A negative number is any number less than zero. The minus sign tells you which side of zero it is on: −6, −0.4 and −3/5 are all negative. Once you connect the signs to a number line, the calculation rules become much easier to understand.
Start with a reference point
Zero separates positive and negative numbers. A sign often describes a position or change relative to a chosen reference: an elevation of −12 m is 12 m below the reference level, while a change of −12 m is a move 12 m downward. Always read what the number represents.
This lesson moves from placing and comparing numbers to calculating with them. You only need basic addition, subtraction, multiplication and division; the later examples extend those skills to fractions and decimals.
What are negative numbers?
On a horizontal number line, negatives sit to the left of 0 and positives sit to the right. The marks must be equally spaced when they represent equal numerical steps.
- Negative integers: −1, −2, −3 and so on. Integers have no fractional part.
- Other negative numbers: −0.4, −2.75 and −3/5. These are negative, but they are not integers.
- Zero: neither positive nor negative. Writing −0 still means 0.
The sign and the size both matter. −3 means “negative three”; 3 − 2 uses a minus sign to mean subtraction. In 3 − (−2), the first minus is an operation and the second belongs to the number being subtracted.
Compare numbers, find opposites and use absolute value
Ordering: farther right means greater
Although 8 is greater than 2, −8 is less than −2: −8 lies farther left. Among two negative numbers, the one closer to zero is greater.
−4 < −1 < −0.5 < 0 < 2
For example, −2.5 < −2.4 because −2.5 is farther left. For fractions, compare equivalent decimals or use a common denominator: −3/4 = −0.75 is less than −1/2 = −0.5.
Opposites: same distance, opposite sides
The opposite of 3 is −3, and the opposite of −3 is 3. A number and its opposite add to zero: 3 + (−3) = 0. The opposite of zero is zero. In symbols, the opposite of a number a is −a; this does not mean −a must be negative, because a could itself be negative.
Absolute value: distance from zero
Vertical bars mean absolute value. Because distance is nonnegative, |−3| = 3, |3| = 3 and |0| = 0. Absolute value is not the same question as “which number is greater?” −8 < −2, but |−8| > |−2|.
Try the absolute value calculator after predicting the result yourself.
How to calculate with negative numbers
1. Addition: follow the second number
Start at the first number. To add a positive number, move right; to add a negative number, move left. Moving can take you across zero.
Add the absolute values and keep the shared sign.
−4 + (−7) = −11
4 + 7 = 11
Subtract the smaller absolute value from the larger. Keep the sign of the number with the larger absolute value.
−8 + 5 = −3
−3 + 5 = 2
If the absolute values are equal and the signs differ, the numbers cancel: −5 + 5 = 0. Notice that two negative addends give a negative sum, not a positive one.
2. Subtraction: add the opposite
Keep the first number. Change subtraction to addition, then replace the second number with its opposite:
a − b = a + (−b)
4 − 7 = 4 + (−7) = −3
4 − (−7) = 4 + 7 = 11
−4 − 7 = −4 + (−7) = −11
−4 − (−7) = −4 + 7 = 3
Subtracting a negative moves right. It does not guarantee a positive final answer: −9 − (−2) = −7.
3. Multiplication: work out the sign and the size
For two nonzero factors, the same signs give a positive product and different signs give a negative product. Multiply their absolute values to find the size.
- Positive × positive: 3 × 4 = 12
- Negative × positive: (−3) × 4 = −12
- Positive × negative: 3 × (−4) = −12
- Negative × negative: (−3) × (−4) = 12
Any number multiplied by zero is zero.
Why is a negative times a negative positive?
The sign rule keeps multiplication consistent with the distributive property: multiplying a sum gives the sum of the products. Since 4 + (−4) = 0, multiplying that sum by −3 must give zero.
(−3) × [4 + (−4)] = 0
−12 + (−3) × (−4) = 0
The missing product must be +12 to cancel −12. So (−3) × (−4) = 12.
The same argument works with any positive numbers a and b: distributing (−a) × [b + (−b)] gives −ab + (−a)(−b) = 0, so (−a)(−b) = ab.
4. Division: use multiplication to check
For nonzero numbers, matching signs give a positive quotient and different signs give a negative quotient. Divide the absolute values, then apply the sign.
−20 ÷ 4 = −5
20 ÷ (−4) = −5
−20 ÷ (−4) = 5
The last answer checks because 5 × (−4) = −20. Zero divided by a nonzero number is zero. Division by zero is undefined, including 0 ÷ 0.
Negative numbers in everyday situations
- Temperature: −4 °C is four degrees below zero Celsius. A negative temperature means below zero on the stated scale, not simply “below freezing.” For example, 20 °F is positive even though it is below water’s usual freezing point of 32 °F.
- Balances: in a simplified account model, a balance of −£25 followed by a £40 deposit becomes −25 + 40 = £15. The sign shows which side of a zero balance you are on.
- Height: a diver at −6 m relative to the water surface who rises 2 m reaches −4 m. Rising does not necessarily mean reaching a positive position.
- Change: a temperature moving from 3 °C to −2 °C has change −2 − 3 = −5 °C. That is a decrease of 5 °C.
Decide what zero and the positive direction mean before writing the calculation. “Below 7” is not the definition of negative; a number must be below zero.
Common mistakes, brackets and powers
- Applying the multiplication rule to addition: (−3) × (−4) = 12, but −3 + (−4) = −7.
- Changing the first number during subtraction: −5 − (−2) becomes −5 + 2, not 5 + 2.
- Confusing order with size: −10 is less than −2, although it is farther from zero.
- Dropping a sign in a fraction: −3/4 means −(3/4). Keep the negative attached to the whole value.
Why −3² and (−3)² have different answers
An exponent applies to its base. Brackets tell you whether the minus sign is part of that base.
−32 = −(3 × 3) = −9
Square 3, then take its opposite.
(−3)2 = (−3) × (−3) = 9
The whole negative number is squared.
A third power has three factors: (−2)3 = (−2) × (−2) × (−2) = −8. In mixed calculations, do brackets and powers first, then multiplication/division from left to right, then addition/subtraction from left to right.
Worked examples with fractions and decimals
Example 1: mixed-sign decimal addition
Calculate −2.8 + 1.5.
- The signs differ, so compare absolute values: 2.8 > 1.5.
- Subtract: 2.8 − 1.5 = 1.3.
- The larger absolute value belonged to the negative number.
−2.8 + 1.5 = −1.3
Example 2: subtract a negative fraction
Calculate −3/4 − (−1/2).
- Add the opposite: −3/4 + 1/2.
- Use quarters: −3/4 + 2/4.
- Add the numerators and keep the denominator: (−3 + 2)/4.
−3/4 − (−1/2) = −1/4
Example 3: divide negative fractions
Calculate (−2/3) ÷ (−4/5).
- Two negative values give a positive quotient.
- Divide the absolute values by multiplying by the reciprocal: (2/3) × (5/4).
- Simplify 10/12 to 5/6.
(−2/3) ÷ (−4/5) = 5/6
Example 4: keep operations in order
Calculate 6 − 2 × (−3).
- Do the multiplication first: 2 × (−3) = −6.
- Subtract the negative result: 6 − (−6) = 6 + 6.
6 − 2 × (−3) = 12
Practice: 12 questions with answers
Try these without looking at the answers. For any wrong answer, identify whether the mistake involved the sign, the size or the order of operations.
- Which are negative: −6, −0.4, 0, 5/8, −3/5?
- Put −2.4, −5/2, −0.3 and 0 in increasing order.
- Insert < or >: −7 ___ −3. Which has the greater absolute value?
- Find the opposite of −0.75, then |−0.75| and |0|.
- Calculate −8 + (−5).
- Calculate −3.6 + 5.1.
- Calculate −3/4 − (−1/2).
- The temperature is −4 °C. It rises 9 °C and then falls 3 °C. What is the final temperature?
- Calculate (−2/3) × (−9/4).
- Calculate −7.2 ÷ 0.6.
- Calculate −18 ÷ (−3) × 2 − 5.
- Evaluate −32, (−3)2 and (−2)3.
Show answers and explanations
- −6, −0.4 and −3/5. Each is less than zero.
- −5/2 < −2.4 < −0.3 < 0. Start by writing −5/2 as −2.5.
- −7 < −3. The number −7 has the greater absolute value: 7 > 3.
- 0.75; 0.75; 0. The opposite reverses the sign; absolute value is distance.
- −13. Add 8 and 5, then keep the negative sign.
- 1.5. Subtract 3.6 from 5.1; the larger absolute value belongs to the positive number.
- −1/4. Rewrite as −3/4 + 2/4.
- 2 °C. −4 + 9 − 3 = 5 − 3 = 2.
- 3/2, or 1.5. The product is positive, and 18/12 simplifies to 3/2.
- −12. Different signs give a negative quotient; (−12) × 0.6 = −7.2.
- 7. Work division/multiplication left to right: 6 × 2 − 5 = 12 − 5.
- −9, 9 and −8. The first minus is outside the power. The other two expressions have negative bases.
Frequently asked questions
Are all negative numbers integers?
No. −5 is an integer, but −0.4 and −3/5 are not. All three are negative because they are below zero.
Does subtracting a negative always give a positive answer?
No. It increases the first number, but the result can still be negative: −9 − (−2) = −7. Rewriting subtraction as addition of the opposite tells you what to do.
Can absolute value be negative?
No. It is distance from zero. It can be zero, so “nonnegative” is more precise than “always positive.”
What happens to an inequality when multiplying by a negative?
If you multiply or divide both sides by the same negative number, reverse the inequality sign. For example, 2 < 5 becomes −2 > −5 after multiplying both sides by −1. The order reverses because the number line is reflected across zero.
Is there a smallest negative number?
No. Subtracting 1 from any negative number gives a smaller one. Also, −1 is the greatest negative integer, but it is not the greatest negative number: −0.5, for example, is greater than −1.
Keep the operation in view
Use the number line for position, order and addition. Use opposites for subtraction, distance for absolute value, and matching or different signs for nonzero multiplication and division. Brackets show exactly which number a power acts on.
When you can explain your answers as well as calculate them, continue with the algebra worksheets and worked examples, where negative numbers appear in expressions and equations.
Sources & References
This original lesson and practice set were checked against the following educational references.
- OpenStax: Introduction to integers — number lines, order, opposites and absolute value.
- Common Core: Grade 6, The Number System — rational numbers, reference points and magnitude.
- Common Core: Grade 7, The Number System — operations on positive and negative rational numbers.
- Illustrative Mathematics: Why is a negative times a negative always positive? — the distributive-property explanation.
- OpenStax: Integers and order of operations — signed arithmetic and negative bases in powers.
- NIST: Temperature units — the Celsius and Fahrenheit scales and water’s freezing point.
Teaching resources and downloadable worksheets
The original resource collection below is retained with its author credits. Links open in a new tab; some lead directly to downloads. Older Textease, PowerPoint and Smart Notebook files may need compatible software. “Order the Negative Numbers” links to the publisher’s resource page rather than a direct program file.
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