NCERT Solutions for Class 9th Maths Chapter 1 Number Systems Exercise 1.3 Question 6

Number Systems

Decimal

Here’s the decimal number system as an example:
digits (or symbols) allowed: 0-9
base (or radix): 10
the order of the digits is significant

345 is really
3 x 100 + 4 x 10 + 5 x 1
3 x 10^2 + 4 x 10^1 + 5 x 10^0
3 is the most significant symbol (it carries the most weight)
5 is the least significant symbol (it carries the least weight)
Binary

Here’s a binary number system:
digits (symbols) allowed: 0, 1
base (radix): 2

each binary digit is called a BIT

the order of the digits is significant

numbering of the digits
msb lsb
n-1 0
where n is the number of digits in the number

msb stands for most significant bit
lsb stands for least significant bit

1001 (base 2) is really
1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 1 x 2^0
9 (base 10)

11000 (base 2) is really
1 x 2^4 + 1 x 2^3 + 0 x 2^2 + 0 x 2^1 + 0 x 2^0
24 (base 10)

Of interest: most assembly languages have no way to represent
binary values!
Octal

Here’s an octal number system:
digits (symbols) allowed: 0-7
base (radix): 8

the order of the digits is significant

345 (base 8) is really
3 x 8^2 + 4 x 8^1 + 5 x 8^0
192 + 32 + 5
229 (base 10)

1001 (base 8) is really
1 x 8^3 + 0 x 8^2 + 0 x 8^1 + 1 x 8^0
512 + 0 + 0 + 1
513 (base 10)

Hexadecimal

here’s a hexadecimal number system:
digits (symbols) allowed: 0-9, a-f
base (radix): 16

the order of the digits is significant

hex decimal binary
0 0 0000
1 1 0001
.
.
.
9 9 1001
a 10 1010
b 11 1011
c 12 1100
d 13 1101
e 14 1110
f 15 1111

a3 (base 16) is really
a x 16^1 + 3 x 16^0
160 + 3
163 (base 10)