Comparing and ordering integers

Comparing and ordering integers

Comparing and ordering integers

The rule for comparing integers is the same as for ordinary numbers — only the picture changes.

The big rule

The integer further to the right on the number line is the bigger one.

That sentence is the whole topic. Memorise it and check it against the number line for every doubt.

A few quick consequences:

  • Every positive number is bigger than every negative number. So +1 > −1000.
  • 0 is bigger than every negative number, but smaller than every positive one.
  • Among two negative numbers, the one closer to zero is bigger. So −2 > −7.

That last point is the trickiest. The numerical-looking "size" (the part after the minus) is backwards for negatives. The minus sign reverses everything.

In the picture, A is at −3 and B is at −2. B sits further to the right, so B > A, or −2 > −3.

Worked examples

Which is bigger: −5 or −9?

−5 is closer to zero (and further to the right on the line). So −5 > −9.

Order from smallest to biggest: 3, −8, 0, −1, 6.

Imagine walking along the number line from left to right. The order is:

−8, −1, 0, 3, 6

The biggest is 6 (rightmost); the smallest is −8 (leftmost). A common shortcut: list the negatives in order first (most negative → least negative), then zero, then the positives in order.

Fill in the symbol: −12 ⃞ −3.

−12 sits much further left than −3, so −12 is the smaller one: −12 < −3.

A common mistake: "the bigger number after the minus wins"

The classic error: thinking that −9 is bigger than −5 because "9 is bigger than 5". The number-line picture shows this is wrong — on the line, −9 sits to the left of −5, so it's smaller.

A helpful reframing: read negatives like temperatures or debts.

  • "I have −9 pounds" means I'm £9 in debt. "I have −5 pounds" means I'm £5 in debt. Being £5 in debt is better (richer) than being £9 in debt — so −5 > −9.
  • "−9 °C" is colder than "−5 °C". The colder temperature is the smaller number.

A puzzle

Order these from smallest to biggest: 2, −4, −10, 0, 5, −1, 8.

Negatives first (most negative → least negative): −10, −4, −1.

Then 0.

Then positives (smallest → biggest): 2, 5, 8.

Full order: −10, −4, −1, 0, 2, 5, 8.

What's next