In mathematics there are different number systems that use symbols, digits, elements or figures that represent all the numbers. Among them are: **decimal system**, **binary**, **hexadecimal** and the **octal**.

**Decimal system**

It is the one that we all know and is the most used, it consists of 10 digits or elements. It is also known as the base 10 system, because we use powers of 10 to represent the numbers.

**Example**:

814 → 8 × 10^{2} + 1 × 10^{1} + 4 × 10^{0}

800 + 10 + 4 = 814

**Binary system**

The binary system is made up of two digits or elements 0 and 1. It is also known as base system 2, since they use powers of two to represent the numbers.

**Example 1**: Transform 1101 to decimal system.

1101 → 1 × 2^{3 }+ 1 × 2^{2 }+ 0 × 2^{1 }+ 1 × 2^{0}

In decimal form we would say:

8 + 4 + 1 = 13 = 1101

In the binary system 1101 represents 13 in the decimal system.

**Example 2**: Transform the decimal number 47_{10 }to binary:

47_{10 }→ 101111_{2}

We must divide the number in decimal system by 2 until it can no longer be divided and the remainders of these divisions will be the digits of our binary number, in the direction of the last division towards the first:

47 ÷ 2 = 23; residue 1

23 ÷ 2 = 11; residue 1

11 ÷ 2 = 5; residue 1

5 ÷ 2 = 2; residue 1

2 ÷ 2 = 1; residue 0

1 ÷ 1 = 1, the latter is taken as the remainder

47 in binary numbers is 101111.

**Hexadecimal system**

It is made up of a group of alpha numerical signs, ranging from 0 to 9 and from letter A to F, where each letter corresponds to a different number. Following the sequence of the numbers the letters would be A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. The base is 16 since this is the one used to represent the numbers.

**Example**:

8DF → 8 × 16^{2 }+ D × 16^{1 }+ F × 10^{0}

8DF → 8 × 16^{2 }+ 13 × 16^{1 }+ 15 × 16^{0}

8DF → 2048 + 208 + 15

8DF_{16 }→ 2271_{10}

We have transformed 8DF to the decimal system, 8DF = 2271.

**Octal system**

It is made up of 8 numbers ranging from 0 to 7, the base used is base 8, since powers of 8 are used to write them.

**Example**:

347_{8 }→ 3 × 8^{2 }+ 4 × 8^{1 }+ 7 × 8^{0}

347_{8 }→ 192 + 32 + 7

347_{8} → 231_{10}

We have transformed the number in octal basis to decimal, therefore, 347 in octal is equal to 231 in decimal base.