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Algebraic proof - Higher

Algebraic proof - Higher

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Tutor: Bilal

Summary

Algebraic proof

​​In a nutshell

Algebraic proof involves using algebraic notation and manipulation to prove a mathematical statement is true. Start by finding an algebraic expression to represent different types of numbers, like even numbers, odd numbers or multiples of five. Then, use these expressions inside a proof to show whether something is true. It is also possible to disprove a statement by using a counterexample that shows a statement to be false.



Write expressions for proof

Write expressions to represent different types of numbers using algebraic notation.


Examples

Here are some examples, for any integer, nn.

DESCRIPTION

EXPRESSION

an even number

2n2n​​

an odd number

2n+12n+1​​

a multiple of 33​​

3n3n​​

an even multiple of 55​​

5(2n)=10n5(2n)=10n​​

an odd mutliple of 77​​

7(2n+1)=14n+77(2n+1)=14n+7​​

 


Algebraic proof

Use algebraic expressions within a mathematical statement to prove something is true.


 Example 1

Prove that the sum of two odd numbers is even.


Answer

First write expressions for the two odd numbers. For any integers, mm and nn, write the odd numbers as

2m+1 and 2n+12m+1 \space and \space 2n+1​​


Perform the calculation to proof the result

2m+1+2n+1=2m+2n+2=2(m+n+1)\begin{aligned}2m+1 + 2n+1 &= 2m+2n+2 \\&=2(m+n+1)\end {aligned}​​


As mm and nn are integers, m+n+1m+n+1 is also an integer. Two multiplied by any integer gives an even number.

Therefore, 2(m+n+1)\underline{2(m+n+1)} is an even number.



Disproof by counter example

It is possible to disprove a statement by finding an example to show that the statement is false.


Example 2

Give a counter example to disprove the statement 'the product of two prime numbers is always odd'. 


Answer

Pick two prime numbers which - when multiplied - would give an even answer. As two is the only even prime number, multiply this number with any other prime number.

2×5=102 \times 5 = 10​​


1010 is not an odd number so

The statement 'The product of two prime numbers is always odd' is false.


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Exercises

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FAQs - Frequently Asked Questions

How do you represent even and odd numbers in proof?

What is disproof by counter example?

What is algebraic proof?

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