# Arithmetic Operations in other bases

a and b are numbers with or without decimals written in the same base n between 2 and 62, for example in binary, decimal, hexadecimal or octal numeral system.

## How to use this calculator ?

This calculator computes arithmetic operations ie addition, subtraction, multiplication and division of two numbers written in the same base (radix). It covers decimal (10), binary (2), hexadecimal (16), octal (8) bases as well as all bases from 2 to 62.

**Used digits**

For bases 2 to 10, we use arabic digits 0123...789.

for bases 10 to 36, we use symbols ABCD...XYZ (uppercase letters) to represent numbers 11,12,13...34,35,36. For these bases (10 to 36), if you input numbers with lower case letters then we automatically convert them to uppercase letters.

For bases 36 to 62, we use lower case letter (abcdef...xyz). For example, base 62 uses digits : 0123456789 ABCDEFGHIJKLMNOPQRSTUVWXYZ abcdefghijklmnopqrstuvwxyz that is a total of 62 symbols. Note that for these bases, letter cases are important. For example, A=11 but a=37.

## Numeration systems

## Arithmetic operations in bases

**Binary addition**

The binary system uses digits 0 and 1 to represent numbers. Here's the binary addition table,

0 + 0 = 0

1 + 0 = 1

0 + 1 = 1

1 + 1 = 10 (which is 0 carry 1)

Example : let's calculate 101 + 111

Carry: | 1 | 1 | 1 | |

1 | 0 | 1 | ||

+ | 1 | 1 | 1 | |

1 | 1 | 0 | 0 |

**Binary subtraction**

The binary subtraction table is as follows,

0 - 0 = 0

0 - 1 = -1 (1 borrow 1)

1 - 0 = 1

1 - 1 = 0

Let's calculate 110 - 101

Borrow: | 1 | |||

1 | 1 | 0 | ||

- | 1 | 0 | 1 | |

0 | 0 | 1 |

**Binay multiplication table**

In binary we have,

0 × 0 = 0

0 × 1 = 0

1 × 0 = 0

1 × 1 = 1

Let's calculate 111 × 110

1 | 1 | 1 | |||

× | 1 | 1 | 0 | ||

0 | |||||

1 | 1 | 1 | |||

1 | 1 | 1 | |||

1 | 0 | 1 | 0 | 1 | 0 |