Skip to main content
Home
plus.maths.org

Secondary menu

  • My list
  • About Plus
  • Sponsors
  • Subscribe
  • Contact Us
  • Log in
  • Main navigation

  • Home
  • Articles
  • Collections
  • Podcasts
  • Maths in a minute
  • Puzzles
  • Videos
  • Topics and tags
  • For

    • cat icon
      Curiosity
    • newspaper icon
      Media
    • graduation icon
      Education
    • briefcase icon
      Policy

      Popular topics and tags

      Shapes

      • Geometry
      • Vectors and matrices
      • Topology
      • Networks and graph theory
      • Fractals

      Numbers

      • Number theory
      • Arithmetic
      • Prime numbers
      • Fermat's last theorem
      • Cryptography

      Computing and information

      • Quantum computing
      • Complexity
      • Information theory
      • Artificial intelligence and machine learning
      • Algorithm

      Data and probability

      • Statistics
      • Probability and uncertainty
      • Randomness

      Abstract structures

      • Symmetry
      • Algebra and group theory
      • Vectors and matrices

      Physics

      • Fluid dynamics
      • Quantum physics
      • General relativity, gravity and black holes
      • Entropy and thermodynamics
      • String theory and quantum gravity

      Arts, humanities and sport

      • History and philosophy of mathematics
      • Art and Music
      • Language
      • Sport

      Logic, proof and strategy

      • Logic
      • Proof
      • Game theory

      Calculus and analysis

      • Differential equations
      • Calculus

      Towards applications

      • Mathematical modelling
      • Dynamical systems and Chaos

      Applications

      • Medicine and health
      • Epidemiology
      • Biology
      • Economics and finance
      • Engineering and architecture
      • Weather forecasting
      • Climate change

      Understanding of mathematics

      • Public understanding of mathematics
      • Education

      Get your maths quickly

      • Maths in a minute

      Main menu

    • Home
    • Articles
    • Collections
    • Podcasts
    • Maths in a minute
    • Puzzles
    • Videos
    • Topics and tags
    • Audiences

      • cat icon
        Curiosity
      • newspaper icon
        Media
      • graduation icon
        Education
      • briefcase icon
        Policy

      Secondary menu

    • My list
    • About Plus
    • Sponsors
    • Subscribe
    • Contact Us
    • Log in
    • Maths in a minute: Simplifying circuits

      28 April, 2016

      Most of us are aware that the work done by our computers is accomplished by breaking tasks down into a series of just a few logical operations. These operations – AND, OR and NOT – can be physically represented in the computer circuits and mathematically representation using Boolean algebra. (You can read more in Maths in a minute: Boolean algebra.) This realisation of the links between circuitry, logic and algebra was first made by Claude Shannon (1916 - 2001), who in 1936 was a 20-year-old student at Massachusetts Institute of Technology writing his Masters thesis.

      Shannon was considering complex circuits of switches and relays that were used in places such as telephone exchanges. Suppose that you have a complicated circuit design, a mess of wires and switches. Shannon realised that if you wrote down the corresponding Boolean algebra expression, you could quickly use the simplification laws to remove redundant components in your circuit.

      For example, consider this circuit:

      a complicated circuit

      This circuit diagram is written in shorthand with symbols to represent the AND, OR and NOT gates that combine the 0/1 values for P and Q.

      The circuit above corresponds to the expression

      ((P x Q + Q') x Q' + P)'

      This expression can be simplified using the laws of Boolean algebra as follows:

      ((P x Q + Q') x Q' + P)'
      = ((P x Q + Q') x Q')' x P' (by De Morgan's laws)
      =((P x Q + Q')' + Q) x P' (by De Morgan's laws)
      =((P x Q)' x Q + Q) x P' (by De Morgan's laws)
      = Q x P' (by the simplification rule A+A x B)

      So analysing the original circuit using Boolean algebra reveals that a simple circuit with just two gates will do the same trick as the original complicated one.

      A simple circuit

      This simple circuit, with just two gates, is equivalent to the original complex one.

      Before Shannon's work, simplifying a circuit design involved writing all the possible positions of the switches in the circuit and following through the chain of events for each, a process he himself described as "very tedious and open to errors." But now, thanks to his insight, any circuit could be easily described and simplified using Boolean algebra.

      • Log in or register to post comments

      Read more about...

      boolean algebra
      computer science
      Maths in a minute
      University of Cambridge logo

      Plus Magazine is part of the family of activities in the Millennium Mathematics Project.
      Copyright © 1997 - 2025. University of Cambridge. All rights reserved.

      Terms