The whole number line

The idea

The rational and irrational numbers together make up the real numbers ℝ, which form the continuous number line; natural numbers lie within the integers, integers within the rationals, and irrationals are separate from rationals.

Any gaps?

We have met natural numbers, integers, rational numbers and irrational numbers. Rational numbers are dense: between any two there is another. But do they fill the whole line?

Do the rational numbers fill the number line completely?

• Yes, with no gaps
• No, there are gaps
• Not sure

Sort the numbers

Each number goes into the smallest ring it belongs to. The irrational numbers sit outside the rational ring.

ℝ

The rational and irrational numbers together make up the real numbers ℝ, which form the continuous number line. Natural numbers lie within the integers, integers within the rationals, and the irrationals are separate from the rationals.

Every length, every temperature, every real physical measurement has a home on this line.

Notes

The rational and irrational numbers together make up the real numbers ℝ, the continuous number line. ℕ lies within ℤ, ℤ within ℚ, and the irrationals are separate from ℚ.

Check yourself

Which set contains both 3/4 and √2?

Which statement is true?

The real numbers ℝ are made up of …

Where does the number 0 belong?

• The rational numbers ℚ. 3/4 is rational, but √2 is not.
• The integers ℤ. Neither √2 nor 3/4 is an integer.
• The real numbers ℝ — correct. Yes. ℝ holds the rational numbers and the irrational numbers together.
• Every integer is a rational number — correct. Yes. Every integer n equals n/1, so it is a ratio of integers.
• Every rational number is an integer. 3/4 is rational but it is not an integer.
• √2 is a rational number. √2 is irrational: it cannot be written as p/q.
• the rational numbers only. The rational numbers leave gaps, and the irrational numbers fill them.
• the rational and the irrational numbers together — correct. Yes. Together they form the continuous number line.
• the integers and the natural numbers. These are only small parts of the line.
• In ℕ, the natural numbers. ℕ = {1, 2, 3, …} does not contain 0.
• Among the irrational numbers. 0 = 0/1 is a ratio of integers, so it is rational.
• In ℤ: it is an integer, but not a natural number — correct. Yes. The natural numbers start at 1.