eolas/zk/Binary_number_system.md

---
title: Binary number system
categories:
  - Mathematics
  - Computer Architecture
tags: [binary, number-systems]
---

# Binary number system

## Decimal (denary) number system

Binary is a **positional number system**, just like the decimal number system.
This means that the value of an individual digit is conferred by its position
relative to other digits. Another way of expressing this is to say that number
systems work on the basis of **place value**.

In the decimal system the columns increase by **powers of 10**. This is because
there are ten total integers in the system:

$1, 2, 3, 4, 5, 6, 7, 8, 9$

When we have completed all the possible intervals between $0$ and $9$, we start
again in a new column.

The principle of counting in decimal:

```
0009
0010
0011
0012
0013
...
0019
0020
0021
```

Thus each column is ten times larger than the previous column:

- Ten ($10^1$) is ten times larger than one ($10^0$)
- A hundred ($10^2$) is ten times larger than ten ($10^1$)

We use this knowledge of the exponents of the base of 10 to read integers that
contain multiple digits (i.e. any number greater than 9).

Thus 264 is the sum of

$$[4 \cdot (10^0)] + [6 \cdot (10^1)] + [2 \cdot 10^2]  $$

## Binary number system

In the binary number system, the columns increase by powers of two. This is
because there are only two integers: 0 and 1. As a result, you are required to
begin a new column every time you complete an interval from 0 to 1.

So instead of:

$$ 10^0, 10^1, 10^2, 10^3 ... (1, 10, 100, 1000) $$

You have:

$$ 2^0, 2^1, 2^2, 2^3, 2^4... (0, 2, 4, 8, 16) $$

When counting in binary, we put zeros as placeholders in the columns we have not
yet filled. This helps to indicate when we need to begin a new column. Thus the
counting sequence:

$$ 1, 2, 3, 4 $$

is equal to:

$$ 0001, 0010, 0011, 0100 $$

Counting in binary:

```
000000
000001
000010
000011
000100
000101
000111
```

## Binary prefix

To distinguish numbers in binary from decimal or
[hexadecimal](/Electronics_and_Hardware/Binary/Hexadecimal_number_system.md)
numbers, it is common to use the prefix `0b`. Thus, e.g, `0b110` for decimal
`6`.

## Converting decimal to binary

Let's convert 6 into binary:

If we have before us the binary place values ($1, 2, 4, 8$). We know that 6 is
equal to: **1 in the two column and 1 in the 4 column → 110**

More clearly:

![](../../img/Pasted_image_20220319135558.png)

And for comparison:

![](../../img/Pasted_image_20220319135805.png)

Or we can express the binary as:

$$ (1 _ 2) + (1 _ 4) $$

Or more concisely as:

$$ 2^1 + 2^2 $$

### Another example

Let's convert 23 into binary:

![](../../img/Pasted_image_20220319135823.png)