The Real Number System

The Number That Broke Ancient Math

The ancient Greek mathematician Hippasus reportedly discovered something terrifying: the diagonal of a square with side length 1 could not be expressed as a fraction. According to legend, his fellow Pythagoreans were so disturbed by this discovery that they threw him off a ship.

Whether or not that story is true, the mathematical discovery is real. The length of that diagonal is

$$\sqrt{2}$$

, and it cannot be written as any fraction — no matter how hard you try. Numbers like this are called irrational, and today you will learn how to work with them.

The Number System: A Nested Family

Think of the real number system like a set of nesting boxes:

• Natural numbers (counting numbers): 1, 2, 3, 4, ...
• Whole numbers: 0, 1, 2, 3, ... (natural numbers plus zero)
• Integers: ..., -3, -2, -1, 0, 1, 2, 3, ... (whole numbers plus negatives)
• Rational numbers: any number expressible as a fraction p/q where p and q are integers and q ≠ 0
• Irrational numbers: real numbers that cannot be expressed as fractions
• Real numbers: all rational and irrational numbers combined

Every natural number is also a whole number. Every whole number is also an integer. Every integer is also a rational number (since 5 = 5/1). But irrational numbers stand apart — they share the number line with rationals but belong to their own category.

Rational Numbers: Terminating and Repeating Decimals

Every rational number, when written as a decimal, either terminates (stops) or repeats in a pattern.

Terminating decimals:

• 3/4 = 0.75 (stops)
• 7/8 = 0.875 (stops)
• 1/5 = 0.2 (stops)

Repeating decimals:

• 1/3 = 0.333... = 0.3̄
• 2/7 = 0.285714285714... = 0.285714̄
• 5/6 = 0.8333... = 0.83̄

The key insight: if you can write a number as a fraction of two integers, its decimal will eventually either stop or repeat. This is a mathematical fact, not a coincidence.

Irrational Numbers: Non-Terminating, Non-Repeating

Irrational numbers are the opposite: their decimal expansions go on forever with no repeating pattern.

• $$\sqrt{2}$$

  = 1.41421356237...

• $$\sqrt{3}$$

  = 1.73205080757...

• $$\pi$$

  = 3.14159265358...

No matter how many digits you write, the pattern never repeats. That is the defining characteristic.

How do you know a number is irrational?

1. If it is a square root of a non-perfect-square integer, it is irrational.
2. Pi (

   $$\pi$$

   ) and e are irrational (you will prove this in later math courses).
3. If the decimal never terminates and you can find no repeating block, it is irrational.

Common perfect squares to memorize: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

If a number is the square root of any of these, it is rational (and in fact a whole number). If not — like

$$\sqrt{7}$$

or

$$\sqrt{50}$$

— it is irrational.

❓

Concept Check

Classify each number as rational or irrational: (a) 0.75, (b) √16, (c) √5, (d) 0.121212..., (e) π

▸

(a) Rational — it is 3/4, a fraction.
(b) Rational — √16 = 4, a whole number.
(c) Irrational — 5 is not a perfect square, so √5 cannot be expressed as a fraction.
(d) Rational — it repeats the block '12', so it equals 12/99 = 4/33.
(e) Irrational — pi's decimal never terminates or repeats.

Approximating Irrational Numbers

You cannot write

$$\sqrt{7}$$

exactly, but you can get very close. Here is how to find its location on the number line.

Step 1: Find the perfect squares on either side.

$$\sqrt{4}=2$$

and

$$\sqrt{9}=3$$

, so

$$\sqrt{7}$$

is between 2 and 3.

Step 2: Narrow it down to tenths.

Try 2.6:

$$2.6^2=6.76$$

(too low) Try 2.7:

$$2.7^2=7.29$$

(too high)

So

$$\sqrt{7}$$

is between 2.6 and 2.7.

Step 3: Narrow to hundredths.

Try 2.64:

$$2.64^2=6.9696$$

(too low) Try 2.65:

$$2.65^2=7.0225$$

(too high)

So

$$\sqrt{7}\approx 2.64$$

(to the nearest hundredth).

The actual value is 2.6457513... — your approximation of 2.64 is only 0.006 off.

Ordering Real Numbers

To compare and order a mix of rational and irrational numbers, convert everything to decimal form.

Order from least to greatest:

$$\sqrt{5},2.1,\frac{9}{4},\sqrt{3}$$

Convert each:

• $$\sqrt{5}\approx 2.236$$

• 2.1 = 2.1 exactly
• 9/4 = 2.25 exactly

• $$\sqrt{3}\approx 1.732$$

Order:

$$\sqrt{3}\approx 1.732<2.1<2.25<2.236$$

Wait — let's recheck: 2.236 > 2.25? No: 2.236 > 2.25 is false. 2.236 < 2.25? Also false. 2.236 > 2.25 means the third decimal place matters: 2.236 vs 2.250. Since 3 < 5, we have 2.236 < 2.25.

Correct order:

$$\sqrt{3}<2.1<\sqrt{5}<\frac{9}{4}$$

Wait, let's verify: 2.1 < 2.236 < 2.25, so the final order is:

$$\sqrt{3}<2.1<\sqrt{5}<\frac{9}{4}$$

❓

Concept Check

Between which two consecutive integers does √50 fall? Then approximate it to the nearest tenth.

▸

Since 7² = 49 and 8² = 64, √50 falls between 7 and 8.
Testing: 7.0² = 49, 7.1² = 50.41.
Since 7.1² = 50.41 > 50 and 7.0² = 49 < 50, we get √50 ≈ 7.1 to the nearest tenth.
(The actual value is 7.0710...)

Looking Ahead

Understanding the real number system sets the foundation for everything in 8th grade math. In Unit 2, you will use this number sense when working with very large and very small numbers in scientific notation. In Unit 9, you will see why irrational numbers like

$$\sqrt{2}$$

appear naturally when you apply the Pythagorean theorem — that Greek discovery is not just ancient history, it shows up every time you calculate a diagonal distance.