Lesson 06: Binary Numbers _ Part 01 Expressing decimal number system in the binary number system
As an example, let us consider the number 3 725. According to what we have learnt in previous grades,
5 denotes the number of 1s (that is, the number of 100 s), 2 denotes the number of 10s (that is, the number of 101 s), 7 denotes the number of 100s (that is, the number of 102 s), 3 denotes the number of 1000s (that is, the number of 103 s)
This can be represented the below image.
Note : 100⁰ = 1. Similarly, any nonzero base raised to the power zero is always equal to one. Accordingly, 20⁰ = 1.
Chronologically, an abacus of a binary number system can be used to convert a given number of decimal number system in to a number of a binary number system.
To understand the method of writing numbers in base two, let us first consider the base ten number 13 as an example. Let us see how we can write 13 as a sum of powers of two.
The first few powers of two are; 1, 2, 4 and 8. Using these numbers which are powers of two, we can write,
13 = 8 + 4 + 1
13 = 23 + 22 + 20
13 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20
(Note that the maximum number of colored balls can be represented on a binary number system abacus is 1 whereas the minimum is 0)
In addition, there are other methods to obtain the binary number.
We need to continue dividing the quotient by 2 and noting down the remainder, until we get 0 as the quotient and 1 as the remainder. The complete division is shown below.
The decimal number 32 can be converted to binary number system as follows:
1. Divide the number by 2.
2. Write down the remainder.
3. Divide the result of the division by 2 and repeat step 2.
4. Continue until the result of the division is 0.
Here is an example of converting 32 to binary:
32 / 2 = 16, remainder 0
16 / 2 = 8, remainder 0
8 / 2 = 4, remainder 0
4 / 2 = 2, remainder 0
2 / 2 = 1, remainder 0
1 / 2 = 0, remainder 1
Reading the remainders from bottom to top, we get 100000. So, 32 in decimal is equal to 100000 in binary.
The decimal number 12345 can be converted to binary number system as follows:
1. Divide the number by 2.
2. Write down the remainder.
3. Divide the result of the division by 2 and repeat step 2.
4. Continue until the result of the division is 0.
Here is an example of converting 12345 to binary:
12345 / 2 = 6172, remainder 1
6172 / 2 = 3086, remainder 0
3086 / 2 = 1543, remainder 0
1543 / 2 = 771, remainder 1
771 / 2 = 385, remainder 1
385 / 2 = 192, remainder 1
192 / 2 = 96, remainder 0
96 / 2 = 48, remainder 0
48 / 2 = 24, remainder 0
24 / 2 = 12, remainder 0
12 / 2 = 6, remainder 0
6 / 2 = 3, remainder 0
3 / 2 = 1, remainder 1
1 / 2 = 0, remainder 1
Reading the remainders from bottom to top, we get 11000000111001. So, 12345 in decimal is equal to 11000000111001 in binary.
As a follow-up to this discussion, I would like you to work on the conversion in the below given figure. This is an important skill that will help you understand mathematical concepts and solve problems more efficiently. Please review the examples and practice questions and try to find the convert the decimal numbers (base ten numbers) given below into binary numbers (base two numbers).
