The Rules of Using Positive and Negative Integers
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Whole numbers, figures that do not have fractions or decimals, are also called integers . They can have one of two values: positive or negative.
- Positive integers have values greater than zero.
- Negative integers have values less than zero.
- Zero is neither positive nor negative.
The rules of how to work with positive and negative numbers are important because you'll encounter them in daily life, such as in balancing a bank account, calculating weight, or preparing recipes.
Tips for Success
Like any subject, succeeding in mathematics takes practice and patience. Some people find numbers easier to work with than others do. Here are a few tips for working with positive and negative integers:
- Context can help you make sense of unfamiliar concepts. Try and think of a practical application like keeping score when you're practicing.
- Using a number line showing both sides of zero is very helpful to help develop the understanding of working with positive and negative numbers/integers.
- It's easier to keep track of the negative numbers if you enclose them in brackets .
Whether you're adding positives or negatives, this is the simplest calculation you can do with integers. In both cases, you're simply calculating the sum of the numbers. For example, if you're adding two positive integers, it looks like this:
- 5 + 4 = 9
If you're calculating the sum of two negative integers, it looks like this:
- (–7) + (–2) = -9
To get the sum of a negative and a positive number, use the sign of the larger number and subtract. For example:
- (–7) + 4 = –3
- 6 + (–9) = –3
- (–3) + 7 = 4
- 5 + (–3) = 2
The sign will be that of the larger number. Remember that adding a negative number is the same as subtracting a positive one.
Subtraction
The rules for subtraction are similar to those for addition. If you've got two positive integers, you subtract the smaller number from the larger one. The result will always be a positive integer:
- 5 – 3 = 2
Likewise, if you were to subtract a positive integer from a negative one, the calculation becomes a matter of addition (with the addition of a negative value):
- (–5) – 3 = –5 + (–3) = –8
If you're subtracting negatives from positives, the two negatives cancel out and it becomes addition:
- 5 – (–3) = 5 + 3 = 8
If you're subtracting a negative from another negative integer, use the sign of the larger number and subtract:
- (–5) – (–3) = (–5) + 3 = –2
- (–3) – (–5) = (–3) + 5 = 2
If you get confused, it often helps to write a positive number in an equation first and then the negative number. This can make it easier to see whether a sign change occurs.
Multiplication
Multiplying integers is fairly simple if you remember the following rule: If both integers are either positive or negative, the total will always be a positive number. For example:
- 3 x 2 = 6
- (–2) x (–8) = 16
However, if you are multiplying a positive integer and a negative one, the result will always be a negative number:
- (–3) x 4 = –12
- 3 x (–4) = –12
If you're multiplying a larger series of positive and negative numbers, you can add up how many are positive and how many are negative. The final sign will be the one in excess.
As with multiplication, the rules for dividing integers follow the same positive/negative guide. Dividing two negatives or two positives yields a positive number:
- 12 / 3 = 4
- (–12) / (–3) = 4
Dividing one negative integer and one positive integer results in a negative number:
- (–12) / 3 = –4
- 12 / (–3) = –4
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Math Problems With Positives and Negatives
How to Make a Negative Sign With the TI-84 Plus
In math, numbers can be classified as positive or negative based on their value in relation to zero and position on the number line. The symbol (-) is placed in front of negative numbers at all times. The symbol (+) may or may not be placed in front of positive numbers, and numbers without a symbol are assumed to be positive. When introduced to problems using negative numbers, a number line is a helpful tool for students to use.
Temperature
Temperature is measured with a thermometer that resembles a number line. Temperatures above zero are considered positive while those below zero are negative. Math problems with temperatures involve real world examples of temperature change. For example, on a cold day the morning temperature is -3 degrees. Ask your students to determine the temperature if it increases by 12 degrees. Students can use the thermometer, as a number line, to count up 12 degrees to see that the new temperature is +9 degrees or 9 degrees above zero.
Problems involving money are useful to reinforce the concept of positive and negative numbers. Saving or depositing money in an account is expressed as addition, and a balance above zero is a positive value. Spending or withdrawing money is expressed as subtraction, and being in debt or owing money is an example of a negative balance. A savings account starts off with a positive balance of $25. If you write a check for $35, the account will show a negative balance of -$10.
Measuring altitude involves positive and negative number applications. Mountains can be measured as above sea level with a positive number while land below sea level can be measured with negative numbers. Give students the following problem: if you are on land at 40 feet above sea level and travel to land that is 10 feet below sea level, how far did you travel? By using a number line, students can determine that they traveled 40 feet to get to sea level and another 10 to get to the distance below sea level. Adding 40 feet to 10 feet results in a total distance traveled of 50 feet.
Modeling With Chips
Students can use manipulatives to model adding and subtracting positive and negative numbers. Using a number line, red chips to model negative numbers and blue chips to model positive numbers, students can add and subtract them. For example, starting with three red chips to represent -3, students can model adding five by first getting back to zero with the three red chips, then using two blue chips. This represents that - 3 plus 5 is equal to +2.
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About the Author
Luann Voza teaches both math and science in an elementary school setting and physical education in a college setting. A former fitness-club owner, Voza has taught group fitness classes in step, aerobics, yoga, Pilates and kickboxing. As a bodybuilder, she held the title of Ms. New Jersey Lightweight Division Winner. Voza has a master's degree in exercise physiology and a doctoral degree in education.
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How to Add and Subtract Positive and Negative Numbers
Numbers can be positive or negative.
This is the Number Line :
No Sign Means Positive
If a number has no sign it usually means that it is a positive number.
Example: 5 is really +5
Play with it!
On the Number Line positive goes to the right and negative to the left.
Try the sliders below and see what happens:
Balloons and Weights
Let us think about numbers as balloons (positive) and weights (negative):
This basket has balloons and weights tied to it:
- The balloons pull up ( positive )
- And the weights drag down ( negative )
Adding a Positive Number
Adding positive numbers is just simple addition.
We can add balloons (we are adding positive value)
the basket gets pulled upwards (positive)
Example: 2 + 3 = 5
is really saying
"Positive 2 plus Positive 3 equals Positive 5"
We could write it as (+2) + (+3) = (+5)
Subtracting A Positive Number
Subtracting positive numbers is just simple subtraction.
We can take away balloons (we are subtracting positive value)
the basket gets pulled downwards (negative)
Example: 6 − 3 = 3
"Positive 6 minus Positive 3 equals Positive 3"
We could write it as (+6) − (+3) = (+3)
Adding A Negative Number
Now let's see what adding and subtracting negative numbers looks like:
We can add weights (we are adding negative values)
Example: 6 + (−3) = 3
"Positive 6 plus Negative 3 equals Positive 3"
We could write it as (+6) + (−3) = (+3)
The last two examples showed us that taking away balloons (subtracting a positive) or adding weights (adding a negative) both make the basket go down.
So these have the same result :
- (+6) − (+3) = (+3)
- (+6) + (−3) = (+3)
In other words subtracting a positive is the same as adding a negative .
Subtracting A Negative Number
Lastly, we can take away weights (we are subtracting negative values)
Example: What is 6 − (−3) ?
6−(−3) = 6 + 3 = 9
Yes indeed! Subtracting a Negative is the same as adding!
Two Negatives make a Positive
What Did We Find?
Adding a positive number is simple addition ..., positive and negative together ..., example: what is 6 − (+3) .
6−(+3) = 6 − 3 = 3
Example: What is 5 + (−7) ?
5+(−7) = 5 − 7 = −2
Subtracting a negative ...
Example: what is 14 − (−4) .
14−(−4) = 14 + 4 = 18
It can all be put into two rules :
They are "like signs" when they are like each other (in other words: the same).
So, all you have to remember is:
Two like signs become a positive sign
Two unlike signs become a negative sign
Example: What is 5+(−2) ?
+(−) are unlike signs (they are not the same), so they become a negative sign .
5+(−2) = 5 − 2 = 3
Example: What is 25−(−4) ?
−(−) are like signs, so they become a positive sign .
25−(−4) = 25+4 = 29
Starting Negative
What if we start with a negative number?
Using The Number Line can help:
Example: What is −3+(+2) ?
+(+) are like signs, so they become a positive sign .
−3+(+2) = −3 + 2
−3+(+2) = −3 + 2 = −1
Example: What is −3+(−2) ?
+(−) are unlike signs, so they become a negative sign .
−3+(−2) = −3 − 2
−3+(−2) = −3 − 2 = −5
Now Play With It!
A common sense explanation.
And there is a "common sense" explanation:
If I say "Eat!" I am encouraging you to eat (positive)
If I say "Do not eat!" I am saying the opposite (negative).
Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).
So, two negatives make a positive, and if that satisfies you, then you are done!
Another Common Sense Explanation
A friend is + , an enemy is −
A Bank Example
Example: last year the bank subtracted $10 from your account by mistake, and they want to fix it..
So the bank must take away a negative $10 .
Let's say your current balance is $80, so you will then have:
$80−(−$10) = $80 + $10 = $90
So you get $10 more in your account.
A Long Example You Might Like
Ally's points.
Ally can be naughty or nice. So Ally's parents have said
"If you are nice we will add 3 points (+3). If you are naughty, we take away 3 points (−3). When you reach 30 Points you get a toy."
So when we subtract a negative, we gain points (i.e. the same as adding points).
See: both " 15 − (+3) " and " 15 + (−3) " result in 12.
It doesn't matter if you subtract positive points or add negative points, you still end up losing points.
Try These Exercises ...
Now try This Worksheet , and see how you go.
And also try these questions:
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Basic Rules for Positive and Negative Numbers
- DESCRIPTION rules for adding and subtracting two numbers positive and negative
- SOURCE Created by Karina Goto for YourDictionary
- PERMISSION Owned by YourDictionary, Copyright YourDictionary
Numbers higher than zero are called positive numbers, and numbers lower than zero are negative numbers. That means they fall at either side of the number line. However, just because they're on the same line doesn't mean they follow the same rules! Keep reading for a list of the basic rules for using positive and negative numbers in math.
Rules for Signed Numbers
When using positive and negative numbers, you use the rules for signed numbers (numbers with positive or negative signs in front of them). Also known as operations for signed numbers, these steps can help you avoid confusion and solve math problems as quickly — and correctly — as possible.
Follow these rules to determine the best way to add, subtract, multiply, and divide positive and negative numbers. Remember, if there is no + or - sign, the number is positive.
Addition: Same Signs, Add the Numbers
When you're adding two numbers together and they have the same sign (two positive or two negative numbers), add the numbers and keep the sign. For example:
- 51 + 32 = 83
- -14 + (-6) = -20
- -196 + (-71) = -267
Notice that equations with two positive numbers have positive sums, and equations with two negative numbers have negative sums. If you're using a number line to solve the problem, adding two positive numbers will go farther to the positive side, and adding two negative numbers will go farther on the negative side.
Addition: Different Signs, Subtract the Numbers
If you're adding positive and negative numbers together, subtract the smaller number from the larger one and use the sign from the larger number. For example:
- 6 + (-5) = 1
- -17 + 22 = 5
- -100 + 54 = -45
- 299 + (-1) = 298
As you can see, adding numbers with different signs is really a form of subtraction. When using a number line, your sum will end up closer to zero.
Subtraction: Switch to Addition
Subtracting positive and negative numbers means that you add the opposite numbers, or additive inverse . Change the subtraction sign to addition and change the sign that follows to its opposite. Then follow the steps for addition. For example:
- -3 - (+5) becomes -3 + (-5) = -8
- 9 - (-7) becomes 9 + (+7) = 16
- -14 - (+8) becomes -14 + (-8) = -22
- 25 - (-90) becomes 25 + (+90) = 115
A good tip is that whenever you see a negative sign and a minus sign together, such as in 9 - (-7), immediately make them positive signs. The negative signs cancel each other out, and the equation becomes an addition problem.
Multiplication and Division: Same Sign, Positive Result
It seems like multiplication and division would be more complicated than addition and subtraction, but they're actually much simpler. The rule for multiplying positive and negative numbers with the same sign (two positive or two negative) is that the product will always be positive. For example:
- (-8) x (-4) = 32
- 10 x 9 = 90
- (-10) x (-9) = 90
The same rule applies for division. When dividing a number by another number with the same sign, the quotient (answer) is positive. For example:
- -12 ÷ (-6) = 2
- 100 ÷ 5 = 20
- -100 ÷ (-5) = 20
Why does multiplying or dividing two negative numbers always equal a positive number? Like subtracting negative numbers, these operations turn the negatives into their opposite (inverse). You are essentially subtracting the negative number several times — and as seen above, subtracting negative numbers results in a positive equation.
- DESCRIPTION rules for multiplying and dividing two numbers positive and negative
- PERMISSION Owned by YourDictionary, Copyright YourDictionary
Multiplication and Division: Opposite Sign, Negative Result
When multiplying a positive and a negative, the product will always be negative. It doesn't matter what order the signs are in. For example:
- 6 x (-7) = -42
- -7 x 6 = -42
- 12 x (-11) = -132
- -11 x 12 = -132
In all of these cases, you first need to multiply or divide the numbers. Then decide whether the product or quotient is positive (two positives or two negatives in the equation) or negative (one positive and one negative in the equation).
Like and Unlike Signs in Addition and Subtraction
Another way to think about adding positive and negative numbers is to look at the signs in a row. Two like signs in a row (++ or --) mean you add the numbers, while two unlike signs in a row (+- or -+) mean that you subtract. For example:
- 7 + (+2) = 9 (++ are like signs, so the equation is addition)
- 9 + (-8) = 1 (+- are unlike signs, so the equation is subtraction)
- 11 - (+13) = 2 (-+ are unlike signs, so the equation is subtraction)
- 15 - (-10) = 25 (-- are like signs, so the equation is addition)
This method follows the same rules as above but might help you solve the problem more quickly if you prefer to work out the signs beforehand. Once you understand positive and negative numbers conceptually, you can decide which method works best for you.
Understanding Math Foundations
Once you know the basics of math and its rules, the entire mathematical world is open to you. Unlike other subjects, math isn't nuanced or up for interpretation — it just is what it is! For more math practice, check out the steps to long division problems (with examples) . You can also review the different types of numbers in math before your next math assignment.
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Integers Worksheets
Welcome to the integers worksheets page at Math-Drills.com where you may have a negative experience, but in the world of integers, that's a good thing! This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers.
If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand. Banks like you to keep negative balances in your accounts, so they can charge you loads of interest. Deep sea divers spend all sorts of time in negative integer territory. There are many reasons why a knowledge of integers is helpful even if you are not going to pursue an accounting or deep sea diving career. One hugely important reason is that there are many high school mathematics topics that will rely on a strong knowledge of integers and the rules associated with them.
We've included a few hundred integers worksheets on this page to help support your students in their pursuit of knowledge. You may also want to get one of those giant integer number lines to post if you are a teacher, or print off a few of our integer number lines. You can also project them on your whiteboard or make an overhead transparency. For homeschoolers or those with only one or a few students, the paper versions should do. The other thing that we highly recommend are integer chips a.k.a. two-color counters. Read more about them below.
Most Popular Integers Worksheets this Week
General Use Printables
General use integers printables including coordinate grid paper and number lines.
Coordinate grid paper
Number lines
Integer number lines can be used for various math activities including operations with integers, counting, comparing, ordering, etc.
Comparing & Ordering Integers Worksheets
Comparing and ordering integers worksheets for learning about ordinality in integers.
Comparing random pairs of integers
For students who are just starting with integers, it is very helpful if they can use an integer number line to compare integers and to see how the placement of integers works. They should quickly realize that negative numbers are counter-intuitive because they are probably quite used to larger absolute values meaning larger numbers. The reverse is the case, of course, with negative numbers. Students should be able to recognize easily that a positive number is always greater than a negative number and that between two negative integers, the one with the lesser absolute value is actually the greater number. Have students practice with these integers worksheets and follow up with the close proximity comparing integers worksheets.
Comparing pairs of integers in close proximity
By close proximity, we mean that the integers being compared differ very little in value. Depending on the range, we have allowed various differences between the two integers being compared. In the first set where the range is -9 to 9, the difference between the two numbers is always 1. With the largest range, a difference of up to 5 is allowed. These worksheets will help students further hone their ability to visualize and conceptualize the idea of negative numbers and will serve as a foundation for all the other worksheets on this page.
Ordering/Sorting Integers
Adding & Subtracting Integers Worksheets
Adding and subtracting integers worksheets in various ranges including a variety of options for parentheses use.
Adding integers worksheets
Have you heard about two-color counters and how they can make your life much easier while helping students understand integers better? Sure, you could just teach them the ++, +-, -+, and -- rules, but then they would have no color in their lives. Two-color counters are usually plastic chips that usually come with yellow on one side and red on the other side. They do come in other colors, so you'll have to use your own colors in our description.
Adding with two-color counters is actually quite easy. You model the first number with a pile of chips flipped to the correct side and you also model the second number with a pile of chips flipped to the correct side, then you mash them all together, take out the zeros (if any) and voila! you have your answer. Since there are a few confused faces in the audience, let us explain a little further.
When we say, the correct side, we mean use red for negative numbers and yellow for positive numbers. You would model -5 with five red chips and 7 with seven yellow chips. Mashing them together should be straight forward. Since you are adding, you put the two groups of chips together, being careful not to flip any of them in the process, of course. Taking out the zeros means removing as many pairs of yellow and red chips as you can. You do this because -1 and 1 when added together equals zero (this is called the zero principle). If you remove the zeros, you don't change the answer at all. The benefit of removing the zeros, however, is that you always end up with only one color and as a consequence, the answer to the integer question.
Vertically arranged integer addition
Scaffolded Integer Addition and Subtraction
These worksheets include groups of questions that all result in positive or negative sums or differences. They can be used to help students see more clearly how certain integer questions end up with positive and negative results. In the case of addition of negative and positive integers, some people suggest looking for the "heavier" value to determine whether the sum will be positive of negative. More technically, it would be the integer with the greater absolute value. For example, in the question (−2) + 5, the absolute value of the positive integer is greater, so the sum will be positive.
In subtraction questions, the focus is on the subtrahend (the value being subtracted). In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the subtrahend has a greater absolute value, the answer will be positive. Vice-versa for both situations. Alternatively, students can always convert subtraction questions to addition questions by changing the signs (e.g. (−5) − (−7) is the same as (−5) + 7; 3 − 5 is the same as 3 + (−5)).
Subtracting integers worksheets
Subtracting with integer chips is a little different. Integer subtraction can be thought of as removing. To subtract with integer chips, begin by modeling the first number (the minuend) with integer chips. Next, remove the chips that would represent the second number from your pile and you will have your answer. Unfortunately, that isn't all there is to it. This works beautifully if you have enough of the right color chip to remove, but often times you don't. For example, 5 - (-5), would require five yellow chips to start and would also require the removal of five red chips, but there aren't any red chips! Thank goodness, we have the zero principle. Adding or subtracting zero (a red chip and a yellow chip) has no effect on the original number, so we could add as many zeros as we wanted to the pile, and the number would still be the same. All that is needed then is to add as many zeros (pairs of red and yellow chips) as needed until there are enough of the correct color chip to remove. In our example 5 - (-5), you would add 5 zeros, so that you could remove five red chips. You would then be left with 10 yellow chips (or +10) which is the answer to the question.
Vertically arranged integer subtraction
Mixed adding and subtracting integers worksheets
Multiplying & Dividing Integers Workshets
Multiplying and dividing integers in various ranges and including worksheets that focus on specific types of integer operations.
Multiplying integers
Multiplying integers is normally where students learn the general rules for multiplying negatives and positives. Summarized, they are ++ = +; -- = +; +- = -; and -+ = -. In other words, multiplying two positives or two negatives together results in a positive products, and multiplying a negative and a positive together results in a negative product.
In order to develop a deeper understanding of these rules, it is nice to think of an example from outside of school such as a bank and its loan clients. For simplicity sake, we'll use low numbers, but the actual numbers will be greater (maybe think in terms of thousands of dollars). Let's say the bank gets 3 new loan clients and each customer borrows $5. From the bank's perspective, they have gained three customers (+3) and lost $5 from each one (-5). In total, they have lost 3 x (-5) = -$15. From the clients' perspective, they each gained $5, so they would all be in positive territory 3 x 5 = $15. If the clients all paid back their loans, the bank would lose the 3 customers
Dividing integers
Mixed Operations with Integers Worksheets
Integers worksheets with a mixture of four operations on the same page.
Mixed operations with integers worksheets
Order of operations with integers
The order of operations worksheets in this section actually reside on the Order of Operations page, but they are included here for your convenience.
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As with multiplication, the rules for dividing integers follow the same positive/negative guide. Dividing two negatives or two positives yields
learn the rules of positive and negative numbers.
The symbol (-) is placed in front of negative numbers at all times. The symbol (+) may or may not be placed in front of positive numbers, and
Play with it! On the Number Line positive goes to the right and negative to the left. Try the sliders below and see what happens: 6 + (−2) = 4. integer
To subtract signed numbers (either positive or negative), ... E. Solve the following word problems using positive and negative numbers.
If you're adding positive and negative numbers together, subtract the smaller number from the larger one and use the sign from the larger number. For example: 6
In this problem, we're investigating the number of steps we would climb up or down to get out of or into the swimming pool. How could you number the steps below
Level Four. Number and Algebra. Numeracy activities. Solve addition and subtraction problems with integers (positive and negative numbers). Resource logo
They will draw random positive or negative integer cards for each problem and use the number line and arrow as a manipulative to solve.
In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the