 ## Binary Number System

Subject: Compulsory Maths

#### Overview

A Binary Number is made up of only 0s and 1s. A method of representing numbers that has 2 as its base and uses only the digits 0 and 1. Each successive digit represents a power of 2. For example, 10011 represents (1 × 24) + (0 × 23) + (0 × 22) + (1 × 21) + (1 × 20), or 16 + 0 + 0 + 2 + 1, or 19.

##### Binary Number System

It is necessary to review the decimal number system at first to understand more about binary number system. Decimal number system refers to base 10 positional notation. It uses ten different symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) arranged using positional notation. Positional notation is used when a number larger then 9 needs to be represented; each position of a digit signifies how many groups of 10, 100, 1000, etc. are contained in that number. For example,

4251

4 $\times$ 1000 + 2 $\times$ 100 + 5 $\times$ 10 + 1

4$\times$ 103+ 2$\times$ 102 + 5$\times$ 101 + 4$\times$ 102

0 and 1 are used in binary number system which is arranged using positional notation (the digit 0 and 1 as a symbol). When a number larger than 1 needs to be represented, the positional notation is used to represent, the positional notation is used to represent how many groups of 2, 4, 8 are contained in the number. For example,  Let's consider the number 30

30 ÷ 2 = 15 Remainder 0

15 ÷ 2 = 7 Remainder 1

7 ÷ 2 = 3 Remainder 1

3 ÷ 2 = 1 Remainder 1

1 ÷ 2 = 0 Remainder 1

##### Things to remember
• A number is a mathematical object used to count, measure and label.
• The binary number system always uses only two different symbols ( the digit 0 and 1) that are arranged using positional notation.
• We can subtract a binary number from the another binary number.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

 2 124 2 62 2 31 2 15 2 7 2 3 2 1

0

Hence,12410=1111100

Remainder

0

0

1

1

1

1

1↑

Solution:

110102

=1×24+1×23+0×22+1×21+0×20

=16+8+0+2+0

=26

Solution:

110102

=1×24+1×23+0×22+1×21+0×20

=16+8+0+2+0

=26

Solution:

1010112

=1×25+0×24+1×23+0×22+1×21+1×20

=32+0+8+0+2+1

=43

Solution:

=1×25+0×24+1×23+0×22+1×21+0×20

=32+0+8+0+2+0

=42

Solution:

1×25+1×24+0×23+0×22+1×21+1×20

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

=51

Solution:

2125

= 2×52+1×51+2×50

= 2×25+1×5+2×1

= 50+5+2

= 5710

Solution:

3145

=3×52+1×51+4×50

= 3×25+1×5+5+4

= 75+5+4

= 8410

Solution:

245

= 2×51+4×5°

= 2×5+4×1

=10+4

=1410

Solution:

 2 105 1 2 52 0 2 26 0 2 13 1 2 6 0 2 3 1 2 1 1 0

∴ 10510 = 11010012

Solution:

 2 1234 0 2 617 1 2 308 0 2 154 0 2 77 1 2 38 0 2 19 1 2 9 1 2 4 0 2 2 0

2 1 1

0

∴ 123410 = 100110100102

Solution:

 2 975 1 2 487 1 2 243 1 2 121 1 2 60 0 2 30 0 2 15 1 2 7 1 2 3 1 1

∴ 97510 = 11110011112

Solution:

111111112

= 1×27+ 1×26 + 1×25+ 1×24 + 1×23 + 1×22 +1×21 + 1×20

= 128 + 64 + 32 + 16 + 8 + 4 +2 + 1

= 225

Solution:

110110110012

= 1×210+1×29+0×28+1×27+1×26+0×25+1×24+1×23 +0×22+0×21+1×20

=1050 +512 +0 +128 +64 +0 +16 +8 +0 +0 +1

= 1751

∴ 110110110012 = 175110

Solution:

= 1×28 + 0×27 + 0×26 + 0×25 + 0×24 + 0×23 + 0×22 + 0×21 + 1×20

= 256 + 0 + 0 +0 +0 +0 +0 +0 +1

= 257

∴ 1000000012 = 25710

Solution:

= 1× 29 + o×28 + 1×27 + 0×26 +1×25 +0×24 + 1×23 + 1×22 + 1×21 + 0×20

= 512 + 0 + 128 +0 + 32 + 0 + 8 + 4 + 2 +0

= 512 + 0 + 128 + 0 + 32 + 0 + 8 + 4 + 2 + 0

= 686

∴ 10101011102 = 68610

Solution:

= 1×29+0×28+0×27+0×26+0×25+1×24+0×23+0×22+0×21 +0×20

= 512 +0 +0+0+0+16+0+0+0 +0

=528

∴ 10000100002 = 52810

Solution:

1011101112

= 1×28+ 0×27 + 1×26 + 1×25 + 1×24 +0×23 + 1×22 + 1×21 + 1×20

= 256 + 0 + 64 + 32 + 16 + 0 + 4 + 2 +1

=359

∴ 1011101112 = 35910

Solution :

1000000012

= 1×28 + 0×27+ 0×26 + 0×25 + 0×24 + 0×23 + 0×22 + 0×21 + 1×20

= 256 + 0 +0 +0 +0 +0 +0+0 +1

= 257

∴ 1000000012 = 25710

Solution:

11001100112

= 1×29 + 1×28 + 0×27 + 0×26 + 1×25 + 1×24 + 0×23 + 0×22 + 1×21 + 1×20

= 512 + 256 + 0 +0 + 32 + 16 + 0 +0 +2 +1

= 819

∴ 11001100112 = 81910