Math101 · Independent Ontario learning library
Math101learn.math101.caIntegers
Integers extend whole numbers to include negative values, opposites, and directed change.
Integers let numbers describe direction as well as size: above/below, gain/loss, and forward/backward.
The integer set
Integers are
They contain negative whole numbers, zero, and positive whole numbers. They do not include fractional or decimal parts.
Number-line order
Numbers farther right are greater. Therefore $-2>-7$ because $-2$ lies closer to zero and farther right.
Negative signs do not mean “small digit”; among negative numbers, the value with greater distance left is smaller.
Opposites and absolute value
Opposites have equal distance from zero in different directions: $5$ and $-5$. Their sum is zero.
Absolute value measures distance from zero:
Absolute value is never negative, but the expression $-|7|$ equals $-7$ because the outside negative is applied afterward.
Adding integers
For the same sign, add magnitudes and keep the sign:
For different signs, subtract the smaller magnitude from the larger and keep the sign of the number with larger magnitude:
A number line or positive/negative tiles can show why opposite pairs cancel.
Subtracting integers
Rewrite subtraction as addition of the opposite:
Do not apply a chant without first identifying which sign belongs to the operation and which belongs to a number.
Multiplying and dividing
The sign rules are:
- same signs produce a positive result;
- different signs produce a negative result.
For example,
The magnitude calculation uses familiar whole-number facts.
Multiple factors
Count negative factors. An even number of negative factors gives a positive product; an odd number gives a negative product.
For
three negatives give a negative result, and $2\cdot3\cdot5=30$, so the product is $-30$.
Order of operations
Grouping is especially important with signs:
while
The exponent applies to what is directly in its base.
Applications
Integers model temperature relative to zero, elevation relative to sea level, account gains and debts, scores above/below a baseline, and directed movement.
Define what positive and negative mean before calculating, then interpret the final sign in that context.
Common mistakes
Thinking $-8>-3$ because $8>3$. Use number-line position.
Treating subtraction of a negative as more negative. Add the opposite.
Using multiplication sign rules for addition. Addition depends on direction and magnitude.
Assuming absolute value keeps a negative sign. Distance is nonnegative.
Ignoring parentheses in powers. $-3^2$ and $(-3)^2$ differ.
Quick self-check
- Where do the numbers lie on a number line?
- Are operation signs and number signs distinguished?
- For addition, which magnitude is larger?
- For multiplication/division, are signs the same or different?
- Does absolute value represent distance?
- Does the final sign make sense in the situation?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Evaluate −7 − (−12).
- −7 − (−12) = −7 + 12
- The result is 5.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
