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

But, computers work internally with only two symbols, because of the straightforward 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 in almost all digital computers.

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

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

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

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

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

**Place value rules in Binary number System :**

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

Value of the place increases as it moves to the left.

Value of the place becomes two times, as we move one place to the left.

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

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

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

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

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

**I Conversion of base-two numerals into base-ten numerals :**

The following examples will make the process clear.

*Example I(1) :*

Find the value of the binary numeral 1001, in the Decimal number system.

*Solution :*

In the given Binary Numeral,

units’ place (Extreme right place) has 1.

Twos’ place (Second place from right) has 0.

Fours’ place (Third place from right) has 0.

Eights’ place (Fourth place from right) has 1.

The value of the given binary numeral (1001) in Decimal Number System

= 1 ones + 0 twos + 0 fours + 1 eights

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

*Example I(2) :*

Write the binary numeral 10010, in the Decimal number system.

*Solution : *

Binary Digit : 0 1 0 0 1

Place Value : 1 2 4 8 16

The Binary numeral 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 numeral 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 numeral 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 numerals into base-two numerals :**

We use division method.

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

We continue the process till the quotient is 0.

The following examples will make the process clear.

*Example II(1) : *

Write the Decimal Numeral 36, in the Binary Number System.

*Solution : *

2 | 36

——

2 | 18 – 0 Units’ place

——

2 | 9 – 0 Twos’ place

——

2 | 4 – 1 Fours’ place

——

2 | 2 – 0 Eights’ place

——

2 | 1 – 0 Sixteens’ place

——

# | 0 – 1 Thirty two’s place

In the above presentation,

first column is having twos with which we are dividing.

Second column is the quotient obtained by dividing with 2.

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

Third column (after ‘-‘) is the remainder (0 or 1) obtained which is the digit taken in successive places starting from units’ place.

Thus, The Decimal Numeral, 36 = 100100 in Binary Number System.

*Example II(2) : *

Write the Decimal Numeral 101, in the Binary Number System.

*Solution : *

2 | 101

——-

2 | 50 – 1 Units’ place

——-

2 | 25 – 0 Twos’ place

——-

2 | 12 – 1 Fours’ place

——-

2 | 6 – 0 Eights’ place

——-

2 | 3 – 0 Sixteens’ place

——-

2 | 1 – 1 Thirty two’s place

——-

# | 0 – 1 Sixty four’s place

Thus, The Decimal Numeral, 101 = 1100101 in Binary Number System.

*Example II(3) :*

Write the Decimal Numeral, 1227 in the Binary Number System.

*Solution : *

2 | 1227

——–

2 | 613 – 1 units’ place

——–

2 | 306 – 1 Twos’ place

——–

2 | 153 – 0 Fours’ place

——–

2 | 76 – 1 Eights’ place

——–

2 | 38 – 0 Sixteens’ place

——–

2 | 19 – 0 Thirty twos’ place

——–

2 | 9 – 1 Sixty Fours’ place

——–

2 | 4 – 1 One hundred twenty eights’ place

——–

2 | 2 – 0 Two hundred fifty six’ place

——–

2 | 1 – 0 Five Hundred twelve’s place

——–

# | 0 – 1 One thousand twenty four’s place

Thus, The Decimal Numeral, 1227 = 10011001011 in Binary Number System.