Binary

View Assign Share Link Report

Binary

Lesson Objectives

The binary representation of numbers
Understand the units of computer memory
Converting denary numbers to a binary system and vice versa
Adding binary numbers
Binary shifts
Bitwise operations
Representing negative numbers in a binary system
Applications of the binary system

Introduction

A computer has many electronic components that work as switches.
These components have two logics as input and output: ON and OFF.
A similar logic is used to represent data in binary form. ON is represented as 1, and OFF is represented as 0.

Place value of the denary system

The denary system has a base value of 10.
It counts in multiples of 10.

Place value of the binary system

The place values of binary numbers are of base 2.

Size of computer memory: base-2

A binary digit is referred to as a bit.
A nibble consists of 4 bits.
A byte consists of 8 bits.
A byte is the smallest unit of memory in the computer system.
The memory sizes available with computers are in multiples of 8 such as 16-bit systems, 32-bit systems, etc.
The memory sizes were originally standardised using the base-2 representation.
In this system, the prefixes kibi-, mebi-, gibi-, tebi- are used to avoid conflicts with the base-10 system.
This representation is now used for representing the size of RAM modules only.

Size of computer memory: base-10

After the standardization of base-10 representation, the memory sizes are now represented as given.

Converting denary to binary

A bidirectional bus consisting of 8, 16, 32, and 64 parallel lines.
This bus is used to transmit data and instructions between the processor, memory, and input-output devices.

A denary number is converted to binary by dividing it by 2 and calculating the remainders.
The binary equivalent is obtained by arranging the remainders in reverse order.

How do you check your answer?

Binary equivalent of denary number 91:

The answer can be checked by:

(0×128)+(1×64)+(0×32)+(1×16)+(1×8)+(0×4)+(1×2)+(1×1)=91

Binary combinations

A one-bit system has a one-place value and can have 2 possible combinations: 0 or 1.
Similarly, a 2-bit system has two-place values and has 4 possible combinations, as shown in the table.

Similarly, a 3-bit system has three-place values and has 8 possible combinations.

Representing numbers

Programmers use many arithmetic operations in a program.
The numbers are either represented as integers or floating point numbers.
Integers are whole numbers, and floating point numbers are used to represent numbers with decimal points.
A 16-bit system can represent integers up to 216-1=65535.
8-bit, 16-bit, 32-bit, and 64-bit are the most common bit lengths.

Adding binary numbers

Binary numbers are added in the column method as the denary numbers are added.
Adding 0101 and 1011 in the table.
Adding binary numbers:

0+0=0

1+0=1

1+1=10 (1 is carried over)

How to check your answer

0101 and 1001 represent the denary numbers 5 and 9.
The sum of 5 and 9 is 14.
Convert the sum obtained to a denary number.
(8×1)+(4×1)+(2×1)+(0×1)=14

The rules are:

0+0=0
1+0=1 (sum 1 and carry 0)
1+1=10 (sum 0 and carry 1 )
1 + 1 + (carry 1) = 11= sum 1 and carry 1

Adding three binary numbers

The rules are:

1 + 0 + 0 = sum 1 and carry 0
1 + 1 + 0 = sum 0 and carry 1
1 + 1 + 1 = sum 1 and carry 1
1 + 1 + 1 + (Carry 1) = 1 0 0
Adding four 1’s results in a sum of 0, and carry 1 is placed two places to the left.

Overflow error

A CPU with an 8-bit register has a capacity of up to 11111111 in binary. If an extra bit is added, it is said to be an overflow error.
The number of bits a register can hold is called word size. Exceeding the capacity of word size in a register results in an overflow error.
Consider the addition of two binary numbers 11101101 and 10000100.
The sum of these two numbers is bigger than 8 bits (an extra bit than the register can hold). The computer thinks that 11101101+10000100=01110001 as it does not have space to store the extra bit.

Binary Shifts

Let us consider the denary number 6. Its binary equivalent is 0110.

What is the binary equivalent of 12? Is it in any way related to the binary equivalent of 6?
The binary equivalent of 12 (6 × 2) is:

It can be noted that the binary equivalent of 6 shifted to the left by one place is the binary equivalent of 12.
The binary equivalent of 12 is:

The binary equivalent of 24 is:

Again, it can be noted that the binary equivalent of 12 shifted to the left is the binary equivalent of 24.
It can be summarised that when a denary number is multiplied by 2, its binary equivalent shifts left by 1 place.
Also, multiplication by 4 results in a shift of binary equivalent by 2 places.
Multiplication by 8 results in a shift of binary equivalent by 3 places and so on.

Representing negative numbers

Signed integers can be either positive or negative.
An extra bit is used to represent the sign of a number in binary representation.
In the case of signed numbers, the leftmost bit is used to represent the sign and is called the sign bit.
0 represents a positive number, and 1 represents a negative number.
The rest of the bits represent the magnitude of the number.
Consider the 8-bit binary number 10001101.

The smallest number that can be represented using 8 bits is 11111111 (-127), and the largest number is 01111111 (+127).
Similarly, the signed number can be represented in 32-bits, 64-bits, and so on.

Finding two's complement

Finding two’s complement is an alternate method to represent negative numbers. This method is used by most computers to perform mathematical operations.
Let us consider an example of representing -5. The binary value of 5 is 101. The leftmost bit is added to represent the positive sign. +5 is 0101.
Each bit is inverted and, hence, the 0101 becomes 1010. 1 is added to this number. 1010 + 1=1011.
Some examples are given.

Sum of numbers: Using two’s complement

Let us consider adding -4 and 3 using two’s complement.
Converting the sum 1111 into a denary number,

-8+4+2+1=-1

Applications of the binary system

Microprocessors in computer use registers to control devices such as robots.
Registers store a group of bits.
Consider a robotic vehicle that has four wheels.
Register used to control this robotic vehicle:

For example: A value of 10100101 means ‘ Motor C is ON, and Motor D is ON, and both wheels are moving together backward.’
As a result of these values, the robotic vehicle moves backward.

Related Worksheets: