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Introduction to Subtraction
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Introduction

In this lesson, you will learn how to subtract. Subtraction is the opposite of addition. While addition combines like things, subtraction separates like things. In both addition and subtraction, the items you work with must be the same.


These videos illustrate the lesson material below. Watching the videos is optional.


Subtraction

Subtraction is taking an amount away from another amount. The steps outlined below will help keep you organized as you start subtracting numbers too big to do in your head.

Consider this: if you have 7 oranges, and you take away 4 apples, how many apples would you have left? This problem doesn’t make sense because you didn’t have any apples to begin with, you only had oranges. If you want to take something away from a group of oranges, you must take away oranges. You cannot take away apples because you didn’t have any.

Two glass bowls on an extremely white background. The bowl on the left contains three apples, and the bowl on the right contains four oranges.

Figure 1

Consider this instead: if you have 7 oranges and you take away 3 oranges, how many would you have left? This problem is solvable because you can take away 3 oranges from 7 oranges, which leaves you with 4 oranges.

Steps for Subtraction:


  1. Stack in columns according to place value and put the bigger number on top.
  2. Regroup when needed.
  3. Subtract in columns by place value, starting on the right and going left.
  4. The strongest wins.

The simplest way to do step 4 is to realize that the bigger number always wins. If the bigger number was negative, the answer is negative. If the bigger number was positive, the answer is positive.

Example 1
\(34-78\)

  • Stack by place value, and make sure the biggest number goes on top. For now, don’t worry about subtraction or the negative sign; the bigger number is 78.

\begin{align*} &78 \\ - &34\\ \hline \end{align*}

  • No regrouping is necessary for this equation. Regrouping happens when the top number in one place value column is smaller than the number on the bottom. In this problem, 8 is bigger than 4 and 7 is bigger than 3, therefore there is no need to regroup.
  • Subtract.

\begin{align*} &78 \\ - &34\\ \hline &44 \end{align*}

  • Determine if the answer is positive or negative.
    • To do so, look at the original equation: \(34-78\).
    • The bigger number has the negative sign, thus the answer will be negative, -44.

This equation could be written like this: \(34+(-78)=-44\)

It might be helpful to think about this in terms of a number line.

34 minus 78. Number line with zero in the middle, numbers in increments of 10 extending to the right until 50 and to the left until negative 50. Two marks on the number line indicate the starting point at 34 and the answer at negative 44. 

Figure 2

The problem begins with 34, so that is where you will begin on the number line. Add -78 by moving to the left instead of to the right, since the number is negative. When you move to the left by -78, you finish at -44.

Example 2
\(34-21\)

  • Stack the problem, with the largest number on top.

\begin{align*}3&4\\+ 2&1\\\hline\end{align*}

  • Check if this problem needs regrouping. This problem does not need to be regrouped. Subtract each place value directly.
  • Determine whether this answer is positive or negative. Remember: the bigger number always wins. In this equation, the 21 is negative and the 34 is positive, therefore the answer is positive: \(34-21=13\)

Add a Negative Number

Subtraction can be thought of as adding in the opposite direction. When you add \(1+4\), start at 1 and add 4 to the right. When you subtract, you add in the opposite direction, to the left. This concept will help you in further math equations and algebra.

A number line with zero in the middle, negative numbers to the left, and positive numbers to the right. It shows numbers from -6 to 0 and 0 to 6, and it is implied that the number line extends onward in both directions.

Figure 3

It might be helpful to write the equation as if it were an addition equation, with parentheses around the negative number. For example: \(5+(-4)=1\).

It also might be helpful to think about these kinds of problems as if they were a Celsius thermometer or temperature gauge.

A celsius thermometer shown as a vertical number line, with 40 degrees at the top and negative 40 at the bottom with increments of 10 shown in between. Negative five is marked with the equation 10 minus 15 equals negative five.

Figure 4

Imagine it is 10 degrees, but you know it’s going to get 15 degrees colder. If it is going to get 15 degrees colder, you can see on a thermometer that it will get down to -5 degrees: \(10-15 = 10+(-15) = -5\).

Subtraction and the Commutative Property of Addition

The commutative property of addition says \(5+4\) is equal to \(4+5\). The numbers can change position, but the answer will remain the same. Subtraction does not follow the same rule.

This can be illustrated with a simple example: \(5-4=1\), but \(4-5=-1\) The results aren’t the same. Subtraction does not have the same property of being able to switch the numbers.

However, you learned in the section above that subtraction can be viewed as adding a negative number. Look at the example above and see if you can make the commutative property of addition apply to the equation.

When you do this, you should rewrite equations with parentheses around the negative number to make sure you attach the negative sign to the correct number.

\(5+(-4)=1\) or \((-4)+5=1\)

This works for subtraction problems. Subtraction is the addition of a negative number, and when viewed this way, you can apply this rule to equations.

Consider another example:
\(9+(-5)=4\) or \((-5)+9=4\)

The Sign of the Strongest Number Wins

At the beginning of this lesson, you learned the steps of subtraction. The last step tells us to determine if the answer is negative or positive. You also learned that this is determined by the biggest number. If the biggest number is positive, the answer is positive. If the biggest number is negative, the answer is negative.

Practice this step by walking through a few subtraction examples.

Example 3
\(-4+9\)

In this example, you will be combining a positive and a negative number. Note that this equation can also be written as \(9+(-4)\) and \(9-4\).

Solve this problem by following the steps of subtraction.

  • Stack the numbers with the larger number on top.

\begin{align*} &9\\ - & 4\\ \hline \end{align*}

  • Subtract.

\begin{align*} &9\\ - & 4\\ \hline &5 \end{align*}

  • Find out if the answer is negative or positive.
    • Look at the original equation: \(-4+9\)
    • 4 is negative and 9 is positive. This means that our answer will be positive 5, since the strongest or biggest number is positive.

Example 4
\(-10 + 3\) or \(3 + (-10)\)

The sign of the strongest number wins. The answer will be negative because 10 is stronger than 3. Therefore, \(-10 + 3 = -7\)

Example 5
\(8-37\)

Again, you are dealing with a positive and a negative number. 8 is positive and 37 is negative. It is helpful to note other ways that you could write this problem: \(8+(-37)\) or \(-37+8\).

  • Stack the numbers. The biggest number will go on top, and the smallest number will go on the bottom. Remember, during this step, don’t consider whether the number is positive or negative.

\begin{align*} 3&7\\ - &8\\ \hline \end{align*}

  • Since 7 is smaller than 8, this problem requires regrouping. Borrow from the 3—turning it into a 2—to make this problem solvable. This will add 10 ones to the 7, which will equal 17.
m02-05_fig06.png

Figure 5

  • Subtract.
m02-05_fig07.png

Figure 6

  • Determine if the answer is negative or positive.
    • Do this by looking again at the original equation: \(8-37\)
    • You have already identified that 8 is positive and 37 is negative. The bigger number is negative, which means that the answer will be negative: \(8-37=-29\)

Things to Remember


  • Subtraction is the addition of a negative number.
  • The steps of subtraction are as follows:
    • Step 1: Stack by place values—with the biggest number on top.
    • Step 2: Regroup as necessary.
    • Step 3: Subtract—right to left.
    • Step 4: Determine whether the number is positive or negative.
  • When determining whether an answer is positive or negative, the biggest or strongest number always wins.

Practice Problems

Evaluate the following expression:
  1. \(9 - 7 =\) ? (
    Solution
    x
    Solution: 2
    Details:
    To represent this problem graphically, here are 9 circles.
    Nine circles.

    Shade 7 circles.
    Nine circles, seven of the circles are shaded orange.

    The difference between the unshaded and shaded circles is 2. The answer to \(9 - 7 = 2\).
    )
  2. \(8 - 4 =\) ? (
    Solution
    x
    Solution:
    4
    )
  3. \(11 - 8 =\) ? (
    Video Solution
    )
  4. \(10 - 2 =\) ? (
    Video Solution
    )
  5. \(17 - 8 =\) ? (
    Solution
    x
    Solution: 9
    Details:
    To show the solution to this problem a number line from 5 to 20 is created. There is an arrow pointing to the number 17. This is the starting point.
    A number line from 5 to 20 with an arrow pointing to the number 17. Number 17 is highlighted orange.

    To subtract 8 from 17, move 8 spaces to the left:
    This is a number line from 5 to 20 with an arrow moving from number 17 to the left for 8 spaces and pointing to the number 9. Number 17 and Number 9 are highlighted in orange.

    When you move 8 spaces to the left, you end up at 9, so \(17 - 8 = 9\).
    )
  6. \(11 - 9 =\) ? (
    Solution
    x
    Solution:
    2
    )

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