# Binary numbers

## Introduction

Binary number system was invented by Gottfried Leibniz. As the word is prefixed with ‘Bi’ which is a Latin word and means ‘two’ in English. This brings us to the first two digits i.e., 0 and 1 which means that while counting in binary you cannot exceed 1. Infact all the numbers which you represent are made up of only two digits i.e., 0 and 1 which is quite interesting. Check out the binary representation of a decimal number (the numbers used for counting i.e., from 0-9) in binary.

``````Example:
Decimal Number :: 25
Binary Number :: 11001
``````

Note: There is no 2, 3, 4, 5, 6, 7, 8 or 9 in Binary!

## Binary counting

### How do we count using binary?

It is just like counting in decimal except we reach 10 much sooner.

Binary Explanation
0 Start at 0
1 Then 1
??? But then there is no symbol for 2 … what to do?

### Well how do we count in Decimal?

Decimal Explanation
0 Start at 0
1 Then 1
2-8 Count 1,2,3,4,5,6,7,8
9 This is the last digit in Decimal
10 Start from back at 0 again, but carry 1 on the left

### The same thing is done in Binary …

Binary Explanation
0 Start at 0
1 Then 1
10 Now start back at 0 again, but carry 1 on the left
11 1 more
??? But NOW what … ?

### What happens in Decimal?

Decimal Explanation
99 When you run out of digits, …
100 … start from back at 0 again, but carry 1 on the left

### And that is what is done in Binary …

Binary Explanation
0 Start at 0
1 Then 1
10 Now start back at 0 again, but carry 1 on the left
11 1 more
100 start back at 0 again, and carry one to the number on the left but that number is already at 1 so it also goes back to 0 and 1 is carried to the next position on the left
101
110
111
1000 Start back at 0 again (for all 3 digits), add 1 on the left

## Binary to decimal demonstration

Let’s tell you something more about conversion. Conversion from Binary to Decimal is quite a simple task. All you need to do is begin from the right. Follow the steps below:

• STEP 1 :: Write the decimal value of each digit on top of them respectively. The value which you seek to write is 2^(place value from right) beginning from 0 i.e., 2^0, 2^1, 2^2 …. continuing up to 2^7.
• STEP 2 :: Now, multiply each digit of binary number with its value.
• STEP 3 :: Add ‘em all.
• STEP 4 :: Result is ready :)

Note: If the number is large, increase bits of the binary number on the left. Keep in mind that it’s value will increase subsequently.

``````Example ::
Decimal number :: 25
You can convert the 1st, 4th, and the 5th digit from the right by tapping on it to convert from 0 to 1.
Further, the respective binary digit is multiplied with the value present on top of each digit. Now add.
In this Case ::
1x16 + 1x8 + 0x4 + 0x2 + 1x1 = 25 which is the decimal equivalent of the binary number 11001
``````

## Signed and unsigned numbers

Currently, we have just looked at unsigned numbers - they can only be positive, as there is no sign. However, sometimes we need to work with negative numbers too. To do this, we add a sign bit on the far left of the binary number, which indicates whether the number is positive (`0`) or negative(`1`).

For example, the number `10000011` would be `131` if the number is unsigned, but if the number is signed, the actual representation would be `-3`

• The first bit `1` represents that the number is negative
• The remaining bits `0000011` represent the actual number, `3`

The downside to using a signed number is that it removes one bit from the actual number representation, halving the maximum value.

• The minimum and maximum values for an `unsigned 8-bit` number would be `0` to `2^8-1` (`0` to `255`)
• The minimum and maximum values for a `signed 8-bit` number would be `-2^7-1` to `2^7-1` (`-127` to `127`)

## Use the Simulator below to get the decimal equivalent of a binary number

Click on the '0' to change it to '1' and vice-versa

Binary
128
0
64
0
32
0
16
0
8
0
4
0
2
0
1
0
Decimal
0
1. Is `0110103` a binary number?
1. No
• Yes
2. What is `10101` as a decimal number?
1. 21
• 10101
• 25
• 1000