Mathematics Year 7: What a rational number is, and the set ℚ
Mathematics, Year 7 (Portugal): A rational number is an integer over a non-zero integer, positive or negative; ℚ holds the integers and the fractions and decimals you know, now with a sign.
An integer over an integer
You already know fractions like 3/4 and integers like −2. A rational number is any number you can write as a fraction whose numerator is an integer and whose denominator is an integer other than 0. It can be positive, like 3/4, or negative, like −3/4, the opposite of 3/4. Zero is rational too (0 = 0/1), and it is neither positive nor negative. All the rational numbers together form the set ℚ.
One set that holds them all
With natural numbers alone you cannot say “it fell below zero”; with integers alone you cannot say “half a degree”. Rational numbers fix both: they measure in parts and they have a sign. And they do not replace what you know, they contain it: every natural number is an integer and every integer is rational, so ℕ ⊂ ℤ ⊂ ℚ. An integer n is rational because n = n/1, and a decimal like 0.6 is 6/10.
Turning numbers into fractions
Show that −5, 0.6 and −1.25 are rational. Step 1: −5 = −5/1, an integer over 1. Step 2: 0.6 = 6/10 = 3/5, dividing numerator and denominator by 2. Step 3: −1.25 = −125/100 and, dividing both by 25, it becomes −5/4. Step 4: where do they live? −5 is in ℤ and in ℚ; 0.6 and −5/4 are in ℚ but not in ℤ, because they lie between two consecutive integers.
“No fraction bar, so not rational”
Many people picture ℤ and ℚ as two separate boxes: “−3 is an integer, so it can't be rational”, or “rational numbers are only the ones with a fraction bar or a decimal point”. Wrong: ℤ sits inside ℚ. What counts is not how a number looks but whether you can write it as an integer over a non-zero integer: −3 = −3/1 = −6/2. Ask yourself: “can I write it as a fraction of integers?” If yes, it is rational.
Where negative rational numbers hide
Negative rational numbers show up whenever something can fall below a reference: a temperature of −3.5 °C, a diver at −7.5 m when the water surface is 0, a bank statement with a balance of −12.50 €. In each case 0 is the reference and the sign tells you on which side of it you are: −7.5 m and 7.5 m are opposite situations. The numbers are not whole because real quantities rarely are: half a degree, a quarter of a metre.
Test what you learned
What has to be true for a number to be rational?
- a) It has to be written with a decimal point or a fraction bar.
- b) It can be written as a fraction with an integer numerator and a non-zero integer denominator.
- c) It has to be positive, because a fraction counts parts of a whole.
- d) It has to lie between 0 and 1.
Correct answer: b) It can be written as a fraction with an integer numerator and a non-zero integer denominator. — Yes, that is the definition. It allows negatives too: 3/4, −3/4, 5 = 5/1 and 0.6 = 6/10 are all rational.
Which fraction, with an integer numerator and an integer denominator, is equal to the rational number −2.4?
- a) −12/5
- b) −2/4
- c) −5/12
- d) 12/5
Correct answer: a) −12/5 — Yes: −2.4 = −24/10, and dividing both by 2 gives −12/5. Check: 12 ÷ 5 = 2.4, with the minus sign.
Rita says: “−3 is not a rational number, because it has no fraction bar.” What is the right reply?
- a) She is right: a rational number always needs a fraction bar or a decimal point.
- b) She is right: −3 is an integer, and integers are not rational.
- c) She is wrong, but only the positive integers, such as 3, are rational.
- d) She is wrong: −3 = −3/1 is an integer over a non-zero integer, so it is rational (ℤ is inside ℚ).
Correct answer: d) She is wrong: −3 = −3/1 is an integer over a non-zero integer, so it is rational (ℤ is inside ℚ). — Yes. What matters is whether the number can be written as an integer over a non-zero integer, and −3 = −3/1 = −6/2 can.
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