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- Binary Number System – Lucid Explanation of Conversion From and to Decimal Number System – Examples
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## Binary Number System – Lucid Explanation of Conversion From and to Decimal Number System – Examples

The base-10 number system or decimal number system is the most popular system used by people around the world.

But computers only work with two symbols internally because digital electronic circuits using logic gates are simple.

Thus, digital computers are based on the base-2 number system or binary number system.

It is used to perform integer arithmetic in almost all digital computers.

The two basic symbols or numbers used in the binary number system are 0 (called zero) and 1 (called one).

We are already familiar with these symbols or numbers in the decimal number system.

We will learn to write numbers using the binary number system.

This system is analogous to the decimal system in following the place value rule.

There, the value of the place becomes ten times as we move one place to the left, and here it becomes double.

**Place value rules in the binary system:**

The right extreme place value is one (1), or unity.

The value of a place increases as it moves to the left.

The value of a place doubles when we move one place to the left.

So the value of the second place from the right is two times one and equals two.

The value of the third place from the right is two times two and equals four.

The value of the fourth place from the right is two times four and equals eight.

The place value, fifth from the right, is two times eight and equals sixteen.

So the next place values are thirty-two, sixty-four, one hundred and twenty-eight, and so on.

**I Converting two base numbers to base ten numbers:**

The following examples make the process clear.

*Example I(1):*

Find the value of the binary number 1001 in the decimal number system.

*Solution:*

Given in binary

the units place (extreme right place) is 1.

The two digit (second digit from the right) is 0.

The fourth place (third place from the right) is 0.

Eighth (fourth place from the right) has 1.

The value of the given binary number (1001) in the decimal number system

= 1 ones + 0 twos + 0 fours + 1 eight

= 1 + 0 + 0 + 8 = 9. Ans.

*Example I(2):*

Write the binary number 10010 in decimal.

*Solution: *

Binary number: 0 1 0 0 1

Place value: 1 2 4 8 16

The binary number 10010 in the decimal number system

= 0 (1) + 1 (2) + 0 (4) + 0 (8) + 1 (16)

= 0 + 2 + 0 + 0 + 16

= 18. Ans.

*Example I(3): *

Write the binary number 1110011 in decimal.

*Solution: *

Binary number: 1 1 0 0 1 1 1

Place value: 1 2 4 8 16 32 64

The binary number 1110011 in the decimal number system

= 1 (1) + 1 (2) + 0 (4) + 0 (8) + 1 (16) + 1 (32) + 1 (64)

= 1 + 2 + 0 + 0 + 16 + 32 + 64

= 115. Ans.

**II Conversion of base ten numbers to base two numbers:**

We use the sharing method.

We successively divide by 2 and take REMAINDER 0 or 1 in successive places, starting from the ones place.

We continue the process until the quotient is 0.

The following examples make the process clear.

*Example II(1): *

Write the decimal number 36 in the binary number system.

*Solution: *

2 | 36

——

2 | 18 – 0 unit place

——

2 | 9 – 0 Double

——

2 | 4:1 Fourth place

——

2 | 2:0 Round of 16

——

2 | 1:0 Round of 16 place

——

# | 0 – 1 Thirty-two places

In the presentation above

the first column has two that we divide by.

The second column is the quotient obtained when dividing by 2.

# indicates the end of the operation if the quotient is 0.

The third column (after ‘-‘) is the resulting remainder (0 or 1), which is a number taken from consecutive places starting from the units place.

So the decimal number, 36 = 100100 in binary.

*Example II(2): *

Write the decimal number 101 in the binary number system.

*Solution: *

2 | 101

——-

2 | 50 – 1 unit place

——-

2 | 25 – 0 Double

——-

2 | 12 – 1 Place of four

——-

2 | 6:0 Round of 16

——-

2 | 3:0 Round of 16 place

——-

2 | 1:1 Thirty-two places

——-

# | 0 – 1 Sixty-fourth place

So, a decimal number, 101 = 1100101 in binary.

*Example II(3):*

Write the decimal number, 1227, in binary.

*Solution: *

2 | 1227

——–

2 | 613 – 1 unit place

——–

2 | 306 – 1 Place of two

——–

2 | 153 – 0 Place of four

——–

2 | 76 – 1 Eighth place

——–

2 | 38 – 0 Round of 16

——–

2 | 19 – 0 Thirty second place

——–

2 | 9:1 Sixty-fourth place

——–

2 | 4 – 1 One hundred and twenty eight places

——–

2 | 2 – 0 Two hundred and fifty six place

——–

2 | 1 – 0 Five hundred and twelve place

——–

# | 0 – 1 One thousand and twenty four places

So, the decimal number, 1227 = 10011001011 in binary.

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