A number without an end

Erklärus showed Liri the orbital period of a distant comet: a number with infinitely many decimal places, with no repetition at all. “I know a number like that – at home some natural constants were described in exactly that way: never ending, never repeating.”

Erklärus: “Those are irrational numbers. Let’s understand where they come from – with square roots.”

√ The square root

The square root of a is the number that gives a again when multiplied by itself.

Liri: “So the reverse of squaring?”
Erklärus: “Exactly. √25 = 5, because 5 × 5 = 25. Important: the number under the root sign must never be negative.”
√ 25 number under the root (never negative!)
Examples

√4=2  ·  √9=3  ·  √16=4  ·  √25=5  ·  √36=6

√49 = ?

♾️ Irrational numbers

Liri: “What about √2? Is there a number that gives exactly 2 when multiplied by itself?”
Erklärus: “Yes, but it can’t be written as a simple fraction. √2 = 1.41421356… – infinitely many decimal places, with no pattern. Such numbers are called irrational.”
Well-known irrational numbers

√2 = 1.41421356…
π = 3.14159265…

🔢 The real numbers ℝ

The real numbers (ℝ) are all the rational AND irrational numbers together. They fill the number line without any gaps.

ℝ – Real numbers √2, π, √3 … ℚ – Rational ½, ¾, −½ … ℤ −1,−2… 0 1,2,3…(ℕ)
Erklärus: “ℕ is contained in ℤ, ℤ in ℚ, and ℚ in ℝ. The irrational numbers fill exactly the gaps that are left between the fractions.”

🧮 Liri’s root calculator – interactive

√ Calculate a square root
a =
🌌 Irrational numbers in the universe

The orbital periods of many heavenly bodies are in irrational ratios to one another – this prevents planets from pulling on each other in exactly repeating patterns and destabilising one another.

❌ Common mistake

√(−4) can not be worked out (among the real numbers)! No real number multiplied by itself gives a negative number – the number under the root sign must never be negative.

📝 Exercise 1
√81 = ?
📝 Exercise 2
Is √3 rational (1) or irrational (2)?
📝 Exercise 3
√100 = ?

Numbers without an end, finally understood

Liri now wrote the comet’s orbital period onto her star map as an irrational number – rounded, but knowing that it really never ends.

Liri: “Some numbers in the universe simply never end – and that is perfectly fine.”
“Square root: √a×√a=a, never negative. Irrational numbers: infinite decimal places with no pattern (√2, π). Real numbers ℝ = rational + irrational numbers.”
What you learned today
ℕ ⊂ ℤ ⊂ ℚ ⊂ ℝ
Real numbers fill the number line without gaps
🌟

Chapter 48 mastered!

Liri now knows square roots and the real numbers.

Erklärus’s notebook – page 48:
“Real numbers = all rational + irrational numbers. Square root: which number times itself gives the value?”
← Back to overview 48 – Real Numbers & Square Roots