Integers: Whole Numbers, Operations and Properties Guide

Integers are the whole numbers extended with their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. They let us describe quantities below zero, such as temperatures below freezing, money owed, or floors below ground level in a building. The set is written as Z = {…, −2, −1, 0, 1, 2, …}.

Integers matter because the real world is not always positive. Without negative numbers you cannot record a loss, a debt or a drop in temperature — and subtraction finally becomes always possible, since 5 − 8 = −3 is a perfectly good integer.

Key Concepts

• Definition: integers are all positive and negative whole numbers, including zero.
• Absolute value: the distance of a number from zero, always non-negative. For example |−7| = 7 and |5| = 5.
• Addition rules: same signs add and keep the sign: (−8) + (−5) = −13; different signs subtract and take the sign of the larger: (−8) + 5 = −3.
• Subtraction rule: subtracting an integer means adding its opposite: 6 − (−4) = 6 + 4 = 10.
• Multiplication rules: same signs give positive, different signs give negative: (−3) × (−4) = 12, but (−3) × 4 = −12.
• Closure: integers are closed under addition, subtraction and multiplication, but not under division (3 ÷ 2 is not an integer).

Formulas

• Sign rules: (−a) + (−b) = −(a + b); (−a) × (−b) = ab; (−a) × b = −ab, for positive a and b.
• Subtraction as addition: a − b = a + (−b) for all integers a, b.
• Absolute value: |a| = a if a ≥ 0, and |a| = −a if a < 0.

Worked Example

Simplify (−8) + (−5) − (−3). First combine the two negatives: (−8) + (−5) = −13. Next, subtracting (−3) means adding its opposite: −13 − (−3) = −13 + 3 = −10. Check with a real situation: the morning temperature is −3°C and it rises by 7°C during the day. The new temperature is (−3) + 7 = 4°C, which makes sense — a rise of 7 degrees from 3 below zero lands 4 above zero.

Common Misconceptions

• "The product of two negative numbers is negative." It is positive: (−3) × (−4) = 12, because the two minus signs cancel.
• "0 is a positive integer." Zero is neither positive nor negative; it sits exactly between them.
• "−5 is bigger than −2." On the number line −5 lies further left, so −5 < −2. The larger-looking digit means a smaller value among negatives.

Real-World Applications

• Weather reports use integers for temperatures above and below 0°C.
• Bank accounts show deposits as positive and withdrawals or debts as negative.
• Altitudes below sea level, such as the Dead Sea shore at about −430 metres, are negative integers.

In short, integers add negatives to the whole numbers so that subtraction always works. Learn the sign rules carefully — especially that minus times minus is plus — and remember that among negative numbers, the one closer to zero is the greater.