells us there are zero tens.
Without that position, the number:
57
means something different.
Zero is therefore more than “nothing.” It can preserve the position of other digits inside a place-value system.
The quantity thirteen can be written in decimal as:
13
Later in this module, you will see that the same quantity can be represented in binary or hexadecimal.
The quantity does not change just because the written symbols change.
That distinction is important in computing.
A number is a value.
A number system is a way of representing that value.
Decimal is familiar, so it gives you a reference point for learning other positional number systems.
When binary feels unfamiliar, the underlying idea is not completely new.
Both decimal and binary use:
The major difference is the base.
Decimal uses powers of 10.
Binary uses powers of 2.
Understanding decimal place value makes that transition much easier.