1.4: Subtraction of Integers
Molly goes shopping with $20. She buys a new notebook for $4 and a soda for $2. How much money does she have left?
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Khan Academy Adding/Subtracting Negative Numbers
Guidance
To subtract one signed number from another, change the problem from a subtraction problem to an addition problem, and change the sign of the number that was originally being subtracted. In other words, to subtract signed numbers simply add the opposite. Once these changes have been made, follow the rules for adding signed numbers.
The subtraction of integers can be represented with manipulatives such as color counters and algebra tiles. A number line can also be used to show the subtraction of integers.
Example A
\begin{align*}7-(-3)=? \quad 7+(+3)=?\end{align*}
The problem can be represented by using color counters. In this case, the red counters represent positive numbers.
The above representation shows the addition of 7 positive counters and 3 positive counters.
The answer is the sum of 7 and 3. The answer takes on the sign of the two digits being added and in this case the digits both have a positive value. Therefore, \begin{align*}7+(+3)=10\end{align*}
Example B
\begin{align*}4-(+6) &= ?\\ 4+(-6) &= ?\end{align*}
Change the problem to an addition problem and change the sign of the original number that was being subtracted.
The above representation shows the addition of 4 positive counters and 6 negative counters.
One positive counter and one negative counter equals zero. \begin{align*}1+(-1)=0\end{align*}
Draw a line through the counters that equal zero.
The remaining counters represent the answer. Therefore, \begin{align*}4-(+6)=-2\end{align*}. The answer is the difference between 6 and 4. The answer takes on the sign of the larger digit and in this case the six has a negative value and it is greater than 4.
Example C
This same method can be extended to adding variables. Algebra tiles can be used to represent positive and negative values.
\begin{align*}5x-(+8x) &= ?\\ 5x+(-8x) &= ?\end{align*}
Add the opposite.
The green algebra tiles represent positive \begin{align*}x\end{align*} and the white tiles represent negative \begin{align*}x\end{align*}. There are 5 positive \begin{align*}x \ tiles\end{align*} and 8 negative \begin{align*}x \ tiles\end{align*}.
The remaining algebra tiles represent the answer. There are three negative \begin{align*}x\end{align*} tiles remaining. Therefore, \begin{align*}(6x)-(+8x)=-3x\end{align*}. The answer is the difference between \begin{align*}8x\end{align*} and \begin{align*}5x\end{align*}. The answer takes on the sign of the larger digit and in this case the eight has a negative value and it is greater than 5.
Example D
\begin{align*}(-4)-(+3) &= ?\\ (-4)+(-3) &= ?\end{align*}
Add the opposite
The solution to this problem can be determined by using the number line.
Indicate the starting point of -4 by using a dot. From this point, add a -3 by moving three places to the left. You will stop at -7.
The point where you stopped is the answer to the problem. Therefore, \begin{align*}(-4)-(+3)=-7\end{align*}
The answer is the sum of 4 and 3. The answer takes on the sign of the digits being added. In this case the 4 and the 3 have negative signs. The answer will be a negative number.
From using models to subtract integers, there are two rules that become obvious. These rules are:
- When you subtract integers you change the question to an addition problem and change the sign of the original number being subtracted.
- Follow the rules for adding integers. When you add two integers with the same sign, add the numbers and use the sign of the digits being added. When you add two integers that have opposite signs, subtract the numbers and use the sign of the larger digit.
Concept Problem Revisited
Molly goes shopping with $20. She buys a new notebook for $4 and a soda for $2. You can figure out how much money she has left by subtracting.
\begin{align*}$20-$4-$2=$14\end{align*}
Vocabulary
- Integer
- All natural numbers, their opposites, and zero are integers. A number in the list ..., -3, -2, -1, 0, 1, 2, 3...
- Number Line
- A number line is a line that matches a set of points and a set of numbers one to one.
It is often used in mathematics to show mathematical computations.
Guided Practice
1. Use a model to answer the problem \begin{align*}(-2)-(-6)=?\end{align*}
2. Use the number line to determine the answer to the problem \begin{align*}7-(+5)=?\end{align*}
3. Determine the answer to \begin{align*}(-8)-(-5)=?\end{align*} and \begin{align*}(-4)-(+9)=?\end{align*} by using the rules for subtracting integers.
Answers:
1. \begin{align*}(-2)-(-6) &= ?\\ (-2)+(+6) &= ?\end{align*}
Cancel the counters that equal zero
There are 4 positive counters left. Therefore, \begin{align*}(-2)-(-6)=4\end{align*}. The answer is the difference between 6 and 2. The answer takes on the sign of the larger digit and in this case the six has a positive value and it is greater than 2.
2. \begin{align*}(7)-(+5) &=?\\ (7)+(-5) &= ?\end{align*}
You begin on 7 and move five places to the left. You stop at 2. Therefore \begin{align*}7-(+5)=2\end{align*}.
The answer is the difference between 7 and 5. The answer takes on the sign of the larger digit and in this case the seven has a positive value and it is greater than 5.
3. \begin{align*}(-8)-(-5) &= ?\\ (-8)+(+5) &= ?\end{align*}
When the problem is written as an addition question and the sign of the original number being subtracted is changed to a positive, the numbers have opposite signs. The numbers must be subtracted and the answer is a negative value since the larger digit of 8 has a negative sign.. Therefore, \begin{align*}(-8)-(-5)=-3\end{align*}
\begin{align*}(-4)-(+9) &= ?\\ (-4)+(-9) &= ?\end{align*}
When the problem is written as an addition question and the sign of the original number being subtracted is changed to a negative, the numbers have the same signs. The numbers must be added and the answer is a negative value. Therefore, \begin{align*}(-4)-(+9)=-13\end{align*}
Practice
Use color counters to represent the following subtraction problems and use that model to determine the answer.
- \begin{align*}(-9)-(-2)\end{align*}
- \begin{align*}(5)-(+8)\end{align*}
- \begin{align*}(5)-(-4)\end{align*}
- \begin{align*}(-7)-(-9)\end{align*}
- \begin{align*}(6)-(+5)\end{align*}
Use a number line to represent the following subtraction problems and use the number line to determine the answer.
- \begin{align*}(8)-(+4)\end{align*}
- \begin{align*}(-2)-(-7)\end{align*}
- \begin{align*}(3)-(+5)\end{align*}
- \begin{align*}(-6)-(-10)\end{align*}
- \begin{align*}(-4)-(-7)\end{align*}
Use the rules that you have learned for subtracting integers to answer the following problems and state the rule that you used.
- \begin{align*}(-13)-(-19)\end{align*}
- \begin{align*}(-6)-(+8)-(-12)\end{align*}
- \begin{align*}(14)-(+8)-(-6)\end{align*}
- \begin{align*}(18)-(+8)-(+3)\end{align*}
- \begin{align*}(10)-(-6)-(+4)-(+2)\end{align*}
For each of the following models, write a subtraction problem and answer the problem. (Hint: You may find it easier to write an addition problem and then to rewrite the problem as a subtraction question)
Image Attributions
Here you will learn to subtract integers by using different representations. You will learn how to subtract integers by using appropriate models and by using the number line. These methods will lead to the formation of two rules for subtracting integers.
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