# Binary Number System in Mathematics (Tutorial)

Recall that Binary is also known as Base two, it is the counting of numerals on base two. Binary System is something that we are not supposed to joke about because it’s been used day by day in computer application.

In Binary System, the highest digit in any given integer is “1”.

### Examples of numbers in base two are:

1, 10, 11, 100, 111, 10010, 110011, 101, 1000, 101111 and so on, provided that its highest digit is 1.

Students can see from the above numbers in base two, the highest digit is 1. From this example we believe you can identify numbers in base two anywhere they are being written.

## OTHER NUMBER BASES

In other number bases, the counting still follows the same method used in Binary System calculation. In base two the highest digit in the integer is 1, but in other bases the highest digit of the integer must be at least 1 lesser than its base.

### Examples of integers in other number bases:

The respective integers can be formed from the illustration given on NUMBER SYSTEM of the listed digits irrespective of any one that comes first, second or last and a digit can also be repeated many times in that same integer;

For base three : 0, 1, 2

The integer can be written as 12three, 22three, 21three, 102three, 120three, 201three, 112three, 222221three and so on. Provided that the highest digit of that particular integer is not greater than 2.

In general, one can be able to identify the range at which any integer’s base is likely to fall in, on looking closely at its make-up (the digits that make the integer)

Note that all those mentioned integers above such as 12, 22, 21, 102 etc. can also be in base 4, base 5, base 6, base 7, base 8, base 9 and base 10. Reason is because the highest digit in those integers is “2” and it is lesser than all those mentioned bases. This is applicable to every other number bases.

## CONVERSION OF NUMBER BASES.

This is simply a method of changing a number base to another.

### CONVERSION FROM BASE TEN TO ANY OTHER BASE.

In this conversion, the number in base ten is to be divided consecutively by the base required. When dividing, the remainders are being recorded beside every quotient gotten at each time of division, which will later be collected from the last to the first after the division is completed, the result gotten after the whole division is the remainder collected from down to top in its REQUIRED BASE.

Examples:

#### Q1. Convert 232ten to a number in base seven.

Solution:

First step is to divide the given integer/number in base ten with the required base till it has a zero quotient and also be recording the remainders at the sides of its respective quotients.

The next step now is to collect the REMAINDERS from the last to the first.

So, 232 base ten = 451 base seven.

Solution:

#### Q3. Express 232ten to a number in base five.

Solution:

Thus, 232 base ten = 1412 base five.

#### Q4. Convert 7ten to a number in base two.

Solution:

7 base ten = 111 base two.

We believe from those examples given with the steps, the students should be able to solve any problems that come their way.