By the end of this lesson, you will be able to:
Can you sort a list of numbers into the two categories?
A classmate claims: "Adding a large enough integer to eventually cancels the irrationality and gives a rational result."
This lesson gives you a proof that settles the question — for every rational number at once.
Sort each into rational or irrational.
Rational: where are integers and
Irrational: real, NOT rational — decimal never terminates, never repeats
Every real number is exactly one: rational or irrational, never both.
This mutual exclusivity is what makes proof by contradiction work.
Classify each as rational or irrational. Show your reasoning.
Remember: simplify first, then classify.
— rational. Does this prove all rational sums are rational?
No. One example shows one case, not all cases.
Proof requires an argument that works for every and — that is what the next slides build.
Let and be any two rational numbers.
Then:
Is rational? We need to verify:
Since is an integer and is a nonzero integer, is rational.
For multiplication:
So is rational.
Verify with an example: . Here : , .
For the sum :
Identify , , , in the general proof.
Template:
Key: To prove X is irrational, assume "X is rational" — never assume what you are trying to prove.
Claim: If is rational and is irrational, then is irrational.
Proof by contradiction.
Assume is rational. Call it .
Then .
But is rational (assumption) and is rational (given), so is rational (difference of two rationals, from the previous proof).
So is rational — contradiction with being irrational.
Therefore, is not rational, so is irrational.
Step 1 — Assume the opposite: suppose where is rational.
Step 2 — Isolate the irrational: .
Step 3 — Ask what category falls in: is rational, is rational, so is rational.
Step 4 — State the contradiction: but is irrational. A number cannot be both rational and irrational.
Step 5 — Conclude: our assumption was wrong. is irrational.
Apply the same argument to prove is irrational.
Step 1: Assume where is rational.
Step 2: Isolate: .
Step 3: Since is rational and is rational, is ________.
Step 4: But is ________. Contradiction.
Step 5: Therefore, is ________.
Goal: prove is irrational.
A student starts: "Assume is irrational."
What goes wrong?
Correct start: Assume is rational.
Fill in the blanks to complete the proof.
Assume _____ where _____ is rational.
Then _____ _____ .
Since _____ is rational and _____ is rational, their difference is _____.
But is _____. Contradiction.
Therefore is _____.
The multiplication proof uses the same template as addition. Answer these before the next slide:
Claim: If is a nonzero rational and is irrational, then is irrational.
Since is rational and is a nonzero rational, is rational (quotient of two rationals, nonzero denominator).
Therefore, is irrational.
Step 3: Since is rational and is a nonzero rational, is rational.
Step 4: But is irrational. Contradiction.
Step 5: Therefore is irrational.
Step 3: Since is rational, is rational (rational times rational).
What happens when ?
is rational.
Where the proof breaks down:
The proof requires dividing by . Division by zero is undefined. So the proof fails when .
The "nonzero" condition is exactly what makes the isolation step valid.
Simplify radicals first, then classify and justify.
Prove that is irrational.
Write the complete proof — no template, no blanks to fill.
Your proof should include:
Mistake 1: — simplify first: , so this is rational + rational.
Mistake 2: Never assume what you want to prove — assume the OPPOSITE ("is rational").
Mistake 3: Product rule requires nonzero : is rational.
The next extension is complex numbers (HSN.CN.A.1):
"What category is this result in, and why?" — the same question drives complex arithmetic.
Click to begin the narrated lesson
Explain operations on rational and irrational numbers