Review of Number Systems

REVIEW OF NUMBER SYSTEMS

A numbers system is a basis for counting various items. Fimilar number system is decimal number with its 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Modern computers communicate and operate with binary numbers which use only two digits 0 and 1, and other number systems are octal and hexadecimal numbers.

Decimal Number System

Decimal number system is familiar number system. It express any decimal number in units, tens, hundreds, thousands and so on. We write a decimal number 4567.8 as

4000 + 500 +60 +7 +0.8 = 4567.8

In power of 10, we can write as

4 × 10³ + 5 × 10² + 6 × 10¹ + 7 × 10⁰ + 8 × 10^-1

The sum of all the digits multiplied by their weights gives the total number being represented. The leftmost digit, which has the greatest weight is called Most Significant Bit (MSB) and rightmost digit, which has the least weight, is called the least significant Bit (LSB). It is shown in figure 4.1.

Problem: 1

Represent decimal number 146.92 in power of 10.

Solution:

N = 1 × 10² + 4 × 10¹ + 6 × 10⁰ + 9 × 10^-1 + 2 × 10^-2

The digit 1 has a weight of 100, the digit 4 has a weight of 10, the digit 6 has a weight of 1, the digit 9 has a weight of 1/10 and the digit 2 has a weight of 1/100.

Binary Number System

The binary system have two possible values '0' and '1'. Each coefficient a_j is multiplied by 2^j. In binary system each binary digit called as bit. Fig 4.2 shows the binary position as a power of 2.

Problem: 2

Represent the binary 1010.101 and find the its decimal equivalent.

1010.101

N ⇒ 1 × 2³ +0 × 2² + 1 × 2¹ + 0 × 2⁰ + 1 × 2^-1 + 0 × 2^-2 + 1 × 2^-3

= 8 + 0 + 2 + 0 + 0.5 + 0 + 0.125

= (10.625)₁₀

(1010.101)₂ ⇒ (10.625)₁₀

Problem: 3

Convert binary to decimal for N = 10111.11.

= 1 × 2⁴ + 0 × 2³ + 1 × 2² + 1 × 2¹ + 1 × 2⁰ + 1 × 2^-1 + 1 × 2^-2

= 16 + 0 + 4 + 2 + 1 + 0.5 + 0.25

= (23.75)₁₀

(10111.11)₂ → (23.75)₁₀

Octal Number System

The octal number system uses first eight digits of decimal number system 0, 1, 2, 3, 4, 5, 6 and 7. It uses 8 digits and base is 8

Problem: 4

Represent octal number 436 in power of 8 and find its decimal value.

4 × 8² + 3 × 8¹ +6 × 8⁰

= 4 × 64 + 3 × 8 + 6 × 1

= 256 + 24 + 6

= 286

(436)₈ → (286)₁₀

Hexadecimal Number System

The hexadecimal number system has a base of 16 having 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

Problem: 5

Convert the hexadecimal number A21 to decimal number.

N = A21

=A × 16² + 2 × 16¹ = 1 × 16⁰

= 10 × 256 + 2 × 16 + 1

= 2560 + 32 + 1

(A21)₁₆ = (2593)₁₀