Mathematics Year 7: Integers: negative numbers on the number line
Mathematics, Year 7 (Portugal): See negative numbers as the other side of zero, place them on the number line, order them and meet ℤ.
The other side of zero
Some situations natural numbers cannot describe: 3 degrees below zero, level 2 of a car park under the ground floor, a debt of 5 euros. For these we use negative numbers: −3, −2, −1. On the number line 0 sits in the middle, positives go to the right and negatives to the left (or up and down, like a thermometer), always with equal spacing. Zero is neither positive nor negative.
A new set: ℤ
With natural numbers, 3 − 5 has no answer and “5 degrees below zero” fits no number. So the set of integers, ℤ, adds zero and the negative of every natural number to what you know: …, −3, −2, −1, 0, 1, 2, 3, … The positive integers are the natural numbers you already use, so every natural number is an integer: ℕ sits inside ℤ. The negatives are the new part.
Six temperatures in order
Six mornings in a row, the temperature was (in °C): −7, 3, −2, 0, 5, −5. From coldest to warmest? Step 1: mark them on the number line. Negatives are left of 0: −7, −5, −2. Positives are right: 3, 5. Step 2: read from left to right, because the one further right is always greater: −7 < −5 < −2 < 0 < 3 < 5. Between −7 and −5 the smaller is −7: it is further from zero, on the cold side.
A bigger size is not a bigger number
The reasonable mistake is to think −9 is greater than −4, because 9 is greater than 4. With natural numbers, the one with the bigger size was the greater. With negatives it is the opposite: the further left of zero, the smaller. −9 is left of −4, so −9 < −4. Think of debts: owing 9 euros is worse than owing 4. Safe rule: every negative number is less than 0 and less than every positive number.
Where negatives appear
Negative numbers are all around you: temperatures below zero on a winter day, the underground levels of a car park (−1, −2, …), the balance of an account when more is spent than is held, the goal difference of a team that conceded more than it scored (−2). In each, zero is the reference: above it is positive, below it is negative, and the number line can be vertical, like a thermometer or a lift panel.
Test what you learned
A thermometer shows 4 degrees below zero. Which integer describes it, and on which side of 0 does it lie on a horizontal number line?
- a) 4, to the left of 0
- b) −4, to the left of 0
- c) −4, to the right of 0
Correct answer: b) −4, to the left of 0 — Yes: below zero is negative, and negatives sit to the left of 0, 4 units away from it.
Put in increasing order, smallest first: 2, −6, 0, −1, −10, 4.
- a) 0, −1, 2, 4, −6, −10
- b) 4, 2, 0, −1, −6, −10
- c) −1, −6, −10, 0, 2, 4
- d) −10, −6, −1, 0, 2, 4
Correct answer: d) −10, −6, −1, 0, 2, 4 — Yes: reading the number line from left to right, −10, −6, −1, then 0, then 2 and 4.
Which is greater, −9 or −4?
- a) −4, because it is further right on the number line
- b) −9, because 9 is greater than 4
- c) They are equal, because both are negative
- d) You cannot compare negative numbers
Correct answer: a) −4, because it is further right on the number line — Yes: −4 is to the right of −9 and closer to zero. Owing 4 euros is better than owing 9.
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