# Rational and Irrational Numbers

Mathematics I · Arithmetic and Number Theory · https://tryals.app/learn/mathematics-i/rational-and-irrational-numbers

## Rational Numbers and Decimals

The **rational numbers** $\mathbb{Q}$ are ratios $\frac{a}{b}$ of integers ($b \neq 0$). Every rational has a **lowest-terms** form where $\gcd(a, b) = 1$.

Fraction arithmetic uses these rules:

| Operation | Formula | Notes |
| --- | --- | --- |
| Addition | $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ | Common denominator |
| Multiplication | $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ | Numerators & denominators |

**Decimal Signature:** A number is rational **iff** its decimal expansion terminates or eventually repeats.

* $\frac{1}{4} = 0.25$ (terminating)
* $\frac{1}{3} = 0.\bar{3}$ (repeating)
* $\frac{1}{7} = 0.\overline{142857}$ (period 6)

## Irrationals and Real Density

The **irrational numbers** are reals that are **not** rational; their decimals run forever without repeating. Examples: $\sqrt{2}$, $\pi$, $e$.

**Proof by Contradiction:** Assume $\sqrt{2} = \frac{a}{b}$ in lowest terms. Then $a^2 = 2b^2$, forcing both $a$ and $b$ to be even, which violates lowest terms.

| Set | Cardinality | Decimal Form |
| --- | --- | --- |
| Rationals | **Countable** | Terminates / Repeats |
| Irrationals | **Uncountable** | Infinite & Non-repeating |

Both sets are **dense** (both appear between any two reals).

**Common Pitfall:** Believing endless decimals are irrational. $0.333\dots$ runs forever, but it **repeats**, making it rational.

## Practice questions

9 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. The classic proof that √2 is irrational assumes a fraction in lowest terms and derives that both numerator and denominator are even. Why is that a contradiction rather than merely an odd result?

A. Because a fraction with both parts even is not a valid fraction and cannot be written down.
B. Because 'lowest terms' means no common factor, and both even share 2.
C. Because the numerator being even forces the denominator to be odd, so the derivation must contain an error.
D. Because even numbers cannot appear in fractions representing irrational numbers.

**Answer:** B. Because 'lowest terms' means no common factor, and both even share 2.

**Why:** The proof makes the assumption as strong as possible and then breaks it: lowest terms means gcd(a, b) = 1, and the algebra forces both to be even. A fraction like 4/6 is perfectly valid, it is simply not in lowest terms. The contradiction is that it cannot be both reduced and reducible, and the argument generalises to every non-square n.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/the-classic-proof-that-2-is-irrational-assumes-a-fraction-in-lowest

### 2. Write 84/126 in lowest terms. What is the numerator of the reduced fraction?

**Answer:** 2

**Why:** gcd(84, 126) = 42, and 84/42 = 2 while 126/42 = 3, so the fraction is 2/3 and the numerator is 2. Dividing by anything smaller than the gcd leaves the fraction reducible; the lowest-terms form is the unique one with coprime numerator and denominator.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/write-84-126-in-lowest-terms-what-is-the-numerator-of-the-reduced

### 3. Sort each number by whether it is rational or irrational.

**Answer:**

- Rational: 0.75, the square root of 16, 0.121212... repeating forever, 22/7
- Irrational: the square root of 10, pi, 0.1010010001... with an extra zero each time

**Why:** The square root of 16 is 4, and 22/7 is a ratio of integers even though it approximates pi, both rational. 0.121212... repeats, so it is rational too. The last decimal runs forever with an ever-growing gap between the 1s, so no block ever repeats: it is irrational. Endlessness alone never decides the question.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/sort-each-number-by-whether-it-is-rational-or-irrational

### 4. Order the steps of the proof by contradiction that the square root of 2 is irrational.

**Answer:**

1. Assume the square root of 2 is rational and write it as a/b in lowest terms
2. Square both sides and rearrange to get a squared = 2 b squared
3. Deduce that a is even, so write a = 2k
4. Substitute to get b squared = 2 k squared, so b is even as well
5. a and b are both even, contradicting the choice of lowest terms

**Why:** The lowest-terms assumption is the trap being laid in step one: the algebra then forces both a and b to be even, so the fraction was reducible after all. Since every rational HAS a lowest-terms form, no such fraction can exist, and the square root of 2 is irrational.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/order-the-steps-of-the-proof-by-contradiction-that-the-square-root-of

### 5. Complete the characterization of irrational numbers.

**Answer:** A real number is irrational exactly when its decimal expansion is **non-terminating and non-repeating**.

**Why:** Irrationality needs BOTH failures: never ending and never settling into a repeating block. Dropping either half misclassifies 1/3 and 1/7, which run forever but repeat. This is the lesson's headline pitfall.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/complete-the-characterization-of-irrational-numbers

### 6. Between any two rationals lies another rational, so they leave no gaps you can point to. Yet they have gaps. How can both be true?

A. Because irrationals are not genuinely numbers, so nothing is missing
B. Because the gaps are infinitely small and therefore not really gaps
C. Because density and completeness are different properties
D. Because the rationals are not actually dense, only appear so at finite precision

**Answer:** C. Because density and completeness are different properties

**Why:** Density says there is no first rational after any other; completeness asks whether every convergent sequence has somewhere to land. A rational sequence can march toward √2 and find nothing there, the hole is at a limit, which is a place density was never looking.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/between-any-two-rationals-lies-another-rational-so-they-leave-no

### 7. Compute 2/3 + 3/4 and write the result in lowest terms. What is the numerator?

**Answer:** 17

**Why:** 2/3 + 3/4 = (2 x 4 + 3 x 3)/12 = 17/12, and gcd(17, 12) = 1 so it is already in lowest terms. The numerator is 17. Adding numerators and denominators separately, the classic error, would give 5/7, which is not even close.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/compute-2-3-3-4-and-write-the-result-in-lowest-terms-what-is-the

### 8. Select every statement that is TRUE.

A. The rational numbers are countable
B. Every non-terminating decimal is irrational
C. The sum of two rational numbers is always rational
D. The sum of two irrational numbers is always irrational
E. Between any two distinct real numbers there is a rational number

**Answer:** A. The rational numbers are countable; C. The sum of two rational numbers is always rational; E. Between any two distinct real numbers there is a rational number

**Why:** Rationals are closed under addition (common denominators keep everything a ratio of integers), they are dense in the line, and they are countable. But the square root of 2 plus its negative is 0, so irrationals are not closed under addition, and 0.333... is a non-terminating rational.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/select-every-statement-that-is-true

### 9. Match each number or term to its defining feature.

**Answer:**

- 1/4 → Terminating decimal
- 1/7 → Repeating decimal with a block of length 6
- The square root of 2 → Irrational, proved by contradiction
- Lowest terms → Numerator and denominator have gcd 1

**Why:** 1/4 = 0.25 terminates because 4 uses only the prime 2; 1/7 repeats with period 6 because 7 divides no power of ten. The square root of 2 escapes the rationals entirely, and lowest terms is the unique reduced representative of a rational.

Page: https://tryals.app/practice/mathematics-i/rational-and-irrational-numbers/match-each-number-or-term-to-its-defining-feature