Algebraic Proofs
Neil Trivedi
Teacher
Algebraic Proofs
Algebraic proof is when we use algebra to show that a statement is always true.
For example, suppose somebody says:
“The sum of any two consecutive integers is always odd.”
We could test this with a few examples:
These examples suggest that the statement is true, but they do not prove that it is always true. To prove something properly, we need to show that it works for every possible case.
To do this, we must first learn how to represent numbers generally. In algebra, we use to represent a number that we do not know yet.
In algebraic proof, we like to use to represent an integer (a whole number that does not have a fractional or decimal component, including positive numbers, negative numbers, and zero) that we are representing generally.
Example 1:
If is an integer, how can we represent the following situations algebraically?
a) An even integer
Single step: Determine what it means to be an even integer.
An even integer is an integer multiple of So, if is a general integer, then an even integer needs to be of the form
b) An odd integer
Single step: Determine what it means to be an odd integer.
An odd integer is one more or one less than an even integer. For example, or
Therefore, we can conclude that the general form of an odd integer is or
c) Two consecutive integers
Single step: Determine what it means to be two consecutive integers.
A consecutive integer is an integer that comes directly before or after another integer. For example, and are consecutive integers. We can say that is one more than We similarly say that and are consecutive with being one less than
Therefore, if is an integer, then its consecutive integers are either one more or one less than it. So, the general form for two consecutive integers is either or
d) Two consecutive even integers
Single step: Determine what it means to be two consecutive even integers.
In part c), we found that consecutive integers are one more or one less than each other. However, with consecutive even integers, they are two more or two less than each other. For example, if we take the even integer then its consecutive even integers are and with being two less than and being two more than
Therefore, if is an even integer, then its consecutive even integers will be and
e) Two consecutive odd integers
Single step: Determine what it means to be two consecutive odd integers.
In part d), we found that consecutive even integers are two more or two less than each other. The same holds true with consecutive odd integers, they are two more or two less than each other. For example, if we take the odd integer then its consecutive odd integers are and with being two less than and being two more than
Therefore, if is an odd integer, then its consecutive odd integers will be and This leaves us with and
If is an integer, then
An even integer
An odd integer or
Consecutive integers and
Consecutive even integers and
Consecutive odd integers and
Divisibility Proofs
A divisibility proof is a proof where we show that an expression is always divisible by a certain number. For example, if we want to show that something is a multiple of then we want to rewrite it in the form
(integer)
This proves it is a multiple of
For example,
So, is always a multiple of
Example 2:
Prove that the sum of three consecutive odd integers is always a multiple of
Step 1: Represent the three consecutive odd integers algebraically.
Three consecutive odd integers can be written as
Notice that each number increases by which is what happens when we move from one odd integer to the next odd integer.
Step 2: Add the three expressions.
The sum is
Removing the brackets,
Collecting the like terms,
Step 3: Factorise to show that it is a multiple of
Since is an integer, is a multiple of
Therefore, the sum of three consecutive odd integers is always a multiple of
Example 3:
Prove that the product of two consecutive even numbers is always a multiple of
Step 1: Represent the two consecutive even numbers algebraically.
Two consecutive even numbers can be written as
and
Step 2: Multiply the two expressions.
The product is
Factorising the expression in the brackets,
So,
Step 3: Explain why this is always a multiple of
The numbers and are consecutive integers.
In any pair of consecutive integers, one of them must be even.
Since either or is even, their product is always even.
So, we can write
where is an integer.
Therefore,
Since this is a multiple of the product of two consecutive even numbers is always a multiple of
General Algebraic Proofs
Not every proof is simply about showing that something is divisible by a number. Sometimes, we may need to prove that an expression is always positive, always even, always odd, or always a square number. The method depends on the wording of the question.
For example:
To prove that something is even, we try to write it as integer.
To prove that something is odd, we try to write it as integer
To prove that something is a multiple of we try to write it as integer.
To prove that something is positive, we may use square numbers, because square numbers are always greater than or equal to
Example 4:
Prove that
is always positive for all values of
Step 1: Complete the square on by first halving the coefficient of writing it in a squared bracket, then subtracting its square. Then, add the constant term at the end.
Here, the coefficient of is We halve that to get and write that in a squared bracket.
Then, we subtract the square of which will be
Then, we add the constant term at the end as it does not affect the square.
For more practice on completing the square, see our Completing the Square Study Note.
Step 2: Use the fact that a square is always non-negative.
Since
add to both sides,
Since the expression is always positive.
Therefore, is always positive for all values of
Example 5:
Show that for any integer
is always odd.
Step 1: Explain why is even.
The numbers and are consecutive integers which means one of them must be even.
Therefore, the product, is even, so we can rewrite it as:
for some integer
Step 2: Add 3.
So,
Odd numbers are of the form so, we rewrite as
Then, factorise from
This is in the form integer So, it is odd.
Therefore, is always odd.
Dealing with Any Two Terms
Sometimes a proof refers to any two numbers, rather than consecutive numbers.
Example 6:
Prove, algebraically, that the sum of the squares of any two odd integers is always even.
Step 1: Represent the two odd integers algebraically.
In this case, we cannot use the same letter for both odd integers. We need two different letters, because the two odd integers may be different.
So, we can write the odd integers as and where and are integers.
We use two different letters, and because the two odd integers do not have to be the same.
If then
If then
So, this allows us to represent any two odd integers, such as and which are not consecutive.
Step 2: Square each odd integer and add them together.
The sum of their squares is
Expanding the first bracket,
Expanding the second bracket,
Adding the two together, then simplifying by collecting the two together,
Step 3: Factorise to show that the expression is even.
We want to show that this is even, so we need to write it as a multiple of by taking out a factor of
The expression inside the bracket is an integer because and are integers.
Therefore, is a multiple of so it is even.
Therefore, the sum of the squares of any two odd integers is always even.
Note: Here, and do not need to be different. If the two odd integers are the same, the proof still works.
Counter Examples
A counter example is a single example that disproves a statement. It is not the same as a proof. It only shows that a statement is false (an anti-proof you could say).
Example 7:
Find counter examples for the following statements.
a) The product of two prime numbers is always odd.
Single step: We need to find two prime numbers whose product is not odd.
A prime number is a positive integer that has two factors, and itself. The number is prime and is prime.
However, is even, not odd. Therefore, and are a counter example.
b) The difference between two square numbers is always even.
Single step: We need to find two square numbers whose difference is not even.
For example, and Their difference is
However, is odd, not even. Therefore, and are a counter example.
c) is prime for all integer values of
Single step: We need to find an integer value of n that makes not prime.
Let’s substitute
However, is not prime because Therefore, is a counter example.
Practice Question