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Integer operations

UPCAT Math Review · Number sense

Integer Operations

Signed numbers — four rules, and a number line that explains all of them.

Evaluate

−2(5 − 8)

Adding: think of it as movement.

Signs	Do	Sign of answer
the same	add the values	keep that sign
different	subtract the values	the bigger one wins

−7 + 12: different signs

12 − 7 = 5, and 12 is bigger

= 5

positive

Subtracting is adding the opposite.

Rewrite every subtraction as an addition and you only need one rule.

−8 − 5 = −8 + (−5)

same signs

= −13

add, keep the minus

5 − (−3) = 5 + 3

the opposite of −3 is +3

= 8

two minuses make a plus

−8 − 5 is not −3

Both moves are downward: eight below zero, then five further down. Treating the second minus as a subtraction that brings you back up is the classic slip.

Multiplying and dividing: count the minuses.

same signs

positive

(−3)(−4) = 12

different signs

negative

−20 ÷ 4 = −5

With several factors: an even number of minuses gives a positive, an odd number gives a negative.

(−2)(−3)(−1)

three minuses

= −6

odd, so negative

Different rules for + and ×

−3 + −4 is −7, but (−3)(−4) is +12. The “two negatives make a positive” line only applies to multiplying and dividing.

Mixed expressions.

−2(5 − 8)

brackets first

= −2(−3)

5 − 8 = −3

= 6

two minuses

5 · −13 · 8 · 12 · −5 · −6 · 6

Keep the order of operations in charge. Signs are settled inside each step, not collected up at the end.

Division and multiplication share a rank, so they go left to right: −36 ÷ (−4) × (−3) = 9 × (−3) = −27. Multiplying first gives −36 ÷ 12 = −3, a different number. The same holds with a power: 7 − 2(1 − 4)² ÷ 6 = 7 − 2(9) ÷ 6 = 7 − 18 ÷ 6 = 7 − 3 = 4.

Powers of negatives.

Expression	Value	Why
(−2)³	−8	three minuses, odd
(−2)⁴	16	four minuses, even
−2⁴	−16	only the 2 is raised
(−1)⁹⁹	−1	odd power

A negative to an even power is positive; to an odd power it stays negative. The bracket decides whether the minus is included at all.

Variations: long chains, zero, and a gap across zero.

A long chain: −15 + 8 − (−6) − 11 + 3.

= −15 + 8 + 6 − 11 + 3

turn − (−6) into + 6 first

positives: 8 + 6 + 3 = 17

collect every plus

negatives: −15 − 11 = −26

collect every minus

17 − 26 = −9

one subtraction at the end

What is different: five terms. Two piles and one subtraction beat five separate moves.

Zero in the mix: (−7)(0)(−3), 0 ÷ (−9) and −9 ÷ 0.

(−7)(0)(−3) = 0

one factor of 0 makes the product 0

0 ÷ (−9) = 0

nothing shared out is still nothing

−9 ÷ 0 is undefined

no number times 0 gives −9

What is different: zero has no sign, so counting the minuses stops mattering once a 0 is a factor.

A gap across zero: the temperature was 9°C at noon and −6°C at midnight. How far did it drop?

drop = 9 − (−6)

higher minus lower

= 9 + 6 = 15°C

9 down to 0, then 6 more

What is different: the two numbers sit on opposite sides of zero, so the distances add.

The drop is not 3°C

Writing 9 − 6 = 3 drops the minus on the −6. A gap that crosses zero is always the sum of the two distances from zero.

Signs without numbers.

If a < 0 < b, a is negative and b is positive. Each expression's sign follows from the rules above — no values needed.

Expression	Reason	Sign
ab	different signs multiplied	negative
b − a	b + (−a): positive plus positive	positive
a³	odd power of a negative	negative
a²b	a² is positive, times positive	positive
−ab	the opposite of a negative	positive

Check any row with a = −2 and b = 3: ab = −6, b − a = 5, a³ = −8, a²b = 12, −ab = 6. For “must be negative”, a row that could go either way — such as a + b — is out.

If a − b = 5 and ab = −6, find a² + b².

(a − b)² = a² − 2ab + b²

square the difference

25 = a² + b² − 2(−6) = a² + b² + 12

minus a negative is a plus

a² + b² = 25 − 12 = 13

check: a = 3, b = −2 gives 9 + 4

The sign of ab decides it

Writing 25 + 12 = 37 treats −2ab as −12 when ab is already negative. Put ab in brackets when you substitute: −2(−6) = +12, which then moves across as − 12.

On the exam

Integer items rarely test one rule alone: they chain signs, powers and brackets in one line, or ask which expression must be negative. Count the minuses in a product, put every substituted negative in brackets, and test a sign question with a = −2, b = 3.

Check: on the number line.

−7 + 12 right 12 from −7 lands on 5 ✓

−8 − 5 left 5 from −8 lands on −13 ✓

5 − (−3) right 3 from 5 lands on 8 ✓

Remember the flow

Rewrite subtraction as adding · Same signs add · Count the minuses

The + and × rules are different.

Now you try

Practice problem

Evaluate

−4² + (−4)²

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