In today's lesson,i will show you the simplest method when it comes to conversion of number base to another. Counting and measuring things have become part of our everyday lifestyle.In our live ,we always counting in tens daily as part of counting system.
From primary section number system has ten digits: 0,1,2,3,4,5,6,7,8,9. Doe, some people thought its start from 1 " wrong".Beside, these numbers are the once used to form other numbers in asthmatics operation.For example, 3456 is a number gotten by combining some numbers in base ten. We have other mathematical method to change a given number to any base not only in base ten.It can be done to base 2, 3, 4 , 5 .....n.
Example1
Let convert Binary numbers to base 10
Convert the following numbers to base 2.
(a) 10011₂
(b) 10111₂
(c) 11000₂
The above numbers are called "binary numbers" .
Solution
(a) 10011₂
10011₂ = ( 1 x 2)⁴ + ( 0 x 2 )³ + ( 0 x 2 )² + ( 1 x 2 )¹ + ( 1 x 2 )⁰
= 1 x 2⁴ + 0 x 2³ + 0 x 2² + 1 x 2¹ + 1 x 2⁰
= 2⁴ + 2 + 1
= 16 + 2 + 1
= 19₁₀
Wow that was awesome! I decided to brake everything up your understanding.The numbers with srikethrough are none value able numbers.. I mean, 0 x 2² the 2 itself has no value because of the zero ( 0 ).Then also in any of the above operations, 1 x 2º the only value able number there is 1 aside that the 2 with power of zero is nothing, so you keep 1 only and through away 2º .
(b) 10111₂
10111₂ = ( 1 x 2)⁴ + (0 x 2)³ + (1 x 2)² + (1 x 2)¹+ (1 x 2)⁰
= 2⁴ + 2² + 2¹ + 1
= 16 + 4 + 2 + 1
= 23₁₀
(c) 11000₂
11000₂ = (1 x 2)⁴ + (1 x 2)³ + (0 x 2)² + (0 x 2)¹ + (0 x 2)⁰
= 2⁴ + 2³
= 16 + 8
=24ten
The above operation is knowing as, conversion of of numbers in base two to numbers in base ten.
Now let convert binary number that has a decimal point.This operation involved the method of place value.
Example2
Convert the following decimal numbers to base 10 (base ten):
(i) 1101.110
(ii) 101.111
Solution
(i) 1101.110₂
1101.110₂ = ( 1 x 2 )³ + ( 1 x 2 )² + (0 x 2)¹ + ( 1 x 2 )⁰ . (1 x 2 )⁻¹ +(1 x 2 )⁻² + ( 0 x 2 )⁻³
1 1
= 2³ + 2² + 1 + ------- + ------- L.C.M 0f 2 an 4 = 4
2
= 13 ----------
4
Converting Numbers From Base ten to Base 2
Now, we will convert number in base ten to base 2 that involve division of of numbers by using (2) as the common divisor.The concept of this operation is dividing numbers that will give remainders to form numbers in base 2.
Example 3
Change/convert the following to base 2 numbers.
(i) 32
(ii) 51
(iii) 42
To solve the above problems you must used 2 as their common divisor.
(i) 32
32
---------- = 16 rem 0
2
16
---------- = 8 rem 0
2
8
---------- = 4 rem 0
4
4
------------ = 2 rem 0
2
2
------------ = 1 rem 0
2
1
---------- = 0 rem 1
2
therefore, 32 = 100000₂
(ii) 51
51
--------- = 25 rem 1
2
25
------------ = 12 rem 1
2
12
----------- = 6 rem 0
2
6
----------- = 3 rem 0
2
3
------------ = 1 rem 1
2
1
------------ = 0 rem 1
2
Therefore, 51 =110011₂
(iii) 42
42
----------- = 21 rem 0
2
21
---------- = 10 rem 1
2
10
------------- = 5 rem 0
2
5
------------ = 2 rem 1
2
2
---------- = 1 rem 0
2
1
---------- = 0 rem 1
2
Therefore, 42 = 101010₂
Note: (Rem) means remainder also 1 divided by 2 is always 0.5 that 5 is 1.After dividing the numbers,your binary start from the bottom to top.
Example 4
In this one, we will convert some digits to a certain base.Any number cab be change to any given base,so it is you that will know how to manipulate the numbers.
(a) Change this to base five.
241 to base five
2 4 1 = (2 x 5)² + (4 x 5)¹ + (1 x 5)⁰
= 2 x 5² + 4 x 5¹ + 1 x 5⁰
= 2 x 25 + 4 x 5 + 1
= 50 + 20 + 1
= 71₅
We were asked to change the number to base five,and we have placed the subscript down as 5. That is how to convert numbers to their base.If you know there are things you don't still understand please ask me questions.
Class Activities
Change the following numbers to their given base.
(a) 10010 to base ten
(b) 1001101 to base ten
(c) 2467 to base four
(d) 762 to base eight
(e) 110.1110 to base ten
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