Binary calculation and Venn diagram

BINARY CALCULATION

The binary system is a numerical system that uses only two symbols i.e 0 and 1. Due to its ease of use in digital electronic circuitry using logic gates, internally all modern computers use the binary system .

The following are some typical conversions between binary values and decimal values.

In binary system the decimal are the following result:

decimal 0 = 0

decimal 1 = 1

decimal 2 = 10

decimal 3 = 2⁰ + 1 = 11

decimal 4 = 2²= 100

decimal 7= 2²+ 2⁰ + 1 = 111

decimal 8 = 2³= 1000

decimal 10 = 2³+ 2⁰ = 1010

Binary Addition

The addition of binaries is similar to the decimal system. The only different is to carry over when the result is 2.

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (0 with carry over1)

For example:

Binary Subtraction

Rules for binary subtraction

1 - 1 = 0

0 - 0 = 0

1 - 0 = 1

0 - 1 = 1

For example:

Binary Multiplication

Rule for binary multiplication

1 x 1 = 1

1 x 0 = 0

0 x 1 = 0

0 x 0 = 0

For example:

Binary Division

Rule for binary division,

1 ÷ 1 = 1

1 ÷ 0 = 0 (not defined)

0 ÷ 1 = 0

0 ÷ 0 = 0 (not defined)

For example:

Some Basic Terms Related with Number System

1. MSB (Most significant Bit)

The left most bit of a number is called MSB. It is also knows as high-order bit. It is considered as having greatest value in the bit of binary digit.

Example: 1 0 1 0

2. LSB ( Least significant Bit)

The right most bit of a number is called LSB. This is the determination of the number whether it is odd or even.

Example: 1 0 1 0

Notes

• Bit: Single binary number either 0 or 1
• Nibble: Combination of 4 binary bits e.g 1001
• Byte: Combination of * binary bits e.g 1001 0111

Venn diagram

Venn diagram is the diagram in which 'Areas' represent 'Operations' or 'Propositions' or 'Statements'. The area within the 'rectangle' represents all the possible proportions and 'shaded area' represents the proportion under the consideration or operation.

The pictorial, the relationships among the variable of a Boolean expression is the Venn diagram. It consists of a rectangle, inside of which are drawn overlapping circles, one for each variable.

Each circle is labeled by a variable and all points inside of a circle as belonging to the named variable and all points outsides a circle as not belonging to the variable.

Venn diagram of the Logic Gates

i.) A' (F=A)

The shaded area represents A.

i.e F=A

ii.) AND (F=A.B)

An electronic circuit which has two or more than to two input values and produces only one out value. As a Venn diagram representation, the current flow in the circuit occurs only when both switches, A and B are closed. A switch has two stable states, ON and OFF. The ON state of logic is taken as 1 and OFF is taken as 0. When both switch switches are ON, the output is ON and when any of switches are OFF and there is no output or the output is 0.

The shaded area represents A.B or A∩B.

iii.) NAND (F=(AB)')

NAND gate is an electronic circuit which have two or more than two input values and produce only one output value. This gate is the combination of AND and NOT gates. This gate produces output 0, when all input are 1, otherwise, the output will be 1.

The shaded area represents (A∩B)'

i.e F=(A.B)'

iv.) OR (F=A+B)

OR gate is an electronic circuit which has two or more than two input values and produces only one output value. OR gate produces 1 output value. OR gate produces 1 output when one of the inputs is 1. If all inputs are 0, then the output will be 0. The logical OR operation can easily be explained taking an example of two switches connected in parallel, as it is shown in below Venn diagram.

v.) NOR (F=(A+B)')

NOR is an electrical circuit which has two or more than two input values and produces only one output value. NOR gate is the combination of OR and NOT gate. It is the complement of the OR function. As shown below figure, the gate produces 1 output when all inputs are 0 or else output will be 0.

vi.) NOT (F=A')

The NOT Gate contains only one input value which produces only one output value. This gate is also known as an inverter. So, this circuit inverts the logical sense of a binary signal. It produces the complemented function. If the input is 1 then this gate will produce 0 as output and vice-versa.

vii.) Exclusive OR ( X-OR) gate (F=A'B+A.B')

Exclusive OR gate in an electronic circuit has two or more than two input values which produces only one output value. The graphical symbol of an X-OR gate is similar to OR gate except for the additional curve line on the input side. This gate produces 1 as output, if any input is 1 and 0 if both inputs are either 1 or 0 otherwise its output is 0.

viii.) Exclusive NOR (X-NOR) gate (F=AB'+A'B')

X-NOR gate contains two or more than two input values which produce only one output value. The X-NOR is the complement of the X-OR, as indicated by the small circle in the graphical symbol. This gate produces 1 output when all inputs are either 0 or 1 otherwise its output value is 0.

Verifying using Venn Diagram

(A+B).(A+C)=A+BC

The Venn diagram of LHS (A+B).(A+C) .......................... The Venn diagram of RHS, A+BC

Comparing the two both Venn diagram, both are same. Hence,it is proved that (A+B).(A+C)=A+BC.

Bibliography

Dilli , S. P., Singh, k. E., Khadka , D., Bhatta, K. D., Baral, N., Saud, R. S., . . . Dangi, R. (2015). Fundamental of Computer Science. Kathmandu: KEC publication and Distribution.

Shrestha,Shrestha & Manandhar, s. (2014). Computer Essential.Kathmandu: Ashmita publication.

Calculator.net. (2015). Retrieved from http://www.calculator.net/binary-calculator.html