Integers

Integers extend whole numbers to include negative values, opposites, and directed change.

Position on a number line encodes order, sign, and distance. A horizontal number line labels consecutive integers from negative three through positive three. Curved arrows from zero point to highlighted positions on opposite sides, emphasizing magnitude as distance rather than direction.

Integers let numbers describe direction as well as size: above/below, gain/loss, and forward/backward.

The integer set

Integers are

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

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:

$$ |-7|=7,qquad|5|=5. $$

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:

$$ -4+(-9)=-13. $$

For different signs, subtract the smaller magnitude from the larger and keep the sign of the number with larger magnitude:

$$ -11+6=-5. $$

A number line or positive/negative tiles can show why opposite pairs cancel.

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?