Positive and Negative Numbers: Calculating and Comparing

Positive, negative, and zero

Numbers can be positive, negative, or zero. Integers have no fractional part, such as −3, 0, and 5. Fractions and decimals can also be negative. On a number line, numbers farther right are greater: −0.25 is greater than −0.8. You will compare these values, calculate with them, and interpret changes such as a temperature rising from −3°C to 5°C.

Position, order, and distance

A number line turns comparison into position: a number farther right is greater. This rule explains why −2 > −7. Although 7 has the larger digit, −7 lies farther left. Between −1 and 0, −0.25 lies to the right of −0.8, so −0.25 is greater.

An opposite has the same distance from zero but the other sign. The opposite of −6 is 6. Absolute value is distance from zero, so |−6| = 6 and |6| = 6. Distance cannot be negative. Do not replace a negative number with its absolute value unless the problem actually uses absolute-value bars.

Distance from zero

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Operate with signed numbers

For addition with the same sign, add magnitudes and keep the sign: −4 + (−3) = −7. For unlike signs, subtract the smaller magnitude from the larger and keep the sign belonging to the larger magnitude: −9 + 5 = −4. Rewrite subtraction as addition of the opposite: 6 − (−2) = 6 + 2 = 8.

For multiplication and division, equal signs give a positive result and different signs give a negative result. This sign rule does not apply to addition. Estimate the direction: adding a negative should move left; subtracting a negative should move right.

Worked examples

Worked example 1: Order −3/2, −0.4, and 1/4. Convert −3/2 to −1.5. On the line, −1.5 is left of −0.4, which is left of 0.25. Therefore −3/2 < −0.4 < 1/4.

Worked example 2: Evaluate −7 − (−10) + 2. Change subtraction to addition: −7 + 10 + 2. The first two terms give 3; then 3 + 2 = 5. A movement check agrees: start at −7, move 10 right, then 2 right.

A reliable way to reason

When several signs appear, separate the job into two questions: what operation is being performed, and what direction should the answer have? Parentheses make that distinction visible. In −4−(−9), the first negative belongs to 4, the middle symbol is subtraction, and the last negative belongs to 9. Rewriting subtraction as addition of the opposite produces −4+9. Now the unlike-sign addition rule gives 5. This is safer than memorizing a phrase without seeing which sign does what.

Magnitude means size without direction. It explains why −12+7 is negative: 12 is the larger magnitude, so the negative direction wins; 12−7 gives magnitude 5. Magnitude also supports an error check. If two negative numbers are added, the result must be farther left than either starting number. If a negative is subtracted, the result must move right. For a real situation, attach meaning to zero. An elevation of −3 meters is below the reference, while a change of +5 meters is upward. The new elevation is 2 meters, not 8 meters, because the signs describe position and change rather than two unsigned distances.

Your quick summary

• Compare values by their position on the number line.
• Compare positions before comparing digit size.
• Treat subtraction as adding the opposite, then check direction.
• Absolute value is distance from zero, so it is nonnegative even when the original number is negative.