IB Mathematics Analysis and Approaches Topic SL…

What A Proof Is, And What It Is Not

A proof is a watertight argument: a chain of justified steps that starts from things already known and ends at the statement you want to establish.
Checking that a claim works for a few numbers is not a proof, because a later case you did not try might still break it.
A deductive proof settles every case at once by arguing with symbols rather than examples.
Two symbols matter here. The equals sign $=$ links expressions that are equal for particular values, which you find by solving.
The identity sign $\equiv$ links expressions equal for all values, so there is nothing to solve; the two sides are always the same thing written differently.

Note

$x + 3 = 7$ is an equation, true only when $x = 4$.
$(x+1)^2 \equiv x^2 + 2x + 1$ is an identity, true for every value of $x$.
Proof is a technique, not a formula, so it has no formula-booklet entry; the marks are for clear, justified reasoning.

How To Lay Out A Left-Hand-Side To Right-Hand-Side Proof

The standard way to prove an identity is to start from one side, usually the more complicated one, and transform it until it becomes the other side.
Write one manipulation per line so a reader can follow, and stop the moment you reach the target expression.

Example

Take the claim $(x-3)^2 + 5 \equiv x^2 - 6x + 14$.
Begin at the left, expand the square, then collect the constants. $$ (x-3)^2 + 5 = (x^2 - 6x + 9) + 5 = x^2 - 6x + 14 $$
The left side has turned into the right side using only known algebra, so the identity is proved.

Example question

Show that $(x+4)(x-4) + 16 \equiv x^2$.

Solution

Start at the LHS and expand the difference of two squares: $(x+4)(x-4) = x^2 - 16$.
Add 16: $x^2 - 16 + 16 = x^2$, which is the RHS, as required.

Common Mistake

Never start by assuming the thing you are trying to prove is already true.
Transform one side into the other; do not work on both sides at once unless every step can be reversed.

Generalising From Numbers To Algebra

A proof often starts with a concrete numerical case that reveals the pattern, which you then rewrite with a variable to cover all cases.

Example

The numerical case: $$\dfrac{1}{4} + \dfrac{1}{12} = \dfrac{3}{12} + \dfrac{1}{12} = \dfrac{4}{12} = \dfrac{1}{3}$$
Notice the second denominator, 12, is the first one, 4, times 3, which hints at the general pattern.

Example question

Prove $\dfrac{1}{m+1} + \dfrac{1}{m^2 + m} \equiv \dfrac{1}{m}$.

Solution

The key first step is to factorise the second denominator: $$m^2 + m = m(m+1)$$
Now both terms share the denominator $m(m+1)$, so rewrite the first fraction over it and add. $$ \frac{1}{m+1} + \frac{1}{m(m+1)} = \frac{m}{m(m+1)} + \frac{1}{m(m+1)} = \frac{m+1}{m(m+1)} = \frac{1}{m} $$
The factor $(m+1)$ cancels top and bottom, leaving $\dfrac{1}{m}$, which is the right-hand side.
Check by substituting a value: $m = 3$ gives $\tfrac{1}{4} + \tfrac{1}{12} = \tfrac{1}{3}$, which matches the numerical case.

Active recall

What is the difference between $=$ and $\equiv$?
Why is testing a few values not a proof of an identity?
In a left-hand-side to right-hand-side proof, which side do you usually start from, and why?
Prove that $(x+2)^2 - 4 \equiv x^2 + 4x$.
What is the first useful step when the denominators in an identity are different?

What A Proof Is, And What It Is Not

A proof is a watertight argument: a chain of justified steps that starts from things already known and ends at the statement you want to establish.
Checking that a claim works for a few numbers is not a proof, because a later case you did not try might still break it.
A deductive proof settles every case at once by arguing with symbols rather than examples.
Two symbols matter here. The equals sign $=$ links expressions that are equal for particular values, which you find by solving.
The identity sign $\equiv$ links expressions equal for all values, so there is nothing to solve; the two sides are always the same thing written differently.

Note

$x + 3 = 7$ is an equation, true only when $x = 4$.
$(x+1)^2 \equiv x^2 + 2x + 1$ is an identity, true for every value of $x$.
Proof is a technique, not a formula, so it has no formula-booklet entry; the marks are for clear, justified reasoning.

How To Lay Out A Left-Hand-Side To Right-Hand-Side Proof

The standard way to prove an identity is to start from one side, usually the more complicated one, and transform it until it becomes the other side.
Write one manipulation per line so a reader can follow, and stop the moment you reach the target expression.

Example

Take the claim $(x-3)^2 + 5 \equiv x^2 - 6x + 14$.
Begin at the left, expand the square, then collect the constants. $$ (x-3)^2 + 5 = (x^2 - 6x + 9) + 5 = x^2 - 6x + 14 $$
The left side has turned into the right side using only known algebra, so the identity is proved.

Example question

Show that $(x+4)(x-4) + 16 \equiv x^2$.

Solution

Start at the LHS and expand the difference of two squares: $(x+4)(x-4) = x^2 - 16$.
Add 16: $x^2 - 16 + 16 = x^2$, which is the RHS, as required.

Common Mistake

Never start by assuming the thing you are trying to prove is already true.
Transform one side into the other; do not work on both sides at once unless every step can be reversed.

Generalising From Numbers To Algebra

A proof often starts with a concrete numerical case that reveals the pattern, which you then rewrite with a variable to cover all cases.

Example

The numerical case: $$\dfrac{1}{4} + \dfrac{1}{12} = \dfrac{3}{12} + \dfrac{1}{12} = \dfrac{4}{12} = \dfrac{1}{3}$$
Notice the second denominator, 12, is the first one, 4, times 3, which hints at the general pattern.

Example question

Prove $\dfrac{1}{m+1} + \dfrac{1}{m^2 + m} \equiv \dfrac{1}{m}$.

Solution

The key first step is to factorise the second denominator: $$m^2 + m = m(m+1)$$
Now both terms share the denominator $m(m+1)$, so rewrite the first fraction over it and add. $$ \frac{1}{m+1} + \frac{1}{m(m+1)} = \frac{m}{m(m+1)} + \frac{1}{m(m+1)} = \frac{m+1}{m(m+1)} = \frac{1}{m} $$
The factor $(m+1)$ cancels top and bottom, leaving $\dfrac{1}{m}$, which is the right-hand side.
Check by substituting a value: $m = 3$ gives $\tfrac{1}{4} + \tfrac{1}{12} = \tfrac{1}{3}$, which matches the numerical case.

Active recall

What is the difference between $=$ and $\equiv$?
Why is testing a few values not a proof of an identity?
In a left-hand-side to right-hand-side proof, which side do you usually start from, and why?
Prove that $(x+2)^2 - 4 \equiv x^2 + 4x$.
What is the first useful step when the denominators in an identity are different?

IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

What A Proof Is, And What It Is Not

How To Lay Out A Left-Hand-Side To Right-Hand-Side Proof

Generalising From Numbers To Algebra

What A Proof Is, And What It Is Not

How To Lay Out A Left-Hand-Side To Right-Hand-Side Proof

Generalising From Numbers To Algebra

IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

IB Mathematics Analysis and Approaches SL 1.6—simple Proof Notes

Number and Algebra

Functions

Geometry & Trigonometry

Statistics & Probability

Calculus

What A Proof Is, And What It Is Not

How To Lay Out A Left-Hand-Side To Right-Hand-Side Proof

Generalising From Numbers To Algebra

What A Proof Is, And What It Is Not

How To Lay Out A Left-Hand-Side To Right-Hand-Side Proof

Generalising From Numbers To Algebra