Mathematics Year 7: Absolute value and the opposite of a number
Mathematics, Year 7 (Portugal): Measure how far a number is from zero, and find the number on the other side at the same distance.
How far, and the other side
The absolute value of a number is the distance between the point that represents it on the number line and zero. It is written |−5| = 5, read “absolute value of minus five”. Also |5| = 5 and |0| = 0: a distance is never negative. The opposite of a number is the one at the same distance from zero on the other side: the opposite of 5 is −5, and the opposite of −5 is 5, written −(−5) = 5. The opposite of 0 is 0 itself.
Distance and direction are two facts
A thermometer at −4 °C and another at 4 °C are both 4 degrees from zero, but on opposite sides. The absolute value keeps only “how far” and the sign keeps “which side”. So |−4| = |4| = 4: a number and its opposite are always at the same distance from zero, because the line is symmetric about 0. Test it on any integer, then justify it: folding the line at 0 lays each number onto its opposite.
Distances and opposites of five numbers
Take −12, 5, −5, 0 and 9. Absolute values: |−12| = 12, |5| = 5, |−5| = 5, |0| = 0, |9| = 9. Opposites: 12, −5, 5, 0 and −9. Which have absolute value 5? Those 5 units from zero, one on each side: 5 and −5. If someone only says “absolute value 5”, there are two possible integers; the sign decides. Compare with the opposites: the opposite of 5 is −5 and the opposite of −5 is 5, so 5 and −5 are each other's opposite.
A minus in front does not always give a negative
The reasonable mistake is to think a “−” in front always gives a negative number, and so write −(−8) = −8. But −(−8) reads “the opposite of −8”, and the opposite of a negative number is positive: −(−8) = 8. Another slip is to think |5| does not exist because 5 has no minus to remove: it does, |5| = 5. And do not mix up opposite and inverse: the opposite of 8 is −8, not 1/8.
Distance without direction
Often you only care about “how much”, not “which way”. Level −3 and level 3 are both 3 floors from the ground floor; a machine that is off by −2 mm or +2 mm is wrong by 2 mm either way; a debt of 20 € and a balance of 20 € have the same absolute value, 20 €, but opposite meanings. Whenever you separate the size from the direction, you are using the absolute value and the opposite.
Test what you learned
On the number line, −7 is 7 units from 0. What is the absolute value of −7?
- a) 7, the distance from −7 to 0
- b) −7, because the absolute value keeps the sign
- c) It has none, because −7 is negative
Correct answer: a) 7, the distance from −7 to 0 — Yes: the absolute value is the distance to zero, and a distance is never negative: |−7| = 7.
Which integers have absolute value 6?
- a) 6
- b) −6
- c) 6, −6
- d) 6, −6, 0
Correct answer: c) 6, −6 — Yes: 6 units to the right of 0 is 6, and 6 units to the left is −6. Two integers, opposites of each other.
What is the opposite of −8?
- a) −8, because it already has a minus
- b) 8
- c) 1/8
Correct answer: b) 8 — Yes: 8 is at the same distance from zero as −8, on the other side. So −(−8) = 8.
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