Mathematics I · No calculator

Operations with Integers (Signed Numbers)

Positive, negative, plus, minus, once you know a few simple sign rules, integers stop being scary.

An integer is just a whole number that can be positive, negative, or zero: numbers like 7, −3, 0, and 42. The little minus sign in front of a number simply means it sits to the left of zero, below sea level, below freezing, in the hole on a bill.

The good news: you already know how to add, subtract, multiply, and divide. The only new thing is keeping track of the sign (the plus or minus). This lesson gives you a small set of rules and a picture in your head, and then we walk through every operation step by step.

Picture a number line

The easiest way to keep signs straight for adding and subtracting is to imagine a number line. Zero sits in the middle. Positive numbers go right, negative numbers go left.

-5-4-3-2-1012345negativepositive
  • Adding a positive number means moving to the right.
  • Adding a negative number (or subtracting) means moving to the left.

Adding and subtracting integers

Here are the two rules that cover every add/subtract problem. Read them once, then watch them work in the examples.

Same signs → add, keep the sign

If both numbers are positive, or both are negative, add the amounts and keep that shared sign. Example: −4 plus −6 = −10.

Different signs → subtract, keep the bigger sign

If one is positive and one is negative, find the difference and take the sign of the number that is larger in size. Example: −9 plus 4 = −5.

One more trick that makes subtraction painless: subtracting a number is the same as adding its opposite. The two minus signs in 5 − (−3) flip a −3 into a +3:

  5 - (-3)   becomes   5 + 3  =  8

  "minus a minus" = "plus"

Worked example #1: −4 + (−6)

  • Check the signs: both are negative, that is the same-signs case.
  • Add the amounts: 4 + 6 = 10.
  • Keep the sign: both were negative, so the answer is negative.
  -4 + (-6)
  = -(4 + 6)
  = -10
Answer: −4 + (−6) = −10. On the number line you started at −4 and moved 6 more steps left, landing on −10.

Worked example #2: 7 − 12

  • Check the signs: think of it as 7 + (−12), one positive, one negative, so this is the different-signs case.
  • Subtract the amounts: 12 − 7 = 5.
  • Keep the bigger sign: 12 is larger than 7 and it was negative, so the answer is negative.
  7 - 12
  = 7 + (-12)
  = -(12 - 7)
  = -5
Answer: 7 − 12 = −5. If you have $7 and spend $12, you are $5 in the hole.

Worked example #3: −8 − (−5)

  • Flip the double minus: −(−5) becomes +5, so the problem turns into −8 + 5.
  • Check the signs: now it is one negative and one positive, different signs.
  • Subtract the amounts: 8 − 5 = 3, and the bigger number (8) was negative.
  -8 - (-5)
  = -8 + 5
  = -(8 - 5)
  = -3
Answer: −8 − (−5) = −3.

Multiplying and dividing integers

Multiplication and division are even simpler, because they share one rule. First do the multiplication or division as if both numbers were positive, then decide the sign:

Same signs → positive

Two positives or two negatives give a positive answer. Example: −6 × −3 = +18.

Different signs → negative

One positive and one negative give a negative answer. Example: −6 × 3 = −18.

A handy way to remember it: "same is happy (+), different is grumpy (−)." This single rule works for both multiplying and dividing.

Worked example #4: −7 × (−8)

  • Multiply the amounts: 7 × 8 = 56.
  • Decide the sign: both are negative, same signs, so the answer is positive.
  -7 x (-8)
  same signs  ->  positive
  = +56
Answer: −7 × (−8) = 56.

Worked example #5: −45 ÷ 9

  • Divide the amounts: 45 ÷ 9 = 5.
  • Decide the sign: one negative and one positive, different signs, so the answer is negative.
  -45 / 9
  different signs  ->  negative
  = -5
Answer: −45 ÷ 9 = −5.

A few things that trip people up

  • Add/subtract rules are different from times/divide rules. For adding, "two negatives" stay negative (−4 + −6 = −10). For multiplying, "two negatives" turn positive (−4 × −6 = +24). Keep them in two separate mental boxes.
  • Turn every subtraction into addition first. Rewriting a − b as a + (−b) means you only ever have to think about adding, which cuts down on slips.
  • Two minus signs side by side make a plus. Whenever you see −(− …), change it to a plus before you do anything else.
  • Sanity-check with the number line. If an answer feels wrong, picture starting at the first number and stepping left (for negatives) or right (for positives).

Your turn: practice problems

Grab a pen and work each one through. No peeking until you've tried.

  1. −6 + (−9)
  2. 3 − 11
  3. −10 − (−4)
  4. −8 × 7
  5. −36 ÷ (−6)
Tap to reveal the answers
  • 1. −6 + (−9) = −15, same signs, so add and keep the negative.
  • 2. 3 − 11 = −8, different signs; 11 − 3 = 8 and the larger number was negative.
  • 3. −10 − (−4) = −10 + 4 = −6, the double minus became a plus.
  • 4. −8 × 7 = −56, different signs give a negative product.
  • 5. −36 ÷ (−6) = 6, same signs give a positive quotient.

Why this matters for the CAEC

Signed numbers show up all over the no-calculator math section: temperature changes, account balances, elevation, and any time a problem says "below," "loss," or "decrease." Getting the sign rules automatic means you can focus on the actual problem instead of second-guessing a minus sign.

Want more practice like this? Our CAEC math guide and the CAEC Ready Workbook are packed with worked examples and practice questions, or start with a free math sample to test yourself.

Disclaimer

This article is a general math tutorial for study purposes. CAEC Ready is an independent study resource and is not affiliated with or endorsed by any government, ministry of education, or official CAEC testing provider.

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