Proof by deduction

Proof by deduction is a process in maths where we show that a statement is true using well-known mathematical principles. With this in mind, try not to confuse it with Proof by Induction or Proof by Exhaustion.

The word deduce means to establish facts through reasoning or make conclusions about a particular instance by referring to a general rule or principle. Furthermore, deduction is the noun associated with the verb deduce. It follows that proof by deduction is the demonstration that something is true by showing that it must be true for all instances that could possibly be considered. Hence, it is not sufficient to check that a statement is true for a few example numbers – this is a mistake that is often made.

In maths, proof by deduction usually requires the use of algebraic symbols to represent certain numbers. For this reason, the following are very useful to know when trying to prove a statement by deduction:

  • Use n to represent any integer. Use n and m to represent any two integers.
  • This means we should use n, n+1 and n+2  to represent 3 consecutive integers. Alternatively, use n-1, n and n+1 or equivalent.
  • In addition, if n represents any integer, then 2n represents any even integer and 2n+1 represents any odd integer.
  • It follows that 2n and 2n+2 represent any two consecutive even numbers; 2n-1 and 2n+1 represent any two consecutive odd numbers. Alternatively, we can use 2m-2 and 2m to represent two consecutive even numbers, or 2m+1 and 2m+3 to represent two consecutive odd numbers.
  • Furthermore, use n^2 and m^2 to represent any two square numbers.
  • n^2 and (n+1)^2 represent any two consecutive square numbers and so on…

Note that a certain amount of algebra is required when completing proofs. For example, expanding (n+1)^2 as (n+1)(n+1)=n^2+2n+1


Prove that the difference between the squares of any two consecutive integers is equal to the sum of those integers.

Firstly, choose n and n+1 to be any two consecutive integers. Next, take the squares of these integers to get n^2 and (n+1)^2 where (n+1)^2=(n+1)(n+1)=n^2+2n+1. The difference between these numbers is n^2+2n+1-n^2=2n+1. Adding together the original two consecutive numbers also gives n+n+1=2n+1. Hence, we have proved by deduction that the difference between the squares of any two consecutive integers is equal to the sum of those integers.

Prove that x^2-4x+9 is always positive.

By completing the square x^2-4x+9 can be written as (x-2)^2+5. Note that (x-2)^2 is positive for any x as it is a square number – adding 5 will retain its positivity.


How to use Proof by Deduction to show that the given expression is always even, regardless of the choice of integer to put in.

Sample Assessment question requiring the use of Proof by Deduction to show that an algebraic inequality holds followed by a Disproof by Counterexample question.

Proof of the Quadratic Formula using Proof by Deduction

By recognising a hidden quadratic, Proof by Deduction can be used to show tricky inequalities.