Algebraic Proof

 

What is Algebraic Proof?

Algebraic proof is the process of proving mathematical statements using algebra. Instead of checking specific cases, we use algebraic expressions to show that a statement is always true for all possible values.

To construct an algebraic proof, you may need to use algebraic skills such as:

• Expanding brackets
• Factorising
• Collecting like terms
• The difference of two squares

 

Representing Integers Algebraically

When proving results about integers, we often express them using algebraic terms. The table below shows some useful representations:

 

Type of Integer	Representation	Notes
Any integer	$n$	A general integer
Consecutive integers	$n,n+1$	Numbers one after another
Any two integers	$n,m$	Different letters for different numbers
An even integer	$2n$	Even numbers are multiples of 2
Consecutive even integers	$2n,2n+2$	Increase by 2 each time
An odd integer	$2n+1$	Odd numbers are one more than even
A multiple of 5	$5n$	A number that can be divided by 5
A multiple of $k$	$kn$	A general multiple of any number $k$
A square number	$n^2$	A number squared
A cube number	$n^3$	A number cubed
A rational number	$\frac{a}{b}$​	$a,b$ are integers, $b \neq 0$

 

Common Proof Techniques

1. Proving Even or Odd Results

To prove that an expression is even, show that it can be written in the form:

$$2k \quad \text{(where$$$k$ is an integer)}

To prove that an expression is odd, show that it can be written in the form:

$$2k + 1$$

 

Example: Show that the sum of two odd numbers is always even.

• Let two odd numbers be $2a + 1$ and $2b + 1$.
• Find their sum: $$(2a + 1) + (2b + 1) = 2a + 2b + 2$$
• Factor out 2: $$2(a + b + 1)$$
• Since $a + b + 1$ is an integer, the result is even.

Conclusion: "This proves that the sum of two odd numbers is always even."

 

2. Proving Divisibility

To show that an expression is divisible by $k$, prove that it can be written in the form: $$k \times \text{(an integer)}$$

Example: Prove that the difference between the squares of two consecutive even numbers is always divisible by 4.

Step 1: Define Two Consecutive Even Numbers

Let the two even numbers be: $$2n \quad \text{and} \quad 2n+2$$

Step 2: Find the Difference of Their Squares $$(2n+2)^2 - (2n)^2$$

Step 3: Expand the Squares $$(4n^2 + 8n + 4) - (4n^2)$$

Step 4: Simplify $$4n^2 + 8n + 4 - 4n^2 = 8n + 4$$

Step 5: Factorise $$8n + 4 = 4(2n + 1)$$>

Since $4(2n + 1)<>$ is a multiple of $4$, the expression is divisible by 4.

Conclusion: "This proves that the difference between the squares of two consecutive even numbers is always divisible by 4."

 

3. Using the Difference of Two Squares

The difference of two squares states that: $$a^2 - b^2 = (a - b)(a + b)$$

Example: Alternative Proof for the Previous Example

Using $a^2 - b^2$: $$(2n+2)^2 - (2n)^2 = [(2n+2) - (2n)] \times [(2n+2) + (2n)]$$

Simplify inside brackets: $$(2) \times (4n + 2)$$

Factor out a 2: $$2 \times 2(2n+1) = 4(2n+1)$$

Since $4(2n+1)$ is a multiple of 4, the result is confirmed.

Conclusion: "This proves that the difference between the squares of two consecutive even numbers is always divisible by 4."

 

Proofs Involving Prime Numbers

• A prime number is only divisible by 1 and itself.
• If $p$ is prime, the only ways to write it as a product are: $$p = 1 \times p \quad \text{or} \quad p = p \times 1$$

Example: Prove that if $p$ is prime, then $p^2 - p$ is always divisible by $p$.

Step 1: Factorise the Expression $$p^2 - p = p(p - 1)$$

Step 2: Show Divisibility

• Since $p(p - 1)$ contains $p$ as a factor, it is always divisible by $p$.

Conclusion: "This proves that $p^2 - p$ is always divisible by $p$ for any prime $p$."