**Two categories of number systems are:**

a) Non-positional number systems (Roman and Greek number systems)

b) Positional number systems (Binary, Octal, Decimal, Hexadecimal number systems)

Positional number system is also known as **place-value** number system.

The **base** or **radix** tells the number of distinct graphic symbols used to represent numbers in a given number system.

The **binary** number system has base 2 as it supports 2 symbols: 0 and 1.

The **octal** number system has base 8 as it supports 8 symbols: 0, 1, 2, 3, 4, 5, 6, 7.

The **hexadecimal** number system has base 16 as it supports 16 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

**LSD:** Least Significant Digit**MSD:** Most Significant Digit**LSB:** Least Significant Bit**MSB:** Most Significant Bit

There are two ways of representing integers in binary:

a) **Unsigned** integers (can represent 0 and positive integers)

b) **Signed** integers (can represent, 0, positive and negative integers)

An unsigned integer containing N bits can have a value in the range **0 to 2 ^{N} – 1**.

There are **three ways of representing signed integers**:

a) Sign-magnitude representation

b) 1’s complement representation

c) 2’s complement representation

In **sign-magnitude representation**, the MSB is the sign bit, with 0 indicating positive and 1 indicating negative. The remaining bits represent the magnitude of the integer.

**Drawbacks of sign-magnitude representation:**

a) There are two representations for the number zero.

b) Positive and negative integers need to be processed separately.

**1’s complement** is calculated by replacing every 0s with 1s and vice-versa.

**Drawbacks of 1’s complement representation:**

a) There are two representations for the number zero.

b) Positive and negative integers need to be processed separately.

**2’s complement** is calculated by adding 1 to its 1’s complement.

**Advantages of using 2’s complement:**

a) There is only one representation for the number zero.

b) Positive and negative integers can be treated together.

Following are the **various ways for representing floating point numbers** in binary:**a) Fixed-point representation:** It splits the bits for representing number to the left of the decimal point and number to the right of the decimal point.**b) Scientific notation:** The number is represented by using mantissa, base and exponent.

1.0_{2} × 2^{-1}**c) Normalized scientific notation:** In this method, only a single non-zero digit before the radix point is used.**d) Mantissa-exponent notation:** In this method, a floating-point binary number is represented in the following form:

Mantissa × 2^{exponent}**e) Excess notation:** It is a means of representing both positive and negative numbers such that the order of the bit patterns is maintained.

Mantissa is also sometimes referred to as **significand**.

**Single precision numbers** include an 8-bit exponent field and a 23-bit fraction, for a total of 32 bits.**Double-precision numbers** include an 11-bit exponent field and a 52-bit fraction, for a total of 64 bits.

**Binary addition:**

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 10

1 + 1 + 1 = 11

**Binary subtraction:**

0 – 0 = 0

1 – 0 = 1

1 – 1 = 0

10 – 1 = 1

0 – 1 = 1 with a borrow of 1

**Binary multiplication:**

0 × 0 = 0

0 × 1 = 0

1 × 0 = 0

1 × 1 = 1

**Binary division:**

0 ÷ 1 = 0

1 ÷ 1 = 1

**8’s complement** is found by first taking 7’s complement and then adding 1 to it.**16’s complement** is found by first taking 15’s complement and then adding 1 to it.

**ASCII** stands for American Standard Code for Information Interchange. It is a 7-bit code, used for representing characters.**EBCDIC** stands for Extended Binary Coded Decimal Interchange Code. It is an 8-bit code for representing various symbols.**Unicode** is a 16-bit code that can represent characters from multiple languages.**ISCII** stands for Indian Standard Code for Information Interchange. It is an 8-bit code that offers coding for Indian scripts, apart from ASCII characters.