1.2: Adding and Subtracting Whole Numbers

Last updated

May 20, 2025
Save as PDF
- 1.1: Place Value and Names for Whole Numbers
- 1.3: Multiplying and Dividing Whole Numbers

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

The Commutative Property of Addition

Let a and b represent two whole numbers. Then,

a + b = b + a.

Grouping Symbols

In mathematics, we use grouping symbols to affect the order in which an expression is evaluated. Whether we use parentheses, brackets, or curly braces, the expression inside any pair of grouping symbols must be evaluated first. For example, note how we first evaluate the sum in the parentheses in the following calculation.

(3 + 4) + 5 = 7 + 5

= 12

The rule is simple: Whatever is inside the parentheses is evaluated first.

Writing Mathematics

When writing mathematical statements, follow the mantra:

One equal sign per line.

We can use brackets instead of parentheses.

5 + [7 + 9] = 5 + 16

= 21

Again, note how the expression inside the brackets is evaluated first.

We can also use curly braces instead of parentheses or brackets.

{2+3} + 4 = 5 + 4

= 9

Again, note how the expression inside the curly braces is evaluated first.

If grouping symbols are nested, we evaluate the innermost parentheses first.

For example,

2 + [3 + (4 + 5)] = 2 + [3 + 9]

= 2 + 12

= 14.

Grouping Symbols

Use parentheses, brackets, or curly braces to delimit the part of an expression you want evaluated first. If grouping symbols are nested, evaluate the expression in the innermost pair of grouping symbols first.

The Associative Property of Addition

Consider the evaluation of the expression (2+3)+4. We evaluate the expression in parentheses first.

(2 + 3) + 4 = 5 + 4

= 9

Now, suppose we change the order of addition to 2 + (3 + 4). Then,

2 + (3 + 4) = 2 + 7

= 9.

Although the grouping has changed, the result is the same. That is,

(2 + 3) + 4 = 2 + (3 + 4).

This property of addition of whole numbers is called the associate property of addition.

Associate Property of Addition

Let a, b, and c represent whole numbers. Then,

(a + b) + c = a + (b + c).

Because of the associate property of addition, when presented with a sum of three numbers, whether you start by adding the first two numbers or the last two numbers, the resulting sum is the same.

The Additive Identity

The Additive Identity Property

The whole number zero is called the additive identity. If a is any whole number, the

a + 0 = a.

The number zero is called the additive identity because if you add zero to any number, you get the identical number back.

Adding Larger Whole Numbers

For completeness, we include two examples of adding larger whole numbers. Hopefully, the algorithm is familiar from previous coursework.

Example 1

Simplify: 1, 234 + 498.

Solution

Align the numbers vertically, then add, starting at the furthest column to the right. Add the digits in the ones column, 4 + 8 = 12. Write the 2, then carry a 1 to the tens column. Next, add the digits in the tens column, 3 + 9 = 12, add the carry to get 13, then write the 3 and carry a 1 to the hundreds column. Continue in this manner, working from right to left

$\begin{array}{r}{11} \\ {1234} \\ {+\quad 498} \\ \hline 1732\end{array}$

Therefore, 1, 234 + 498 = 1, 732

Exercise

Simplify: 1,286 + 349.

Answer: 1635

Add three or more numbers in the same manner.

Example 2

Simplify: 256 + 322 + 418.

Solution

Align the numbers vertically, then add, starting at the furthest column to the right. Add the digits in the ones column, 6 + 2 + 8 = 16. Write the 6, then carry a 1 to the tens column. Continue in this manner, working from right to left.

$\begin{array}{r}{256} \\ {322} \\ {+418} \\ \hline 996\end{array}$

Therefore, 256 + 322 + 418 = 996.

Exercise

Simplify: 256 + 342 + 283

Answer: 881

Subtraction of Whole Numbers

The key idea is this: Subtraction is the opposite of addition. For example, consider the difference 7 - 4 depicted on the number line in Figure 1.5.

Screen Shot 2019-08-05 at 8.43.27 PM.png — Figure 1.5: Subtraction means *add the opposite*.

If we were adding 7 and 4, we first draw an arrow starting at zero pointing to the right with magnitude (length) seven. Then, to add 4, we would draw a second arrow of magnitude (length) 4, attached to the end of the first arrow and pointing to the right.

However, because subtraction is the opposite of addition, in Figure 1.5 we attach an arrow of magnitude (length) four to the end of the first arrow, but pointing in the opposite direction (to the left). Note that this last arrow ends at the answer, which is a shaded dot on the number line at 3. That is, 7 − 4 = 3.

Note that subtraction is not commutative; that is, it make no sense to say that 7 − 5 is the same as 5 − 7.

Subtraction is not associative. It is not the case that (9 − 5) − 2 is the same as 9 − (5 − 2). On the one hand,

(9 − 5) − 2 = 4 − 2

= 2,

but

9 − (5 − 2) = 9 − 3

= 6.

Subtracting Larger Whole Numbers

Much as we did with adding larger whole numbers, to subtract two large whole numbers, align them vertically then subtract, working from right to left. You may have to “borrow” to complete the subtraction at any step.

Example 3

Simplify: 1, 755 − 328.

Solution

Align the numbers vertically, then subtract, starting at the ones column, then working right to left. At the ones column, we cannot subtract 8 from 5, so we borrow from the previous column. Now, 8 from 15 is 7. Continue in this manner, working from right to left.

$imageedit_14_9019801375.png$

Therefore, 1, 755 − 328 = 1, 427.

Exercise

Simplify: 5,635 - 288.

Answer: 5,347

Order of Operations

In the absence of grouping symbols, it is important to understand that addition holds no precedence over subtraction, and vice-versa.

Perform all additions and subtractions in the order presented, moving left to right.

Let’s look at an example.

Example 4

Simplify the expression $15 − 8 + 4$ .

Solution

This example can be trickier than it seems. However, if we follow the rule (perform all additions and subtractions in the order presented, moving left to right), we should have no trouble. First comes fifteen minus eight, which is seven. Then seven plus four is eleven.

$\begin{align*}15 − 8 + 4 &= 7 + 4 \\[4pt] &= 11. \end{align*}$

Exercise

Simplify: $25 − 10 + 8$ .

Answer: 23

Caution! Incorrect answer ahead!

Note that it is possible to arrive at a different (but incorrect) answer if we favor addition over subtraction in Example 4. If we first add eight and four, then 15 − 8 + 4 becomes 15 − 12, which is 3. However, note that this is incorrect, because it violates the rule “perform all additions and subtractions in the order presented, moving left to right.”

Exercises

Determine which property of addition is depicted by the given identity.

1. (51 + 66) + 88 = 51 + (66 + 88)

2. 64 + 39 = 39 + 64

3. 79 + 0 = 79

Simplify the given expression.

4. 16 − 8+2

5. 15 − 5+8

Find the sum.

6. 8583 + 592

7. 841 + 368 + 919

8. (86 + 557) + 80

Find the difference.

9. 8338 − 7366

10. 881 − 606

11. 3838 − (777 − 241)

12. Water Subsidies. Since the drought began in 2007, California farms have received $79 million in water subsidies. California cotton and rice farmers received an additional $439 million. How much total water subsidies have farmers received? Associated Press Times-Standard 4/15/09

13. Sun Frost. Arcata, CA is home to Sun Frost, a manufacturer of highly efficient refrigerators and freezers. The AC model RF12 refrigerator/freezer costs $2,279 while an R16 model refrigerator/freezer costs $3,017. How much more does the R16 model cost? Source: www.sunfrost.com/retail pricelist.html

14. Earth’s Orbit. Earth orbits the sun in an ellipse. When earth is at its closest to the sun, called perihelion, earth is about 147 million kilometers. When earth is at its furthest point from the sun, called aphelion, earth is about 152 million kilometers from the sun. What’s the difference in millions of kilometers between aphelion and perihelion?

15. Jobs. The Times-Standard reports that over the next year, the credit and debit card processing business Humboldt Merchant Services expects to cut 36 of its 80 jobs, but then turn around and hire another 21. How many people will be working for the company then? Times-Standard 5/6/09

Answers

1. Associative property of addition

2. Commutative property of addition

3. Additive identity property of addition.

4. 10

5. 18

6. 9175

7. 2128

8. 723

9. 972

10. 275

11. 3302

12. $518 million

13. $738

14. 5 million kilometers

15. 65 people

Search

Text Color

Text Size

Margin Size

Font Type

The Commutative Property of Addition

Grouping Symbols

The Associative Property of Addition

The Additive Identity

Adding Larger Whole Numbers

Subtraction of Whole Numbers

Subtracting Larger Whole Numbers

Order of Operations

Exercises

Answers

Support Center

How can we help?