RK Rajesh Kumar Physical Science

12 Mar 2026

Basic Electronics: Boolean Algebra and Number Systems

Boolean variables, basic laws, identities, decimal-binary conversion method, and exam-style answer points.

msc semester-iii open-elective electronics boolean-algebra number-systems

Digital electronics works with two stable states, usually represented by 0 and 1. Boolean algebra is the mathematical language used to describe operations on these two-state variables.

In circuits, 0 and 1 may represent low and high voltage levels. Boolean algebra therefore provides the bridge between logical statements and digital hardware.

Boolean variable

A Boolean variable can have only two possible values:

\[0 \quad \text{or} \quad 1.\]

These values may also be interpreted as false/true, off/on, or low/high.

Basic operations

Boolean algebra is based mainly on three operations:

Operation Symbol Meaning
AND $A\cdot B$ Output is 1 only when both inputs are 1
OR $A+B$ Output is 1 when at least one input is 1
NOT $\overline{A}$ Output is the complement of input

The physical implementation of these operations is done using logic gates.

Addition laws

In Boolean algebra, addition means OR operation. The addition laws are:

\[A+0=A,\] \[A+1=1,\] \[A+A=A,\] \[A+\overline{A}=1.\]

These laws are different from ordinary algebra because Boolean variables can take only the values 0 and 1.

Multiplication laws

Boolean multiplication means AND operation. The multiplication laws are:

\[A\cdot 1=A,\] \[A\cdot 0=0,\] \[A\cdot A=A,\] \[A\cdot\overline{A}=0.\]

Other useful identities

Distributive laws:

\[A(B+C)=AB+AC,\] \[A+BC=(A+B)(A+C).\]

Absorption laws:

\[A+AB=A,\] \[A(A+B)=A.\]

De Morgan’s theorems:

\[\overline{A+B}=\overline{A}\cdot\overline{B},\] \[\overline{AB}=\overline{A}+\overline{B}.\]

Decimal and binary numbers

The decimal system uses base 10 and digits 0 to 9. The binary system uses base 2 and digits 0 and 1. Binary numbers are used in digital electronics because circuits naturally work with two states.

Each place in a binary number represents a power of 2.

For example, the binary number

\[(b_3b_2b_1b_0)_2\]

represents

\[b_3 2^3+b_2 2^2+b_1 2^1+b_0 2^0.\]

Decimal to binary method

To convert a decimal integer into binary:

  1. divide the decimal number by 2;
  2. record the remainder;
  3. continue dividing the quotient by 2;
  4. read the remainders from bottom to top.

This method is commonly asked in university examinations. The important part in an answer is to show the division steps neatly.

Binary to decimal method

To convert a binary number into decimal:

  1. write each binary digit with its place value;
  2. multiply each digit by the corresponding power of 2;
  3. add all non-zero contributions.

For fractional binary parts, use negative powers:

\[2^{-1},\quad 2^{-2},\quad 2^{-3},\ldots\]

Small reference table

Decimal Binary
0 0
1 1
2 10
3 11
4 100
5 101
6 110
7 111
8 1000

University questions

The 2019 open elective paper asked:

  1. “Explain addition laws in Boolean algebra.”
  2. “Convert 23 into binary number.”
  3. “Convert 1101.01 into decimal number.”

For conversion questions, write the division or place-value steps clearly; the examiner gives credit for the method as well as the final representation.

Practice questions:

  1. Define Boolean algebra.
  2. Explain Boolean addition laws.
  3. State De Morgan’s theorems.
  4. Why is the binary system used in digital electronics?
  5. Explain the method of decimal-to-binary conversion.

Answer points

Discussion

Share This Page