Learn with Amar
Teaching video
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Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Use a lowest-terms contradiction to establish irrationality.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Explain why √2 cannot equal a fraction of integers.
Why this math matters
Use definitions and contradiction to distinguish a proof from numerical evidence. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Prove a number is irrational
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Explain why √2 cannot equal a fraction of integers.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Assume √2=a/b in lowest terms, b≠0
A rational representation can be reduced before reasoning about its factors.
Develop the calculation
a²=2b² makes a even; write a=2k
An odd integer has an odd square, so an even square requires an even integer.
Check and interpret
Then b²=2k², so b is even too: contradiction
Both numerator and denominator would have a factor two, violating lowest terms.
The result
Then b²=2k², so b is even too: contradiction
Both numerator and denominator would have a factor two, violating lowest terms.
Common mistakes to catch
- Nonterminating alone does not establish irrationality; repeating decimals are rational.
- A finite calculator display cannot prove a number's exact rationality.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Is √49 irrational?
Show a hint
Check for a perfect square.
Reveal answer and explanation
No; it is 7
An integer is rational because it equals itself divided by one.
Practice 2
Can a decimal that continues forever still be rational?
Show a hint
Consider a repeating expansion.
Reveal answer and explanation
Yes
1/3=0.333… repeats forever but is an integer ratio.
Take the idea with you
Use definitions and contradiction to distinguish a proof from numerical evidence.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Distribute a negative factor carefully
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