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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 humans across the world.

But, computers work internally with only two symbols, due to the simple implementation in digital electronic circuits using logic gates.

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

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

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

We already know these symbols or digits in the decimal number system.

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

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

There the square value becomes ten times, when we move one square to the left, and here it becomes twice.

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

The value of the far right place is one (1) or one.

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

The square value becomes twice, as we move one square to the left.

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

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

The value of the square, fourth from the right is twice four and equals eight.

The value of the square, fifth from the right is twice eight and is equal to sixteen.

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

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

The following examples will make the process clear.

*Example I(1):*

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

*Solution :*

In the given binary digit,

the units square (extreme right square) has 1.

The place of twos (second place from the right) has 0.

Place de Fours (third place from the right) has 0.

The eights place (fourth place from the right) has 1.

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

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

= 1 + 0 + 0 + 8 = 9. Rep.

*Example I(2):*

Write the binary number 10010, in the decimal number system.

*Solution : *

Binary digit: 0 1 0 0 1

Place value: 1 2 4 8 16

The binary digit 10010, in the decimal number system

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

= 0 + 2 + 0 + 0 + 16

= 18. Rep.

*Example I(3): *

Write the binary number 1110011, in the decimal number system.

*Solution : *

Binary digit: 1 1 0 0 1 1 1

Place value: 1 2 4 8 16 32 64

The binary digit 1110011, in the decimal numeral system

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

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

= 115. Rep.

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

We use the division method.

We divide successively by 2 and we take the REMAINDER 0 or 1 by successive places starting from the place of the units.

We continue the process until the quotient is 0.

The following examples will make the process clear.

*Example II(1): *

Write the decimal digit 36, in the binary number system.

*Solution : *

2 | 36

——

2 | 18 – 0 Unit Square

——

2 | 9 – 0 Square of both

——

2 | 4 – 1 Place des Fours

——

2 | 2 – 0 Square of Eight

——

2 | 1 – 0 Sixteen Square

——

# | 0 – 1 Thirty-two place

In the presentation above,

first column has two that we are dividing with.

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

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

The third column (after ‘-‘) is the remainder (0 or 1) obtained which is the figure taken in successive places starting from the place of the units.

So, the decimal digit, 36 = 100100 in the binary number system.

*Example II(2): *

Write the decimal digit 101, in the binary number system.

*Solution : *

2 | 101

——-

2 | 50 – 1 Unit Square

——-

2 | 25 – 0 Square of both

——-

2 | 12 – 1 Place des Fours

——-

2 | 6 – 0 Square of Eight

——-

2 | 3 – 0 Sixteen Square

——-

2 | 1 – 1 Thirty-two place

——-

# | 0 – 1 Sixty-four Square

So, the decimal digit, 101 = 1100101 in the binary number system.

*Example II(3):*

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

*Solution : *

2 | 1227

——–

2 | 613 – Place of 1 units

——–

2 | 306 – 1 Place des Deux

——–

2 | 153 – 0 Place des Fours

——–

2 | 76 – 1 Place of the Eight

——–

2 | 38 – 0 Place des Sixteen

——–

2 | 19 – 0 Thirty-two

——–

2 | 9 – 1 Place des Sixty Fours

——–

2 | 4 – 1 Place one hundred and twenty eight

——–

2 | 2 – 0 Two hundred fifty-six’ place

——–

2 | 1 – 0 Five hundred and twelve place

——–

# | 0 – 1 thousand twenty-four place

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

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