What are negative numbers and how do you add, subtract and multiply them?

Negative numbers are numbers less than zero, written with a minus sign, such as −3, and they sit to the left of 0 on the number line. To add a negative number, move left; to subtract a negative number, add its opposite, so 5 − (−3) = 8. To multiply or divide, same signs give a positive answer and different signs give a negative one.

Are negative numbers whole numbers?

Negative numbers are not whole numbers. The whole numbers are 0, 1, 2, 3 and so on, with no negatives. Negative numbers such as −1, −2 and −3 belong to the integers, which are the whole numbers together with their opposites, and a negative fraction such as −1/2 is not an integer at all.

Are negative numbers rational numbers?

Negative integers and negative fractions are rational numbers. A rational number is any number that can be written as a fraction p/q of two integers with q not zero, so −3 = −3/1 and −3/4 are both rational. Negative irrational numbers also exist, such as the opposite of the square root of 2.

Are negative numbers real numbers?

Negative numbers are real numbers. The real numbers fill the whole number line, and every negative number has a point to the left of 0 on that line. Negative integers, negative fractions, negative decimals and negative irrational numbers are all real.

Why is a negative times a negative a positive?

A negative times a negative is positive because the multiplication pattern has to continue past zero. In the list −3 × 2 = −6, −3 × 1 = −3 and −3 × 0 = 0, each answer goes up by 3, so the next line must be −3 × (−1) = 3. Common Core standard 7.NS.A.2a ties the rule (−1)(−1) = 1 to the distributive property.

What is a negative minus a negative?

A negative minus a negative is found by adding the opposite of the second number. For −4 − (−6), change the subtraction of −6 into the addition of 6, so −4 + 6 = 2. For −9 − (−2), the same move gives −9 + 2 = −7, so the answer can be positive or negative depending on which number is farther from zero.

Negative Numbers: Rules for Adding and Subtracting

A seventh grader stares at 5 − (−3) and writes 2, the thermometer reads −5 °F, and the bank app shows a balance of −$30. Negative numbers sit behind all three, and the rules for adding, subtracting and multiplying negative numbers follow from one picture: the number line. Once that picture is clear, subtracting negative numbers, the sign rules and the classic "negative minus negative" question all become the same move.

What are negative numbers?

A negative number is a number less than zero, written with a minus sign in front, such as −1, −7 or −3/4. On a horizontal number line, positive numbers sit to the right of 0 and negative numbers sit to the left, so every negative number is smaller than 0 and smaller than every positive number.

Negative numbers describe amounts below a starting point. OpenStax Prealgebra uses cold weather ("degrees below zero") and elevation, where a place 1,302 feet below sea level is written −1,302 feet. A bank balance of −$30 means you owe $30.

Common Core standard 6.NS.C.5 asks sixth graders to "understand that positive and negative numbers are used together to describe quantities having opposite directions or values", and lists temperature, elevation, credits and debits as examples.

Opposites and absolute value

The opposite of a number is the number at the same distance from 0 on the other side, so the opposite of 4 is −4 and the opposite of −4 is 4. Standard 6.NS.C.6a states it with an example: "the opposite of the opposite of a number is the number itself, e.g., −(−3) = 3, and that 0 is its own opposite."

The absolute value of a number is its distance from 0, written with two bars. Both |−7| and |7| equal 7, because each is 7 steps from 0. Standard 6.NS.C.7c uses a debt: "for an account balance of −30 dollars, write |−30| = 30 to describe the size of the debt in dollars."

Ordering follows the line. A number to the right is always greater, so −3 > −8 even though 8 is larger than 3. Standard 6.NS.C.7b gives the reading "−3 °C is warmer than −7 °C."

How to add and subtract negative numbers, step by step

Adding and subtracting negative numbers means moving along the number line: adding a positive number moves right, adding a negative number moves left, and subtracting a number moves the opposite way from adding it. Standard 7.NS.A.1c names the rule: "understand subtraction of rational numbers as adding the additive inverse, p − q = p + (−q)."

Place the first number on the number line. For −4 + 6, put your finger on −4.
Turn every subtraction into an addition of the opposite. 5 − (−3) becomes 5 + 3, and 2 − 7 becomes 2 + (−7).
Move right for a positive number and left for a negative one. From −4, adding 6 means 6 steps right, which lands on 2.
Use absolute values when the numbers are large. With the same signs, add the absolute values and keep the sign: −20 + (−15) = −35. With different signs, subtract the smaller absolute value from the larger and keep the sign of the number farther from 0: −20 + 15 = −5.
Check the answer with a quick estimate. If you start below zero and add a smaller positive number, the answer must still be below zero.

A number line from −10 to 10 with two moves: a jump of 3 to the right from 5 to 8 for 5 − (−3) = 5 + 3 = 8, and a jump of 6 to the right from −4 to 2 for −4 + 6 = 2.

Subtracting a negative: the part people get wrong

Subtracting a negative number is the step most students miss, because "take away" sounds like the answer should get smaller. For 5 − (−3), rewrite it as 5 + 3, and the answer is 8, which is larger than 5.

A bank error makes the reason visible. If your balance is $5 and the bank removes a $3 charge it made by mistake, taking away −3 leaves you with $8. Negative minus negative works the same way: −4 − (−6) = −4 + 6 = 2, and −9 − (−2) = −9 + 2 = −7.

Whole-number subtraction still carries the arithmetic underneath, and the guide to subtraction with regrouping covers that layer.

The rules for negative numbers in one table

The sign rules for negative numbers fit in five rows: two for addition, one for subtraction and two for multiplication and division. Each row gives the rule, a worked example and the matching move on the number line.

Operation	Rule	Example	Number-line move
Add, same signs	Add the absolute values, keep the sign	−2 + (−3) = −5	Start at −2, move 3 left
Add, different signs	Subtract the absolute values, keep the sign of the number farther from 0	−7 + 4 = −3	Start at −7, move 4 right
Subtract a negative	Add its opposite	−4 − (−6) = 2	Start at −4, move 6 right
Multiply or divide, same signs	The answer is positive	(−3) × (−4) = 12	Mirror of 3 jumps of 4 to the left
Multiply or divide, different signs	The answer is negative	−12 ÷ 3 = −4	3 equal jumps from 0 to −12

Multiplying negative numbers takes two moves: multiply the absolute values, then choose the sign. Same signs give a positive product and different signs give a negative product, so 3 × (−4) = −12 and (−3) × (−4) = 12.

OpenStax puts division under the same rule: "Division of signed numbers follows the same rules as multiplication." So −12 ÷ 3 = −4, 12 ÷ (−3) = −4 and −12 ÷ (−3) = 4.

With more than two factors, count the negative signs. An even count gives a positive answer and an odd count gives a negative one, so (−1) × (−2) × (−3) = −6.

A two-by-two grid of sign rules for multiplication: plus times plus is plus with 3 × 4 = 12, plus times minus is minus with 3 × (−4) = −12, minus times plus is minus with (−3) × 4 = −12, and minus times minus is plus with (−3) × (−4) = 12.

Why a negative times a negative is positive

A negative times a negative is positive because the multiplication pattern has to keep going past zero. Write the times table for −3 with the second factor dropping by 1 each line:

−3 × 2 = −6
−3 × 1 = −3
−3 × 0 = 0
−3 × (−1) = 3
−3 × (−2) = 6

Each answer rises by 3 as the second factor drops by 1, so the line after 0 must be 3. Any other answer would break a pattern that holds on every other line.

Standard 7.NS.A.2a gives the formal reason: multiplication is extended to negative numbers "by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (−1)(−1) = 1." In symbols, 0 = −1 × (1 + (−1)) = −1 + (−1)(−1), so (−1)(−1) has to equal 1.

The same chain of rules sits under the order of operations, and the guide to order of operations shows where signs fit among brackets, powers and products.

Negative numbers in everyday life

Negative numbers turn up wherever a quantity can fall below a starting point. Temperature, elevators, sea level and money are the four you meet most.

Temperature: If the morning reads −5 °F and the afternoon reads 12 °F, the rise is 12 − (−5) = 12 + 5 = 17 degrees.
Elevators: Floors below the ground floor are often numbered −1 and −2. From floor 3 down to floor −2 is 3 − (−2) = 5 floors.
Money: A balance of −$30 followed by a $50 deposit gives −30 + 50 = $20.
Sea level: A diver at −40 feet who rises 15 feet ends at −40 + 15 = −25 feet.

England's national curriculum uses the same contexts: Year 6 pupils "use negative numbers in context, and calculate intervals across 0", and its guidance names temperature as the model.

Distance between two numbers

The distance between two numbers on the number line is the absolute value of their difference. Standard 7.NS.A.1c asks students to "show that the distance between two rational numbers on the number line is the absolute value of their difference." From −5 to 12 that is |12 − (−5)| = 17, the same 17 degrees as the thermometer.

How to check that negative numbers have landed

Five short questions show whether a child or an adult can work with negative numbers. Ask for the reasoning as well as the answer.

Ordering: Which is greater, −8 or −3? The answer is −3, because it lies to the right of −8.
Adding: What is −6 + 4? The answer is −2.
Subtracting a negative: What is 7 − (−5)? The answer is 12, from 7 + 5.
Sign rules: What are (−6) × (−5) and 24 ÷ (−8)? The answers are 30 and −3.
Context: The temperature falls from 4 °F to −9 °F. How far did it fall? The answer is 4 − (−9) = 13 degrees.

A two-minute test

A two-minute test of the five questions, mixed in order, tells you more than a page of one type. A student who gets the addition right but writes 2 for 7 − (−5) needs the "add the opposite" step, and a student who ranks −8 above −3 needs the number line again.

Grade 6 and grade 7 expectations around these skills are laid out in the guide to what math each grade should know. For quick practice with one-step equations such as x − 4 = 9, the solve-for-x game plays free in the browser.

Play free in your browserSolve for XGrade 6+

Common mistakes with negative numbers

Four mistakes cause most wrong answers with negative numbers, and each has a short fix.

Over-applying "two negatives make a positive": −2 + (−3) is −5, because adding a negative moves left. The positive-product rule belongs to multiplication and division only.
Ordering by digits: −8 is less than −3, even though 8 is greater than 3. Picture the number line and pick the number farther right.
Dropping the brackets: 5 − (−3) written as 5 − 3 gives 2 instead of 8. Keep the brackets until you rewrite the problem as an addition.
Losing the sign in a long problem: In −4 × 3 + 10, the product is −12, so the answer is −12 + 10 = −2. Circle each sign before you calculate.

Edge case: −3² and (−3)²

An exponent applies only to what sits directly under it, so −3² and (−3)² give different answers. −3² means −(3 × 3) = −9, while (−3)² means (−3) × (−3) = 9. The guide to exponents covers the rule and negative exponents, which are a separate idea.

Your negative numbers checklist and next steps

Picture: sketch a number line before you calculate.
Order: the number farther right is greater, so −3 > −8.
Subtract: rewrite p − q as p + (−q) before you move.
Add: same signs add and keep the sign; different signs subtract and keep the sign of the number farther from 0.
Multiply and divide: same signs give a positive answer, different signs a negative one.
Exponents: watch the brackets in −3² and (−3)².

Parents of sixth graders can find grade-level practice on the math game for 6th graders page. Adults rebuilding their number sense can read how to get better at math as an adult or try the math game for adults.

MathIt is a free app for iOS and Android with about 90 game types, adaptive difficulty that raises the level as answers come out right, daily practice and 2–5 player real-time battles, in 33 languages. Twenty-two of its games play free in the browser with no sign-up.