HOW TO CONVERT BINARY NUMBERS TO BASE 10 AND OTHER BASES
Historians believe that the binary system was invented by an Indian Mathematician called Pingala. The Mathematician Gottfried Leibnitz then further developed the system by explaining how to use the number system in more details.
The binary number system also called the base 2 number system, has two digits to represent all values, namely 0 and 1. This means the numbers you can see in a binary system are 0's and 1's. Binary numbers are used in computers, representing when digital circuits are either on (1) or off (0).
Binary digits are sometimes called bits which evolved from the first and last two letters of the word "binary digits".
Examples of binary digits are: 1101₂ , 1001₂ , 1111₂...etc. The subscript (₂) means base 2 or base two. It is essential to put the subscript because without it, it's a base 10 digit.
At the end of this post, you should be able to Convert base 10 numbers to base 2 and vice versa.
CONVERTING BASE 10 NUMBERS TO BINARY NUMBERS
NB: Base 10 numbers are natural numbers i.e they are counting numbers e.g 0,1,2,3,4,5,6,7,8,9,...
Example
(a) Convert 73 to a binary number
(b) Convert 01001 to a base 10 number
Solution
(a) The best way and fastest way to convert 73 to base 2 is to continue dividing by 2. So far you divide by 2, the remainders will be only 0's and 1's.
Which is what we want.
Take a look
If you go through the above solution, we divided by 2 (because we're converting to base 2) from 73 which gave us 36 remainder 1, then divided 36 again to get 18 remainder 0, and so on...till we got to 1 divided by 2, which doesn't make sense in binary so we wrote 0 remainder what we tried to divide, which is 1.
Now that you understand how to convert base 10 number to base 2, how are you going to write your answer? Well, stick with me.
Take a look at the solution one more time, look at the right side, immediately after the remainders, you'll see an arrow pointing up. That indicates that you write your answer from bottom to top.
Therefore 73===> base two = 1001001₂
WHEN WILL THE ARROW POINT DOWNWARDS?
Well, the arrow should always point upward, there's no pointing down. It indicates the answer is from bottom to top. Just like when subtracting numbers, you subtract from back to front.
WHAT IF 2 CAN'T DIVIDE A NUMBER?
All numbers can divide all numbers. That is 2 can divide all numbers, both even and odd. When dividing even number, the remainder is 0, when dividing odd number, the remainder will always be 1.
Let's proceed...
You already understand base 10 to base 2, lets try base 2 to base 10. Ofcourse it's not the same way.
CONVERTING BINARY NUMBERS TO BASE 10 NUMBERS
(b) Convert 01001₂ to base 10.
0⁴ 1³ 0² 0¹ 1⁰₂
= (0 x 2⁴) + (1 x 2³) + (0 x 2²) + (0 x 2¹) + (1 x 2⁰)
= (0 x 16) + (1 x 8) + (0 x 4) + (0 x 2) + (1 x 1)
= 0 + 8 + 0 + 0 + 1
= 8 + 1
= 9
Now lets analyse the above solution.
We started by giving every number a representation. Starting from 0 and also from the back. It's always from the back.
0⁴ 1³ 0² 0¹ 1⁰₂
The next step was multiply each number by 2 (which is base 2) and by its representative.
0 1 0 0 1
(0 x 2⁴) + (1 x 2³) + (0 x 2²) + (0 x 2¹) + (1 x 2⁰)
Now let's solve our result.
(0 x 16) + (1 x 8) + (0 x 4) + (0 x 2) + (1 x 1)
Let me reiterate this;
From the above result, we can simply understand that;
2⁴ = 16
2³ = 8
2² = 4
2¹ = 2
2⁰ = 1 •••{that's right, any number raised to the power of 0 is 1}
Back to where we stopped
(0 x 16) + (1 x 8) + (0 x 4) + (0 x 2) + (1 x 1)
= 0 + 8 + 0 + 0 + 1
= 8 + 1
= 9
We have successfully converted 01001₂ to base 10, which gave us 9.
WHY 9 DOESN'T HAVE ANY SUBSCRIPT?
Base 10 doesn't have any subscript because normal counting numbers are in base 10 and no body puts base 10 after counting numbers. You can always put subscript 10, but it's not necessary.
Want to know how counting numbers are base 10 numbers? Let me know in the comment section.
WHY IS CONVERTING FROM BASE 2 TO BASE 10 VOLUMINOUS?
Well, to destroy something is easy, to repair is difficult.
NB: converting doesn't change the value of numbers, it just simply means 01001 in base 2 is also the same as 9 in base 10.
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great post
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