Every number as p over q

The idea

A rational number is any number that can be expressed as p/q, where p and q are integers and q ≠ 0; the set ℚ includes the natural numbers, whole numbers and integers.

One family?

5 is a whole number. 3/4 is a fraction. −10 is a negative integer. Is there one family that holds all of them?

Can the whole number 5 be written as a fraction?

• Yes
• No, fractions are only for parts of a whole
• Only 5/5

Write it as p over q

Each number is on a card. Set p and q with the sliders so that p over q is that number.

ℚ holds them all

A rational number is any number that can be written as p/q, where p and q are integers and q ≠ 0. The set of rational numbers is ℚ (for *quotient*).

5 = 5/1 and −10 = −10/1. So ℚ also contains all the natural numbers, whole numbers and integers.

Notes

A rational number is a number that can be written as p/q, with p and q integers and q ≠ 0. ℚ contains the natural numbers, whole numbers and integers.

Check yourself

Which of these is not a rational number?

Is every integer a rational number?

Write 7 as a fraction with denominator 1. What is the numerator?

Answer: 7

7 = 7/1, so the numerator is 7.

In p/q, which value is not allowed?

• 5/0 — correct. Yes. The denominator q can never be 0, so 5/0 is not a number.
• −10. −10 = −10/1. Both are integers and q = 1 ≠ 0, so it is rational.
• 3/4. 3 and 4 are integers and 4 ≠ 0, so 3/4 is rational.
• No. Only fractions are rational. An integer n is the fraction n/1, so it is rational too.
• Yes. For any integer n we can write n = n/1 — correct. Yes. Both n and 1 are integers and 1 ≠ 0.
• Only the positive integers. Negative integers work too: −10 = −10/1.
• p = 0. p = 0 is allowed: 0/5 = 0, which is a rational number.
• q = 1. q = 1 is allowed: p/1 = p.
• q = 0 — correct. Yes. We cannot divide by 0, so q ≠ 0.