Introduction to Integers

Locate Positive and Negative Numbers on the Number Line Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than 0. Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example, −1°F (read as “negative one degree Fahrenheit”) is 1degree below 0. A minus sign is shown before a number to indicate that it is negative. shows −20°F, which is 20degrees below 0.

Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative. Elevations can also be represented by negative numbers. The elevation at sea level is 0 feet. Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about 1,302feet below sea level, so the elevation of the Dead Sea can be represented as −1,302feet. See .

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of 500feet. Its position would then be −500feet as labeled in .

Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at 0 and showed the counting numbers increasing to the right as shown in . The counting numbers (1, 2, 3, …) on the number line are all positive. We could write a plus sign, +, before a positive number such as +2 or +3, but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with −1 at the first mark to the left of 0,−2 at the next mark, and so on. See .

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number. Doing the Manipulative Mathematics activity "Number Line-part 2" will help you develop a better understanding of integers. Plot the numbers on a number line: ⓐ 3 ⓑ −3 ⓒ −2 Solution Draw a number line. Mark 0 in the center and label several units to the left and right. ⓐ To plot 3, start at 0 and count three units to the right. Place a point as shown in .

ⓑ To plot −3, start at 0 and count three units to the left. Place a point as shown in .

Plot the numbers on a number line. ⓐ 1 ⓑ −1 ⓒ −4

Plot the numbers on a number line. ⓐ −4 ⓑ 4 ⓐ −1

Order Positive and Negative Numbers We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See .

Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation a<b (read a is less than b) when a is to the left of b on the number line. We write a>b (read a is greater than b) when a is to the right of b on the number line. This is shown for the numbers 3 and 5 in .

The numbers lines to follow show a few more examples. ⓐ

4 is to the right of 1 on the number line, so 4>1. 1 is to the left of 4 on the number line, so 1<4. ⓑ

−2 is to the left of 1 on the number line, so −2<1. 1 is to the right of −2 on the number line, so 1>−2. ⓒ

−1 is to the right of −3 on the number line, so −1>−3. −3 is to the left of −1 on the number line, so −3<−1. Order each of the following pairs of numbers using < or >: ⓐ 14___6 ⓑ −1___9 ⓒ −1___−4 ⓓ 2___−20 Solution Begin by plotting the numbers on a number line as shown in .

ⓐ Compare 14 and 6. 14___6 14 is to the right of 6 on the number line. 14>6

ⓑ Compare −1 and 9. −1___9 −1 is to the left of 9 on the number line. −1<9

ⓓ Compare 2 and −20. 2___−20 2 is to the right of −20 on the number line. 2>−20

Order each of the following pairs of numbers using < or >. ⓐ 15___7 ⓑ −2___5 ⓒ −3___−7 ⓓ 5___−17 ⓐ > ⓑ < ⓒ > ⓓ > Order each of the following pairs of numbers using < or >. ⓐ 8___13 ⓑ 3___−4 ⓒ −5___−2 ⓓ 9___−21 ⓐ < ⓑ > ⓒ < ⓓ >

Find Opposites On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers 2 and −2 are the same distance from zero, they are called opposites. The opposite of 2 is −2, and the opposite of −2 is 2 as shown in (a). Similarly, 3 and −3 are opposites as shown in (b).

Opposite The opposite of a number is the number that is the same distance from zero on the number line, but on the opposite side of zero. Find the opposite of each number: ⓐ 7 ⓑ −10 Solution ⓐ The number −7 is the same distance from 0 as 7, but on the opposite side of 0. So −7 is the opposite of 7 as shown in .

ⓑ The number 10 is the same distance from 0 as −10, but on the opposite side of 0. So 10 is the opposite of −10 as shown in .

Find the opposite of each number: ⓐ 4 ⓑ −3 ⓐ −4 ⓑ 3 Find the opposite of each number: ⓐ 8 ⓑ −5 ⓐ −8 ⓑ 5

Opposite Notation Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol “−”, in three different ways. 10−4 Between two numbers, the symbol indicates the operation of subtraction. We read 10−4 as 10 minus 4. −8 In front of a number, the symbol indicates a negative number. We read −8 as negative eight. −x In front of a variable or a number, it indicates the opposite.We read−x as the opposite of x. −(−2) Here we have two signs. The sign in the parentheses indicates that the number is negative 2. The sign outside the parentheses indicates the opposite. We read −(−2) as the opposite of −2.

Opposite Notation −a means the opposite of the number a The notation −a is read the opposite of a. Simplify: −(−6). Solution −(−6)The opposite of −6 is 6. 6

Simplify: −(−1) 1 Simplify: −(−5) 5

Integers The set of counting numbers, their opposites, and 0 is the set of integers. Integers Integers are counting numbers, their opposites, and zero.…−3,−2,−1,0,1,2,3… We must be very careful with the signs when evaluating the opposite of a variable. Evaluate −x: ⓐ when x=8 ⓑ when x=−8. ⓐ To evaluate −x when x=8, substitute 8 for x. −x

Simplify. −8

ⓑ To evaluate −x when x=−8, substitute −8 for x. −x

Simplify. 8

Evaluate −n: ⓐ whenn=4 ⓑ whenn=−4 ⓐ −4 ⓑ 4 Evaluate: −m: ⓐ whenm=11 ⓑ whenm=−11 ⓐ −11 ⓑ 11

Simplify Expressions with Absolute Value We saw that numbers such as 5 and −5 are opposites because they are the same distance from 0 on the number line. They are both five units from 0. The distance between 0 and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative. The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of 5 is written as |5|, and the absolute value of −5 is written as |−5| as shown in .

Absolute Value The absolute value of a number is its distance from 0 on the number line. The absolute value of a number n is written as |n|. |n|≥0for all numbers Simplify: ⓐ |3| ⓑ |−44| ⓒ |0| Solution ⓐ |3| 3 is 3 units from zero. 3

ⓑ |−44| −44 is 44 units from zero. 44

Simplify: ⓐ |12| ⓑ −|−28| ⓐ 12 ⓑ −28 Simplify: ⓐ |9| ⓑ −|37| ⓐ 9 ⓑ −37 We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first. Evaluate: ⓐ |x|whenx=−35 ⓑ |−y|wheny=−20 ⓒ −|u|whenu=12 ⓓ −|p|whenp=−14 Solution ⓐ To find |x| when x=−35: |x|

Take the absolute value. 35

ⓑ To find |−y| when y=−20: |−y|

Simplify. |20| Take the absolute value. 20

Take the absolute value. −12

ⓓ To find −|p| when p=−14: −|p|

Take the absolute value. −14

Notice that the result is negative only when there is a negative sign outside the absolute value symbol. Evaluate: ⓐ |x|whenx=−17 ⓑ |−y|wheny=−39 ⓒ −|m|whenm=22 ⓓ −|p|whenp=−11 ⓐ 17 ⓑ 39 ⓒ −22 ⓓ −11 ⓐ |y|wheny=−23 ⓑ |−y|wheny=−21 ⓒ −|n|whenn=37 ⓓ −|q|whenq=−49 ⓐ 23 ⓑ 21 ⓒ −37 ⓓ −49 Fill in <,>,or= for each of the following: ⓐ |−5|___−|−5| ⓑ 8___−|−8| ⓒ −9___−|−9| ⓓ −|−7|___−7 Solution To compare two expressions, simplify each one first. Then compare. ⓐ |−5|___−|−5| Simplify. 5___−5 Order. 5>−5

ⓑ 8___−|−8| Simplify. 8___−8 Order. 8>−8

ⓓ −|−7|___−7 Simplify. −7___−7 Order. −7=−7

Fill in <,>,or=for each of the following: ⓐ |−9|___−|−9| ⓑ 2___−|−2| ⓒ −8___|−8| ⓓ −|−5|___−5 ⓐ > ⓑ > ⓒ < ⓓ = Fill in <,>,or= for each of the following: ⓐ 7___−|−7| ⓑ −|−11|___−11 ⓒ |−4|___−|−4| ⓓ −1___|−1| ⓐ > ⓑ = ⓒ > ⓓ < Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols. Simplify: ⓐ |9−3| ⓑ 4|−2| Solution For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses. ⓐ |9−3| Simplify inside the absolute value sign. |6| Take the absolute value. 6

ⓑ 4|−2| Take the absolute value. 4⋅2 Multiply. 8

Simplify: ⓐ |12−9| ⓑ 3|−6| ⓐ 3 ⓑ 18 Simplify: ⓐ |27−16| ⓑ 9|−7| ⓐ 11 ⓑ 63 Simplify: |8+7|−|5+6|. Solution For each expression, follow the order of operations. Begin inside the absolute value symbols just as with parentheses. |8+7|−|5+6| Simplify inside each absolute value sign. |15|−|11| Subtract. 4

Simplify: |1+8|−|2+5| 2 Simplify: |9−5|−|7−6| 3 Simplify: 24−|19−3(6−2)|. Solution We use the order of operations. Remember to simplify grouping symbols first, so parentheses inside absolute value symbols would be first. 24−|19−3(6−2)| Simplify in the parentheses first. 24−|19−3(4)| Multiply 3(4). 24−|19−12| Subtract inside the absolute value sign. 24−|7| Take the absolute value. 24−7 Subtract. 17

Simplify: 19−|11−4(3−1)| 16 Simplify: 9−|8−4(7−5)| 9

Translate Word Phrases into Expressions with Integers Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates −20. So does the word opposite in “the opposite of 20.” Translate each phrase into an expression with integers: ⓐ the opposite of positive fourteen ⓑ the opposite of −11 ⓒ negative sixteen ⓓ two minus negative seven Solution ⓐ the opposite of fourteen −14 ⓑ the opposite of −11 −(−11)=11 ⓒ negative sixteen −16 ⓓ two minus negative seven 2−(−7) Translate each phrase into an expression with integers: ⓐ the opposite of positive nine ⓑ the opposite of −15 ⓒ negative twenty ⓓ eleven minus negative four ⓐ −9 ⓑ 15 ⓒ −20 ⓓ 11−(−4) Translate each phrase into an expression with integers: ⓐ the opposite of negative nineteen ⓑ the opposite of twenty-two ⓒ negative nine ⓓ negative eight minus negative five ⓐ 19 ⓑ −22 ⓒ −9 ⓓ −8−(−5) As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example. Translate into an expression with integers: ⓐ The temperature is 12degrees Fahrenheit below zero. ⓑ The football team had a gain of 3yards. ⓒ The elevation of the Dead Sea is 1,302feet below sea level. ⓓ A checking account is overdrawn by $40. Solution Look for key phrases in each sentence. Then look for words that indicate negative signs. Don’t forget to include units of measurement described in the sentence. ⓐ The temperature is 12 degrees Fahrenheit below zero. Below zero tells us that 12 is a negative number. −12ºF

ⓑ The football team had a gain of 3 yards. A gain tells us that 3 is a positive number. 3 yards

ⓓ A checking account is overdrawn by $40. Overdrawn tells us that 40 is a negative number. −$40

Translate into an expression with integers: The football team had a gain of 5yards. 5 yards Translate into an expression with integers: The scuba diver was 30feet below the surface of the water. −30 feet ACCESS ADDITIONAL ONLINE RESOURCES Introduction to Integers Simplifying the Opposites of Negative Integers Comparing Absolute Value of Integers Comparing Integers Using Inequalities

Practice Makes PerfectLocate Positive and Negative Numbers on the Number Line For the following exercises, draw a number line and locate and label the given points on that number line. ⓐ 2 ⓑ −2 ⓒ −5

ⓐ −7 ⓑ 7 ⓒ −1 Order Positive and Negative Numbers on the Number Line In the following exercises, order each of the following pairs of numbers, using < or >. ⓐ 9__4 ⓑ −3__6 ⓒ −8__−2 ⓓ 1__−10 ⓐ > ⓑ < ⓒ < ⓓ > ⓐ 6__2; ⓑ −7__4; ⓒ −9__−1; ⓓ 9__−3 ⓐ −5__1; ⓑ −4__−9; ⓒ 6__10; ⓓ 3__−8 ⓐ < ⓑ > ⓒ < ⓓ > ⓐ −7__3; ⓑ −10__−5; ⓒ 2__−6; ⓓ 8__9 Find Opposites In the following exercises, find the opposite of each number. ⓐ 2 ⓑ −6 ⓐ −2 ⓑ 6 ⓐ 9 ⓑ −4 ⓐ −8 ⓑ 1 ⓐ 8 ⓑ −1 ⓐ −2 ⓑ 6 In the following exercises, simplify. −(−4) 4 −(−8) −(−15) 15 −(−11) In the following exercises, evaluate. −mwhen ⓐ m=3 ⓑ m=−3 ⓐ −3 ⓑ 3 −pwhen ⓐ p=6 ⓑ p=−6 −cwhen ⓐ c=12 ⓑ c=−12 ⓐ −12; ⓑ 12 −dwhen ⓐ d=21 ⓑ d=−21 Simplify Expressions with Absolute Value In the following exercises, simplify each absolute value expression. ⓐ |7| ⓑ |−25| ⓒ|0| ⓐ 7 ⓑ 25 ⓒ 0 ⓐ |5| ⓑ |20| ⓒ |−19| ⓐ |−32| ⓑ |−18| ⓒ |16| ⓐ 32 ⓑ 18 ⓒ 16 ⓐ |−41| ⓑ |−40| ⓒ |22| In the following exercises, evaluate each absolute value expression. ⓐ |x|whenx=−28 ⓑ |−u|whenu=−15 ⓐ 28 ⓑ 15 ⓐ |y|wheny=−37 ⓑ |−z|whenz=−24 ⓐ −|p|whenp=19 ⓑ −|q|whenq=−33 ⓐ −19 ⓑ −33 ⓐ −|a|whena=60 ⓑ −|b|whenb=−12 In the following exercises, fill in <,>,or= to compare each expression. ⓐ −6__|−6| ⓑ −|−3|__−3 ⓐ < ⓑ = ⓐ −8__|−8| ⓑ −|−2|__−2 ⓐ |−3|__−|−3| ⓑ 4__−|−4| ⓐ > ⓑ > ⓐ |−5|__−|−5| ⓑ9__−|−9| In the following exercises, simplify each expression. |8−4| 4 |9−6| 8|−7| 56 5|−5| |15−7|−|14−6| 0 |17−8|−|13−4| 18−|2(8−3)| 8 15−|3(8−5)| 8(14−2|−2|) 80 6(13−4|−2|) Translate Word Phrases into Expressions with Integers Translate each phrase into an expression with integers. Do not simplify. ⓐ the opposite of 8 ⓑ the opposite of −6 ⓒ negative three ⓓ 4 minus negative 3 ⓐ −8 ⓑ −(−6), or 6 ⓒ −3 ⓓ 4−(−3) ⓐ the opposite of 11 ⓑ the opposite of −4 ⓒ negative nine ⓓ 8 minus negative 2 ⓐ the opposite of 20 ⓑ the opposite of −5 ⓒ negative twelve ⓓ 18 minus negative 7 ⓐ −20 ⓑ −(−5), or 5 ⓒ −12 ⓓ 18−(−7) ⓐ the opposite of 15 ⓑ the opposite of −9 ⓒ negative sixty ⓓ 12 minus 5 a temperature of 6degrees below zero −6 degrees a temperature of 14degrees below zero an elevation of 40feet below sea level −40 feet an elevation of 65feet below sea level a football play loss of 12yards −12 yards a football play gain of 4yards a stock gain of $3 $3 a stock loss of $5 a golf score one above par +1 a golf score of 3 below par

Everyday Math Elevation The highest elevation in the United States is Mount McKinley, Alaska, at 20,320feet above sea level. The lowest elevation is Death Valley, California, at 282feet below sea level. Use integers to write the elevation of: ⓐ Mount McKinley ⓑ Death Valley ⓐ 20,320 feet ⓑ −282 feet Extreme temperatures The highest recorded temperature on Earth is 57° Celsius. The lowest recorded temperature is 90° below 0° Celsius. Use integers to write the: ⓐ highest recorded temperature ⓑ lowest recorded temperature State budgets In June, 2011, the state of Pennsylvania estimated it would have a budget surplus of $540 million. That same month, Texas estimated it would have a budget deficit of $27 billion. Use integers to write the budget: ⓐ surplus ⓑ deficit ⓐ $540 million ⓑ −$27 billion College enrollments Across the United States, community college enrollment grew by 1,400,000 students from 2007 to 2010. In California, community college enrollment declined by 110,171 students from 2009 to 2010. Use integers to write the change in enrollment: ⓐ growth ⓑ decline

Writing Exercises Give an example of a negative number from your life experience. Sample answer: I have experienced negative temperatures. What are the three uses of the “−” sign in algebra? Explain how they differ.

Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were: …confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific. …with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved? …no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.