Operations With Integers
Integers
Integers are whole numbers (no decimals) that can be either positive or negative (i.e, \(+2, -10, +52, -1\) ). Positive and negative integers appear in our lives everyday: the temperature, depositing or withdrawing money at the bank, walking up a hill or down a hill. You can think of numbers on a number line where \(0\) is in the middle, positive numbers are to the right and negative numbers are to the left.
Operation With Integers
We can add, subtract, multiply and divide positive and negative integers. There are a few rules we will need to follow.
When adding and subtracting integers, think about moving on the number line. Moving in the positive direction means moving to the right. Moving in the negative direction means moving to the left.
We can also simplify expressions that have multiple signs (\(+\) or \(-\)) beside each other. If the signs are the same, replace it with a \(+\) sign. If the signs are different, replace it with a \(-\) sign. You can also switch the order of a \(+\) and \(-\) sign.
| Operation | Rule | General Example | Example |
| Addition | When you add integers with the same sign, you add the numbers and keep the sign. | \((+) + (+) = (+)\) | \( (+5) + (+7) = +12\) |
| \((-) + (-) = (-)\) | \( (-3) + (-2) = -5 \) | ||
| When you add integers with different signs, you subtract the numbers and keep the sign of the bigger number. | \((+) + \large{(-)} = (-)\) | \( (+5) + (-7) \) \(-(7-5) = -2\) |
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| \(\large{(+)} + \normalsize{(-)} = \large{(+)}\) | \( (+15) + (-10)\) \( +(15-10) = +5\) |
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| Subtraction | When you subtract integers with the same signs, switch the sign of the second integer and turn the question into addition. Then use the rules for addition. | \((+) - (+) = (+) + (-)\) | \( (+5) - (+7) \) \(= (+5) + (-7) = -2\) |
| \((-) - (-) = (-) + (+) \) | \( (-15) - (-10)\) \( =(-15) + (+10) = -5\) |
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| When you subtract integers with different signs, switch the sign of the second integer and turn the question into addition. Then use the rules for addition. | \((-) - (+) = (-) + (-)\) | \( (-5) - (+7)\) \( = (-5) + (-7) = -12\) |
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| \((+) - (-) = (+) + (+) \) | \( (+15) - (-10)\) \( = (+15) + (+10) = +25\) |
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| Multiplication | When you multiply integers with the same sign, the answer is positive. | \( (+) * (+) = (+) \) | \( (+5) * (+2) = (+10) \) |
| \( (-) * (-) = (+) \) | \( (-5) * (-2) = (+10) \) | ||
| When you multiply integers with different signs, the answer is negative. | \( (-) * (+) = (-) \) | \( (-5) * (+2) = (-10) \) | |
| \( (+) * (-) = (-) \) | \( (+5) * (-2) = (-10) \) | ||
| Division | When you divide integers with the same sign, the answer is positive. | \( (+) / (+) = (+) \) | \( (+10) / (+2) = (+5) \) |
| \( (-) / (-) = (+) \) | \( (-10) / (-2) = (+5) \) | ||
| When you divide integers with different signs, the answer is negative. | \( (-) / (+) = (-) \) | \( (-10) / (+2) = (-5) \) | |
| \( (+) / (-) = (-) \) | \( (+10) / (-2) = (-5) \) |
\( 12 - (-3)\)
We are subtracting a negative number so we can rewrite as adding a positive number:
\(= (+12) + (+3)\)
Now we are adding two integers with the same sign, the answer is positive:
\(= (+15)\)
Therefore, we can evaluate the above expression as \(\boldsymbol{(+15)}\).
\(4*(-2) \)
Since we are multiplying integers with opposite signs, the answer will be negative:
\(= (+4) * (-2) \)
\(= (-8)\)
Therefore, we can evaluate the above expression as \(\boldsymbol{(-8)}\).
Order of Operations
While we are talking about operations with integers, it is helpful to remind you about order of operations. When you have an expression like \( (4 - (-3)) * 2\) there is an order that we need to follow.
Two common acrynoms are helpful for remembering the order of operations BEDMAS and PEMDAS:
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These serve as reminders for which part of the expression we should do first. One important feature is that multiplication/division and addition/subtraction happen at the same time, just in the order they occur from left to right.
We will start inside the brackets. Within the brackets, multiplication comes before subtraction:
\((-5)^2 + \cfrac{(2*3-(-5))}{4} \)
\((-5)^2 + \cfrac{(6-(-5))}{4} \)
\((-5)^2 + \cfrac{(6+(+5))}{4} \)
\((-5)^2 + \cfrac{11}{4}\)
Next, handle the exponent. Notice that a negative number squared is positive!
\((-5)(-5) + \cfrac{11}{4}\)
\(25 + \cfrac{11}{4}\)
It is usually best to leave the answer as a fraction. So instead of dividing \(11\) by \(4\), leave it and do the addition:
\(25 + \cfrac{11}{4}\)
\(\cfrac{100}{4} + \cfrac{11}{4} \)
\(\cfrac{111}{4}\)
Finally, we will convert to a mixed number:
\(27\cfrac{3}{4}\)
If you forget how to deal with fractions, check out the next lesson a review on fractions or operations with fractions.