Negative numbers and absolute value

What does a value below zero mean?

A thermometer reads −3 °C in the morning and 4 °C later. How can the first reading be below zero, and how far apart are the readings? The minus sign tells us that the first temperature is three degrees below 0 °C. Zero is a reference on this scale; it does not mean that there is no temperature.

A survey could record a point three metres below a chosen reference level as −3 m, and a point four metres above it as +4 m. State the reference, positive direction and unit. A negative value describes a position or quantity in that context. It does not mean that something is automatically bad, missing or impossible.

Extend the line with integers

Keep the same unit spacing on both sides of zero. Moving left from zero gives −1, −2, −3 and so on; moving right gives 1, 2, 3 and so on. The integers are these positive numbers, their negative counterparts, and zero: …, −3, −2, −1, 0, 1, 2, 3, …. The line continues in both directions beyond any displayed view.

Read −3 as “negative three” or “minus three.” The positive number 3 can also be written +3. Zero is neither positive nor negative. Locate −3 by counting three unit intervals left from zero; counting three intervals right locates +3. Read the scale first, since one visible gap need not represent one unit.

A minus sign can have different jobs

In −3, the sign belongs to the written number and tells you which side of zero it occupies. In 5 − 3, the minus sign between two numbers asks you to subtract. Saying “five minus three” describes an operation; saying “negative three” names a value.

In 5 − (−3), the first minus is the subtraction operation and the second marks the negative number being subtracted. Parentheses make that distinction visible. Read the expression as “five subtract negative three.” Identify each sign’s job before calculating; two minus marks in an expression are not two identical instructions.

Find an opposite by reflecting across zero

The numbers −4 and 4 are opposites. Each is four units from zero, on a different side. Imagine reflecting the point at −4 across zero: it lands at 4. Reflecting again returns to −4. Opposites have the same distance from zero, while their signs differ when they are nonzero.

The opposite of 0 is 0: reflecting the origin leaves it in place. Opposites are not neighbouring integers. For example, −4 and −3 are neighbours one unit apart, but they are not opposites. Adding an opposite cancels a signed change: a four-unit step left and a four-unit step right return you to the starting position.

Compare positions, including negative positions

Values still increase to the right. Therefore −8 < −3: negative eight lies farther left. Reversing the written order gives −3 > −8. Every negative integer lies left of zero, so it is less than zero and less than every positive integer.

On the Celsius scale, −3 °C is warmer than −8 °C. Looking only at the digits eight and three would miss the signs and reverse this comparison. A number can be farther from zero without being the greater value. Ask whether you are comparing signed values or distances before choosing < or >.

Absolute value measures distance from zero

A number’s absolute value is its distance from zero on the number line. Write it with a vertical bar on each side: |−4| = 4 and |4| = 4. Both points are four units from zero. Also |0| = 0, because zero is already at the origin. Absolute value is never negative; it can be zero.

Keep the question clear. The signed values −8 and −3 satisfy −8 < −3, but their absolute values are 8 and 3, so |−8| > |−3|. Absolute value records how far from zero a point is, without recording which side. Knowing only that the absolute value is four does not distinguish −4 from 4.

Compare distances to zero and between points

In the explorer, set A = −3 and B = 4. Predict each absolute value and compare the signed values, then check the positions and distance markers. The distance from −3 to zero is three units; the distance from zero to 4 is four. Together, these make seven unit intervals between −3 and 4. Count the gaps: −3 to −2, then to −1, 0, 1, 2, 3 and 4.

The distance between the points is seven units in either direction. It is not |−3| alone or |4| alone: those measure distances to zero. Try the preset with both points at −3; their distance apart is zero although each is three units from zero. The line’s unit determines numerical distance, regardless of the drawing’s width.

Use the reflection control to make B the opposite of A. Try A = −4: B becomes 4, their absolute values match, and their distance apart is eight units. At A = 0, reflection gives B = 0. For the −8 and −3 preset, count five unit intervals between them. Adding their absolute values gives eleven, which is not their distance apart: both points are on the same side of zero. You can choose integers from −10 to 10 here; these display limits are not the ends of the number line.

Use addition for a signed change

To model addition, start at the first number. Adding a positive integer moves right by that many units; adding a negative integer moves left by its absolute value. Thus −3 + 4 = 1: start at −3 and take four unit steps right. Also 4 + (−3) = 1: start at 4 and take three unit steps left.

Adding does not always make a value greater. For −3 + (−2), move two units left from −3 to reach −5. Adding zero leaves the starting position unchanged. A point’s location and a directed change are different roles for a number; name the starting value, the change and the resulting value.

Subtraction reverses the specified change

Subtracting a number has the same effect as adding its opposite. For 4 − 3, move three units left from 4 to reach 1. For 4 − (−3), reverse the three-unit leftward change: move three units right from 4 to reach 7. The result increases here because the number subtracted is negative.

Check subtraction by adding back the number subtracted: 7 + (−3) = 4. A negative answer is possible too: −2 − 3 = −5, checked by −5 + 3 = −2. Explain the direction instead of applying a slogan about signs. Multiplication and division involving negative numbers need their own explanations; we will study them later.

Worked examples

Try it yourself

1. A point is six metres below a marked reference level. Upward is positive. Write its signed height and explain the sign.

   Show the answer and reasoning

   −6 m relative to the marked level.

   The minus sign identifies the side below the chosen zero, and metres give the unit. Six metres is the point’s distance from the reference level.

2. Put −8, 2, −3 and 0 in increasing order. Which of the two negative numbers is greater?

   Show the answer and reasoning

   −8 < −3 < 0 < 2. Of the negative numbers, −3 is greater.

   Read the positions from left to right. −3 lies right of −8 even though its distance from zero is smaller.

3. Find the opposite and absolute value of −6. Then find the opposite and absolute value of 0.

   Show the answer and reasoning

   For −6: opposite 6, absolute value 6. For 0: opposite 0, absolute value 0.

   Reflection sends −6 to 6 and leaves zero fixed. Absolute value is the distance from zero, so the zero case gives zero rather than a positive number.

4. Complete both comparisons: −7 __ −2 and |−7| __ |−2|. Explain why the symbols differ.

   Show the answer and reasoning

   −7 < −2, but |−7| > |−2|.

   −7 is farther left, making its signed value smaller. Its distance from zero is seven units, greater than the two-unit distance of −2.

5. How far apart are −2 and 3 on a unit-spaced line? What if both points are at −2?

   Show the answer and reasoning

   Five units for −2 and 3; zero units for two points both at −2.

   Count two intervals to zero and three more to 3. Coincident points have no gap between them, although each has an absolute value of two.

6. Find −2 + 5, −2 + (−3), −2 − 3 and −2 − (−3). Check the last answer by addition.

   Show the answer and reasoning

   3, −5, −5 and 1 respectively. Check: 1 + (−3) = −2.

   Adding 5 moves right five units. Adding −3 or subtracting 3 moves left three units. Subtracting −3 reverses that change, so move right three units to 1. Adding back −3 returns to −2.

What to carry forward

• A negative value needs a meaning: state the reference, direction and unit when describing a situation.
• Integers extend the number line both ways; values increase to the right and zero is neither positive nor negative.
• Opposites reflect across zero. Absolute value measures distance to zero and is always nonnegative, including |0| = 0.
• Compare signed values separately from their absolute values. Count unit intervals to find the distance between points.
• Addition follows a signed change; subtraction reverses it. Check a subtraction by adding back the number subtracted.