The World of Numbers · Lesson 7 of 7
Chapter Summary and Practice
“See whether the entire number family still seems rational.”
• Recall how the number system expanded from natural numbers to real numbers. • Distinguish rational and irrational numbers using definitions and decimals. • Apply rational-number arithmetic and number-line ideas. • Use irrationality proofs and square-root constructions. • Solve mixed problems involving decimal expansions and rational representations.
The story of numbers is a story of mathematical expansion. Natural numbers answered the need to count. Zero represented absence and completed place value. Negative numbers made debts and directed quantities possible. Fractions filled the spaces between integers. Irrational numbers showed that fractions still did not fill the entire line. Rational and irrational numbers together formed the real numbers.
Natural Numbers
Natural numbers are the basic counting numbers. They are closed under addition but not under subtraction.
Zero and Integers
Zero acts as the additive identity. Integers extend the number line in both directions and include negative values.
Rational Numbers
Rational numbers include integers and fractions. They are dense: between any two distinct rational numbers there are infinitely many rational numbers.
Irrational Numbers
Irrational numbers cannot be written as p/q. Examples include √2 and π. Their decimal expansions are non-terminating and non-repeating.
Real Numbers
Every rational and irrational number lies on the real number line. Together they form a continuous number system used for ordinary measurement.
Decimal Signatures
| Number type | Decimal behaviour |
|---|---|
| Rational | Terminating or repeating |
| Irrational | Non-terminating and non-repeating |
How to Choose the Right Idea
| Problem asks about | Useful idea |
|---|---|
| Counting | Natural numbers |
| Debt, loss, below zero | Integers |
| Part of a whole or ratio | Rational numbers |
| Between two rationals | Average or common-denominator method |
| Square root of non-perfect square | Consider irrationality |
| Decimal terminates or repeats | Check denominator prime factors |
| Repeating decimal to fraction | Shift by powers of 10 and subtract |
| Exact geometric irrational length | Use Pythagorean construction |
• Treating 0 as a natural number without checking the convention being used, • Allowing denominator 0 in p/q, • Forgetting to reduce p/q to lowest terms before predicting decimal behaviour, • Calling every non-terminating decimal irrational, • Forgetting that repeating decimals are rational, • Assuming dense rational numbers fill the entire real line, • Confusing an approximation such as 3.1416 with the exact value of π,
Guided Practice
Problem
Classify √81 and √12.
- 1.√81 = 9, which is an integer and therefore rational.
- 2.12 is not a perfect square.
- 3.√12 = 2√3, and √3 is irrational.
- 4.Therefore √12 is irrational.
Problem
Find a rational number between 2/5 and 3/5.
- 1.Take the average: [(2/5)+(3/5)]/2.
- 2.The sum is 5/5 = 1.
- 3.1/2 lies between 2/5 and 3/5.
- 4.Therefore 1/2 is one such rational number.
Problem
Will 18/125 terminate?
- 1.125 = 5³.
- 2.The reduced denominator contains only the prime factor 5.
- 3.Therefore the decimal terminates.
- 4.To write it explicitly: 18/125 = 144/1000 = 0.144.
Problem
Convert 0.232323... to p/q.
- 1.Let x = 0.232323...
- 2.Two digits repeat, so 100x = 23.232323...
- 3.Subtract x: 99x = 23.
- 4.Therefore x = 23/99.
Problem
Outline a proof that √5 is irrational.
- 1.Assume √5 = p/q in lowest terms.
- 2.Square: p² = 5q².
- 3.Then p² is divisible by 5, so p is divisible by 5. Write p = 5k.
- 4.Substitute: 25k² = 5q², so q² = 5k².
- 5.Therefore q is also divisible by 5.
- 6.This contradicts p/q being in lowest terms.
- 7.Hence √5 is irrational.
Practice Problems
- Convert 3/50 and 2/9 to decimal form by long division and classify each decimal.
- Prove that √5 is irrational.
- Convert these decimals into p/q form: 12.6, 0.0120, 3.052, 1.235, 0.232323..., 2.05050....
- Locate 0.532 and 1.15 on a number line.
- Find six rational numbers between 3 and 4.
- Find five rational numbers between 2/5 and 3/5.
- Find five rational numbers between 1/6 and 2/5.
- Solve x/3 + x/5 = 16/15.
- Let a and b be non-zero rational numbers satisfying a + 1/b = 0. Determine whether ab is positive or negative and justify.
- A rational number has a terminating decimal with its last non-zero digit in the fourth decimal place. Explain why it can be written with denominator 10⁴ before reduction.
- Without division, determine whether 18/125 has a terminating decimal and state the number of decimal places.
- A reduced rational number has denominator 2³ × 5. Determine the number of decimal places in its terminating decimal.
- Show that (a+b)/2 lies between distinct rational numbers a and b.
- Find the hypotenuse lengths in the first several triangles of a square-root spiral.
Quiz
Which set contains negative whole numbers, zero and positive whole numbers?
Which denominator guarantees a terminating decimal after the fraction is reduced?
Which statement is true?
Why is √2 irrational?
What is 0.999...?
Make sure you can explain why each larger number system was needed, classify numbers correctly, work with rational numbers, locate them on the number line, understand density, follow a proof of irrationality, predict decimal behaviour from denominator factors, and convert repeating decimals back into fractions.
Key Takeaways
• Number systems expanded whenever earlier systems could not answer new mathematical needs. • Natural numbers lead to integers, integers lie inside the rational numbers, and rational plus irrational numbers form the real numbers. • Rational numbers have terminating or repeating decimals. • Irrational numbers have non-terminating, non-repeating decimals. • Rational numbers are dense but do not alone fill the real number line. • Zero and negative numbers fundamentally changed arithmetic. • The real number line provides a home for ordinary measurable quantities.
This completes The World of Numbers. Continue by practising how definitions, number-line representations, proofs and decimal behaviour connect the different number systems.
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Real Numbers: Decimals and Cyclic Patterns
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