Staying inside

The idea

Adding, subtracting or multiplying two rational numbers gives a rational number, and so does dividing one by another, as long as one does not divide by zero.

Leaving the family?

Add two natural numbers and you get a natural number. But 3 − 5 is not a natural number. Can adding, subtracting, multiplying or dividing rational numbers ever lead you outside the rational numbers?

Take any two rational numbers and subtract them. Is the answer a rational number?

• Yes, always
• No, sometimes it is not
• Not sure

Try every operation

Choose two numbers A and B. Press an operation and look at the answer.

Closed

Rational numbers are closed under addition, subtraction and multiplication: the answer is again a rational number. They are also closed under division, as long as we do not divide by zero.

Whatever two rational numbers you start with, the answer is p/q with p and q integers and q ≠ 0. The only operation that breaks this is dividing by 0.

Notes

Rational numbers are closed under addition, subtraction and multiplication, and under division as long as we do not divide by zero.

Check yourself

3/4 − 7/8 = −1/8. Is the answer a rational number?

For which of these is the answer not a rational number?

Multiply 5/2 × 4/5. The answer is a rational number that is also a whole number. What is it?

Answer: 2

20/10 = 2. A whole number is a rational number too: 2 = 2/1.

“Rational numbers are closed under division.” What does that mean?

• Yes. p = −1 and q = 8 are integers and q ≠ 0 — correct. Yes. Subtracting two rational numbers gives a rational number.
• No, because it is negative. Negative fractions are rational numbers too: p can be negative.
• No, because the denominator changed. The denominator may change. What matters is that p and q are integers and q ≠ 0.
• 5/6 × 0. 5/6 × 0 = 0, a rational number.
• 5/6 ÷ 0 — correct. Yes. We cannot divide by 0, so there is no answer at all.
• 0 ÷ 5/6. 0 ÷ 5/6 = 0. Dividing zero by a non-zero number is fine.
• You can divide by any number, including 0. Dividing by 0 is not allowed.
• You can never divide rational numbers. You can: 3/4 ÷ 1/2 = 3/2.
• Dividing one rational number by another gives a rational number, as long as we do not divide by zero — correct. Yes. The only exception is dividing by 0.