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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 all over the world.

However, computers work internally with only two symbols, because of the direct implementation in digital electronic circuitry using logic gates.

Thus, the base-2 number system or binary number system is the basis for digital computers.

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

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

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

Let’s learn how to write numbers using the binary number system.

This system is similar to the decimal number system that follows the place value rule.

There, the value of the place is tenfold, when we move one place to the left, and here it is doubled.

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

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

Moving to the left increases the location value.

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

Therefore, the value of the second place from the right is equal to two times one and two.

The third place value from the right is equal to two times two and four.

The place value, fourth from the right is equal to two times four and eight.

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

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

**I Convert base-two digits to base-ten numbers:**

The following examples will illustrate the process.

*Example I(1):*

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

*Solution:*

In a given binary digit,

1 is in the units place (extreme place).

The position of two (second position from the right) is 0.

The place of four (third place from the right) is 0.

Eight’s place (fourth place from right) is 1.

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

= 1 one + 0 two + 0 four + 1 eight

= 1 + 0 + 0 + 8 = 9. Answer.

*Example I(2):*

Write the binary number 10010 in decimal number system.

*Solution: *

Binary digit: 0 1 0 0 1

Place value: 1 2 4 8 16

Binary digit 10010 in 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 number system.

*Solution: *

Binary digit: 1 1 0 0 1 1 1

Place value: 1 2 4 8 16 32 64

Binary digit 1110011 in 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 digits to base-two digits:**

We use the division method.

We successively divide by 2 and carry the REMAINDER 0 or 1 from the units place to successive places.

We continue the process until the quotient is 0.

The following examples will illustrate the process.

*Example II(1): *

Write the decimal number 36 in binary number system.

*Solution: *

2 | 36

——

2 | Location of 18 – 0 units

——

2 | 9 – 0 two places

——

2 | 4 – 1 square space

——

2 | 2 – 0 eight place

——

2 | 1 – 0 Sixteenth place

——

# | 0 – 1 thirty-two places

In the presentation above,

The first column has two that we are dividing.

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

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

The third column (after ‘-‘) is the obtained remainder (0 or 1) which is the digit taken in successive positions starting from the unit’s position.

Thus, a decimal number, 36 = 100100 in binary number system.

*Example II(2): *

Write the decimal digit 101 in the binary number system.

*Solution: *

2 | 101

——-

2 | Location of 50 – 1 units

——-

2 | 25 – 0 two places

——-

2 | 12 – 1 square space

——-

2 | 6 – 0 eight place

——-

2 | 3 – 0 sixteenth place

——-

2 | 1-1 thirty second place

——-

# | 0-1 place of sixty-four

Thus, a decimal number, 101 = 1100101 in binary number system.

*Example II(3):*

Write the decimal number, 1227, in the binary number system.

*Solution: *

2 | 1227

——–

2 | 613 – 1 unit location

——–

2 | 306 – 1 two places

——–

2 | 153 – 0 square place

——–

2 | 76 – 1 eight place

——–

2 | 38 – 0 sixteenth place

——–

2 | 19 – 0 thirty second place

——–

2 | 9 – 1 Sixth Square Place

——–

2 | 4 – 1 one hundred and twenty eight places

——–

2 | 2 – 0 two hundred and fifty-six’ places

——–

2 | 1 – 0 five hundred and twelve places

——–

# | 0 – 1 position of one thousand and twenty four

Thus, the decimal number, 1227 = 10011001011 in binary number system.

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