Rational and Irrational Numbers

Grade 8 · 8.NS.A.1 · Lesson 1 of 1

In this lesson:

• Classify numbers as rational or irrational
• Understand why rational decimals always repeat
• Convert repeating decimals to exact fractions

Learning Objectives

By the end of this lesson, you should be able to:

1. Define rational and irrational numbers and classify given numbers correctly
2. Explain informally that every real number has a decimal expansion
3. Show why long division of any fraction must produce a repeating decimal
4. Convert a repeating decimal to a fraction using the algebraic subtraction technique
5. Recognize that and have decimal expansions that neither terminate nor repeat

What Numbers Do You Know?

You've worked with all of these:

• Integers: ..., −2, −1, 0, 1, 2, 3, ...
• Fractions: , ,
• Decimals: 0.25, 0.333..., −4.1717...

All of these are rational numbers.

Now try √2 on a calculator:

1.41421356237309504880168872...

Do you see any repeating block?

Rational Numbers

A number is rational if it can be written as where and are integers and

All of these are rational:

• Integers: , ,
• Fractions: ,
• Terminating:
• Repeating:

Two Kinds of Real Numbers

Every real number is one or the other — never both

Irrational Numbers

A number is irrational if it cannot be written as — for any integers and

Key examples:

The decimal expansion is infinite and non-repeating

"Irrational" means "not a ratio" — the prefix ir negates rational

Classification Examples

The key test: Can it be expressed as ?

Quick Check: Rational or Irrational?

Classify each — think before the next slide:

Which two are the trickiest?

Answers

1. → Rational (terminates)
2. → Rational (perfect square)
3. → Irrational (non-repeating)
4. → Rational (it's already a fraction!)
5. → Irrational (≠ 22/7)

— close to , but not equal to

Every Real Number Has a Decimal Expansion

Rational = terminating or repeating
Irrational = neither terminates nor repeats

Long Division: 3 ÷ 8

Divide 3.000 by 8 — tracking each remainder:

Remainder = 0 means the decimal terminates

Long Division: 1 ÷ 7

Repeating block: 142857 →

Why Every Fraction Must Repeat

When you divide by :

• Possible remainders: — only choices
• After at most steps, a remainder must repeat
• When a remainder repeats, the decimal pattern repeats

Every Fraction Produces a Repeating Decimal

The general rule: For any fraction :

• At most steps before a remainder repeats
• Maximum repeating block length:
• If remainder 0 appears first → terminates

Irrationals: No Cycle, Ever

If had a repeating decimal, it would mean:

But cannot be expressed as any fraction — proven by mathematicians

Therefore its decimal never repeats:

Proof that is irrational uses proof by contradiction — you'll see it in high school

Try It Yourself

Long divide : record the remainder at each step

At which step does a remainder first repeat?
What is the repeating block?

Quick Check: Decimal Expansion Type

Classify the decimal expansion of each:

Terminating · Repeating · Non-repeating (irrational)

For #4: What is the maximum possible block length before the repeat is guaranteed?

Answers

1. → Terminating (rational)
2. → Repeating (rational)
3. → Non-repeating (irrational)
4. → Repeating — period 6, max possible 12

Terminating decimals are rational — they repeat with block

The Reverse Direction: Decimal → Fraction

So far: fraction → decimal (by long division)

Now: repeating decimal → fraction (by algebra)

The three-step technique:

1. Set the decimal equal to
2. Multiply to align the repeating tails
3. Subtract to eliminate the infinite tail — solve for

If a decimal repeats, it must equal an exact fraction

Example 1: Single-Digit Repeating Block

Multiply by 10 (block length = 1, so ):

Subtract:

Check: 5 ÷ 9 = 0.555... ✓

Choosing the Right Multiplier

Rule: If the repeating block has digits, multiply by

Why: Shifting exactly one full block aligns the repeating tails for subtraction

Using for a 2-digit block leaves the tails misaligned — the subtraction will not produce a whole number

Example 2: Two-Digit Repeating Block

Block "27" has length 2, so multiply by :

Subtract:

Check: 3 ÷ 11 = 0.272727... ✓

Quick Check: Which Multiplier?

For each repeating decimal, choose the correct multiplier:

1. — block is "7"
2. — block is "36"
3. — block is "142"

· ·

The multiplier matches the block length, not the number of decimal places shown

When Initial Digits Don't Repeat

For — only "16" repeats; the 4 does not

Strategy: Use two multiplications to isolate the repeating tail

1. Multiply by 10 to shift past the non-repeating digit:
2. Multiply by 1000 to shift one full block further:
3. Subtract: — tails cancel

Choose multipliers so the repeating tails are identical in both equations

Example 3: Non-Repeating Then Repeating

Subtract:

Check: 5 ÷ 12 = 0.41666... ✓

Your Turn

Convert these repeating decimals to exact fractions:

For each:

• Write the decimal
• Choose the multiplier ( for block length )
• Subtract and solve for
• Verify by long division

Try both before the next slide

Answers

1. — multiply by 10:

2. — multiply by 100:

Check: 8 ÷ 9 = 0.888...; 4 ÷ 33 = 0.121212... ✓

Challenge: try 0.3166... where only the 6 repeats

The Three-Step Procedure

Every repeating decimal equals an exact fraction — confirming it is rational

Key Takeaways

✓ Every real number is either rational or irrational — never both
✓ Rational decimals terminate or repeat; irrational decimals do neither
✓ Every fraction must repeat within steps (pigeonhole)
✓ Every repeating decimal converts to an exact fraction (algebraic subtraction)
✓ Irrational numbers are real and exact — the decimal is our imperfect notation

Watch out: A long decimal is not automatically irrational — check for a repeating block
Watch out: Terminating decimals ARE rational — they repeat with block
Watch out: Multiply by where = block length (not always ×10)
Watch out: Irrational numbers are not approximations — they are exact values

What's Next: 8.NS.A.2

You can now classify any real number as rational or irrational.

Next lesson: 8.NS.A.2 — Approximating Irrationals

• Locate irrational numbers on a number line
• Compare irrational and rational numbers using decimal approximations
• Find rational approximations of , , ,

Everything in 8.NS.A.2 depends on the framework you built today