Rational and Irrational Numbers

🔢 Rational and Irrational Numbers

Part 1 of 5 — What Makes a Number Rational

Topics in This Part

🔑 Key Concept: A rational number is any number you can write as a fraction $\dfrac{a}{b}$ where $a$ and $b$ are integers and $b \ne 0$. By the end of this part you'll be able to spot one instantly.

The Definition of a Rational Number

A number is rational if it can be written as a ratio of two integers:

$\text{rational} = \frac{a}{b}, \quad a, b \text{ are integers}, \quad b \ne 0$

The word rational literally contains the word ratio — a fraction. That's the whole idea.

What counts as rational?

🔑 Key Idea: Every integer is rational, because any whole number $n$ equals $\dfrac{n}{1}$. The denominator can never be $0$, though — division by zero is undefined.

Concept Check 🎯

Decimals That Are Rational

A decimal is rational when it either stops or repeats forever in a pattern.

Terminating decimals end after a finite number of digits: $0.5 = \frac{1}{2}, \qquad 0.75 = \frac{3}{4}, \qquad 0.125 = \frac{1}{8}$

Repeating decimals have a block of digits that repeats without end. We draw a bar over the repeating block: $0.\overline{3} = 0.3333\ldots = \frac{1}{3}, \qquad 0.\overline{6} = 0.6666\ldots = \frac{2}{3}$

💡 Both terminating and repeating decimals are rational. We'll learn to turn a repeating decimal into a fraction in Part 3.

Classify Each Number 🔽

Decide whether each value is rational. (Every number here is rational — choose the reason.)

Write It as a Fraction 🧮

Every rational number is a ratio of integers. Write each as a fraction in lowest terms. (Type like 3/4.)

1) $6 = \,?$ (use denominator $1$) 2) $0.5 = \,?$ 3) $-2 = \,?$ (use denominator $1$)

🔢 Rational and Irrational Numbers

Part 2 of 5 — Irrational Numbers

🔑 The Idea: An irrational number cannot be written as a fraction of integers. Its decimal goes on forever with no repeating pattern. The prefix ir- means "not," so irrational = "not a ratio."

What Makes a Number Irrational

A decimal is irrational when it is non-terminating (never ends) and non-repeating (never settles into a pattern).

The three families you'll meet most

1) The square root of a number that isn't a perfect square. $\sqrt{2} = 1.41421356\ldots \qquad \sqrt{3} = 1.7320508\ldots \qquad \sqrt{10} = 3.16227\ldots$

2) The number $\pi$ (ratio of a circle's circumference to its diameter): $\pi = 3.14159265\ldots$

3) Special constants like $e = 2.71828\ldots$ (you'll meet $e$ in later courses).

⚠️ Careful: $3.14$ and $\dfrac{22}{7}$ are rational approximations of $\pi$ — they are not equal to $\pi$. The true value $\pi$ never ends and never repeats, so $\pi$ itself is irrational.

Perfect Squares vs. the Rest

A perfect square is a number whose square root is a whole number. Its root is rational. The square root of any non-perfect-square is irrational.

It helps to know the perfect squares by heart: $1,\; 4,\; 9,\; 16,\; 25,\; 36,\; 49,\; 64,\; 81,\; 100,\; 121,\; 144$

🔑 Quick test for $\sqrt{n}$: If $n$ is in the list above (a perfect square), $\sqrt{n}$ is rational. Otherwise $\sqrt{n}$ is irrational.

Concept Check 🎯

Rational or Irrational? 🔽

Label each value.

Evaluate the Perfect Squares 🧮

Each of these IS a perfect square, so its root is a whole number. Find each value.

1) $\sqrt{49} = \,?$ 2) $\sqrt{81} = \,?$ 3) $\sqrt{144} = \,?$

🔢 Rational and Irrational Numbers

Part 3 of 5 — Turning Repeating Decimals into Fractions

🔑 Why this matters: Every repeating decimal is rational, which means it must equal some fraction. This part shows the algebra trick that finds that fraction every time.

The "Let $x$ Equal It" Method

To convert a repeating decimal to a fraction:

1. Let $x$ equal the decimal.
2. Multiply by $10$, $100$, or $1000$ — enough to slide the repeating block left by one full block.
3. Subtract the original equation to cancel the repeating tail.
4. Solve for $x$ and simplify.

Worked Example: $0.\overline{7}$ (one repeating digit)

$x = 0.7777\ldots$

Multiply by $10$ (one repeating digit → one zero): $10x = 7.7777\ldots$

Subtract the first equation from the second: $10x - x = 7.7777\ldots - 0.7777\ldots \;\Rightarrow\; 9x = 7$

$x = \frac{7}{9}$

✅ Check: $7 \div 9 = 0.7777\ldots = 0.\overline{7}$ ✓

Worked Example: $0.\overline{36}$ (two repeating digits)

$x = 0.363636\ldots$

There are two repeating digits, so multiply by $100$: $100x = 36.363636\ldots$

Subtract: $100x - x = 36.363636\ldots - 0.363636\ldots \;\Rightarrow\; 99x = 36$

$x = \frac{36}{99} = \frac{4}{11}$

💡 Shortcut pattern: For a decimal made of only a repeating block, the fraction is (the block) over as many $9$s as there are repeating digits. $0.\overline{7} = \dfrac{7}{9}$, $\;0.\overline{36} = \dfrac{36}{99} = \dfrac{4}{11}$, $\;0.\overline{452} = \dfrac{452}{999}$.

Order the Steps 🔽

You're converting $0.\overline{5}$ to a fraction. Choose what happens at each stage.

Convert to a Fraction 🧮

Use the method (or the $9$s shortcut). Give each answer in lowest terms (type like 2/3).

1) $0.\overline{8} = \,?$ 2) $0.\overline{45} = \,?$ (reduce it!) 3) $0.\overline{1} = \,?$

Concept Check 🎯

🔢 Rational and Irrational Numbers

Part 4 of 5 — Estimating & Ordering on a Number Line

🔑 Big Payoff: You can't write an irrational like $\sqrt{2}$ as an exact decimal, but you can trap it between two whole numbers and place it on a number line. That's how we compare and order real numbers.

Trapping a Square Root Between Whole Numbers

To estimate $\sqrt{n}$, find the two nearest perfect squares — one below $n$ and one above.

Example: Estimate $\sqrt{50}$

The perfect squares around $50$ are $49$ and $64$: $49 < 50 < 64 \;\Rightarrow\; \sqrt{49} < \sqrt{50} < \sqrt{64} \;\Rightarrow\; 7 < \sqrt{50} < 8$

Since $50$ is very close to $49$, $\sqrt{50}$ is just above $7$ (in fact $\sqrt{50} \approx 7.07$).

Example: Estimate $\sqrt{20}$

$16 < 20 < 25 \;\Rightarrow\; 4 < \sqrt{20} < 5$

$20$ is closer to $16$ than to $25$, so $\sqrt{20} \approx 4.5$ (actually $4.47$).

💡 To pick which whole number it's nearer to: compare $n$ to the two perfect squares. Closer to the lower square → root is nearer the lower whole number.

Trap the Root 🧮

Each $\sqrt{n}$ falls between two consecutive whole numbers. Enter the whole number just below each root.

1) $\sqrt{40}$ is between ___ and the next whole number. Lower bound $= \,?$ 2) $\sqrt{30}$ → lower bound $= \,?$ 3) $\sqrt{8}$ → lower bound $= \,?$

Concept Check 🎯

Ordering Rationals and Irrationals Together

To order a mixed list, turn everything into a decimal estimate, then compare.

Example: Order $\;2.5, \;\sqrt{8}, \;\dfrac{7}{3}, \;\pi\;$ from least to greatest.

Reading the estimates in order: $\frac{7}{3} < 2.5 < \sqrt{8} < \pi$

⚠️ Don't guess from the symbols. $\sqrt{8}$ looks small but $\approx 2.83$, larger than $2.5$. Always convert to decimals first, then compare.

Compare Each Pair 🔽

Choose the correct symbol so each statement is true. (Estimate as decimals first.)

🔢 Rational and Irrational Numbers

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) define and spot rational numbers, (2) recognize irrationals, (3) convert repeating decimals to fractions, and (4) estimate and order roots on a number line. Let's put it all together.

Quick Reference

⚠️ Two classic traps: (1) $\pi \ne 3.14$ and $\pi \ne \frac{22}{7}$ — those are approximations. (2) A square root is only irrational when the radicand is not a perfect square: $\sqrt{9}=3$ is rational.

Mixed Practice 🎯

One More Round 🧮

1) Convert $0.\overline{4}$ to a fraction in lowest terms. $0.\overline{4} = \,?$ (type like 2/9) 2) What is the largest whole number less than $\sqrt{75}$? 3) Evaluate $\sqrt{121} = \,?$ (it's a perfect square)

Exit Quiz ✅

Answer all three to finish the lesson.