Ordering Integers | Moosa Academy

Comparing positive numbers is second nature. Negatives are where intuition tends to fail: $-1000$ looks enormous, yet it is smaller than $-1$ . One rule settles every case, and it works for positives and negatives alike.

Concept What the integers are

The integers are the whole numbers together with their negatives and zero — no fractions, no decimals.

Included:

\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots

Not included:

1.5

-2.3

\tfrac{1}{2}

Concept The one rule you need

On the number line, the number further right is always the greater one.

Reading left to right, the values only ever increase. To compare any two integers, ask which one sits further right.

Concept Why negatives feel backwards

Among negatives, the bigger the digits, the further left the number sits — and so the smaller it is.

-1 > -2

, because

-1

is nearer to zero.

-2 > -3

, for the same reason.

The minus sign reverses the usual comparison: a larger digit means a deeper move into negative territory.

Concept The thermometer picture

If the rule ever feels slippery, think of temperature:

-10^\circ

is colder than

-5^\circ

, so

-10 < -5

-5^\circ

is colder than

0^\circ

, so

-5 < 0

Nobody hesitates over which day was colder. That everyday judgement is exactly the mathematical ordering.

Example Comparing two negatives

Which is greater, $-5$ or $-2$ ?

Were they positive, 5 would beat 2.

But

-5

lies further left than

-2

⟹ −2 > −5

Example An extreme case

Which is greater, $-1000$ or $-1$ ?

-1000

is a very long way to the left of zero.

-1

is just one step to the left.

⟹ −1 > −1000, by a wide margin

The size of the digits is a poor guide here. Position on the line is the only thing that decides.

Example Sorting a mixed list

Put these in ascending order: $3,\ -1,\ 0,\ -5,\ 2$ .

Place each on the number line and read left to right.

The negatives come first, deepest one leading:

-5

, then

-1

Then zero, then the positives in the usual order.

⟹ −5, −1, 0, 2, 3

Note Mistakes to avoid

Judging negatives by digit size:

-5

is smaller than

-2

, not larger.

Forgetting that every negative is below zero and below every positive.

Including fractions or decimals among the integers.

Reading the number line right to left when ordering ascending.

Ignoring zero when it appears in the list.

Summary

Integers are whole numbers, their negatives and zero — no fractions.
On the number line, the number further right is greater.
Among negatives, larger digits mean a smaller value.
Every negative is less than zero and less than every positive.
Thinking of temperatures makes the ordering obvious.