Number System

Numbers and their representations

Imagine a herder in ancient times trying to count their sheep. They only have a certain number of stones—12:

However, they can only count up to three and arrange the total into groups of three:

The picture above is a representation (a kind of notation) of the number of stones. We have one group that is made up of three groups of three stones each and a separate group that has three stones. If they could count up to ten, we would see a different representation of the same number of stones—we would have one group of ten stones and another group of two stones.

There are many kinds of representations of numbers:

• Decimal representation
• Ternary representation
• Binary representation
• Hexadecimal representation

Decimal representation (base ten)

Let’s imagine our shepherd can count to ten, not three. They would then organize the same number of stones in an entirely different way. They would have one group of ten stones and another group of two stones.

Let’s now see how 12 stones are represented in arithmetic notation if we can count up to ten, with some examples.

Example 1

We have one group of ten numbers plus two:

$$12_{\text{dec}} =1*10+2 \space\space\space\space\space\space or\space\space\space\space\space\space1*10^1 +2*10^0$$

$12_{\text{dec}}$ is $12$ units or we can say $1$ ten plus $2$ units.

Example 2

Here is another exercise with 123 stones. We have a group of ten by ten stones, another group of two sets, each comprising ten stones, and the last group of three stones:

$$123_{\text{dec}} = 1 * 10*10 + 2 * 10 + 3\space\space\space or\space\space\space\ 1 * 10^2 + 2 * 10^1 + 3 * 10^0$$

$123_{\text{dec}}$ is $123$ units, or $12$ tens plus $3$ units, or we can say $1$ hundred plus $2$ tens and $3$ units.

Note: $n$ represents the number of digits that need to convert, so we find the summation till $n−1$ in the formula because we started from $0$.

Formulas

We can formalize it in the following summation notation till the number $n-1$:

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