Signed Numbers, Powers, & Roots Chapter 2 Sections 2.1-2.6 Chapter 1C Section 1.15

Signed Numbers, Powers, & Roots

Chapter 2 Sections 2.1-2.6

Chapter 1C Section 1.15

Copyright © Cengage Learning. All rights reserved.

Addition of Signed Numbers2.1

• Technicians use negative numbers in many ways. In an experiment using low temperatures, for example, you would record 10 below zero as –10. Or consider sea level as zero altitude. If a submarine dives 75 m, you could consider its depth as –75 m (75 m below sea level). See Figure 2.1.

Addition of Signed Numbers

(a) (b)Figure 2.1

• These measurements indicate a need for numbers other than positive integers, which are the only numbers that we have used up to now.

• To illustrate the graphical relationship of these numbers, we draw a number line as in Figure 2.2 with a point representing zero and with evenly spaced points that represent the positive integers (1, 2, 3, . . .) to the right as shown.


The real number line

Figure 2.2

• Then we mark off similarly evenly spaced points to the left of zero.

• These points correspond to the negative integers (–1, –2, –3, . . .) as shown.

• The negative integers are preceded by a negative (–) sign; –3 is read “negative 3,” and –5 is read “negative 5.”

• Each positive integer corresponds to a negative integer.

•For example, 3 and –3 are corresponding integers. Note that the distances from 0 to 3 and from 0 to –3 are equal.


• The rational numbers are defined as those numbers

that can be written as the ratio of two integers; that is,

a/b, where b ≠ 0.

The irrational numbers are those numbers that cannot

be written as the ratio of two integers, such as ,

, or the square root of any nonperfect square; ; and

several other kinds of numbers.


• The real numbers consist of the rational and irrational numbers and are represented on the real number line as shown in Figure 2.2.

• The real number line is dense or full with real numbers;that is, each point on the number line represents a distinctreal number, and each real number is represented by adistinct point on the number line.



Figure 2.2

• Examples of real numbers as illustrated in

Your text on page 109


• The absolute value of a number is its distance from zero on the

number line.

• Because distance is always considered positive, the absolute value

of a number is never negative.

• We write the absolute value of a number x as | x |; it is read “the

absolute value of x.”

Thus, | x | 0. (“” means “is greater than or equal to.”) For

example, | +6 | = 6, | 4 | = 4, and | 0 | = 0.


• However, if a number is less than 0 (negative), its absolute value is

the corresponding positive number.

• For example, | –6 | = 6 and | –7 | = 7.

Remember:


• Find the absolute value of each number: a. +3, b. –5, c. 0, d. –10, e. 15.

• a. |+3| = 3

• b. | –5 | = 5

• c. | 0 | = 0

• d. | –10 | = 10

• e. | 15 | = | +15 | = 15

Example 1

The distance between 0 and +3 on the number line is 3 units.

The distance between 0 and –5 on the number line is 5 units.

The distance is 0 units.

The distance between 0 and –10 on the number line is 10 units.

The distance between 0 and +15 on the number line is 15 units.

• One number is larger than another number if the first number is to the right of the second on the number line in Figure 2.2.

Thus, 5 is larger than 1, 0 is larger than –3, and 2 is larger than –4. Similarly, one number is smaller than another if the first number is to the left of the second on the number line in Figure 2.2. Thus, 0 is smaller than 3,

–1 is smaller than 4, and –5 is smaller than –2.



Figure 2.2

• The use of signed numbers (positive and negative numbers) is one

of the most important operations that we will study.

Adding Two Numbers with Like Signs (the Same Signs)

1. To add two positive numbers, add their absolute values.

The result is positive. A positive sign may or may not be

used before the result. It is usually omitted.

•

2. To add two negative numbers, add their absolute values

and place a negative sign before the result.


• Add:

• a. (+2) + (+3) = +5

• b. (–4) + (–6) = –10

• c. (+4) + (+5) = +9

• d. (–8) + (–3) = –11

Example 2

• Adding Two Numbers with Unlike Signs

• To add a negative number and a positive

number, find the difference of their

absolute values.

• The sign of the number having the larger

absolute value is placed before the result.


• Add:

• a. (+4) + (–7) = –3

• b. (–3) + (+8) = +5

• c. (+6) + (–1) = +5

• d. (–8) + (+6) = –2

• e. (–2) + (+5) = +3

• f. (+3) + (–11) = –8

Example 3

• Adding Three or More Signed Numbers

• Step 1: Add the positive numbers.

• Step 2: Add the negative numbers.

• Step 3: Add the sums from Steps 1 and 2

according to the

rules for addition of two signed

numbers.


• Add (–8) + (+12) + (–7) + (–10) + (+3).

• Step 1: (+12) + (+3) = +15

• Step 2: (–8) + (–7) + (–10) = –25

• Step 3: (+15) + (–25) = –10

• Therefore, (–8) + (+12) + (–7) + (–10) + (+3) = –10.

Example 4

Add the positive numbers.

Add the negative numbers.

Add the sums from Steps 1 and 2.


Subtraction of Signed Numbers2.2

• Subtracting Two Signed Numbers

• To subtract two signed numbers, change

(or reverse) the sign of the number being

subtracted (second number) and add

according to the rules for addition of

signed numbers. This is because

subtraction reverses the direction we

travel on the number line.

Subtraction of Signed Numbers

• Subtract:

• a. (+2) – (+5) = (+2) + (–5)

• = –3

• b. (–7) – (–6) = (–7) + (+6)

• = –1

• c. (+6) – (–4) = (+6) + (+4)

• = +10

Example 1

To subtract, change the sign of thenumber being subtracted, +5, and add.

To subtract, change the sign of thenumber being subtracted, –6, and add.

To subtract, change the sign of thenumber being subtracted, –4, and add.

• d. (+1) – (+6) = (+1) + (–6)

• = –5

• e. (–8) – (–10) = (–8) + (+10)

• = +2

• f. (+9) – (–6) = (+9) + (+6)

• = +15

• g. (–4) – (+7) = (–4) + (–7)

• = –11

Example 1 cont’d

To subtract, change the sign of thenumber being subtracted, +6, and add.

• Subtract: • (–4) – (–6) – (+2) – (–5) – (+7)• = (–4) + (+6) + (–2) + (+5) + (–7)

• Step 1: (+6) + (+5) = +11

• Step 2: (–4) + (–2) + (–7) = –13

• Step 3: (+11) + (–13) = –2

• Therefore, (–4) – (–6) – (+2) – (–5) – (+7) = –2.

Example 2

Change the sign of each number being subtracted and add the resulting signed numbers.

• Adding and Subtracting Combinations

of Signed Numbers

• When combinations of additions and

subtractions of signed numbers occur in

the same problem, change only the sign of

each number being subtracted. Then add

the resulting signed numbers.

Subtraction of Signed Numbers

• Perform the indicated operations:• (+4) – (–5) + (–6) – (+8) – (–2) + (+5) – (+1)• = (+4) + (+5) + (–6) + (–8) + (+2) + (+5) + (–1)

• Step 1: (+4) + (+5) + (+2) + (+5) = +16

• Step 2: (–6) + (–8) + (–1) = –15

Example 3

Change only the sign of each number being subtracted and add the resulting signed numbers.

• Step 3: (+16) + (–15) = +1

• Therefore,

• (+4) – (–5) + (–6) – (+8) – (–2) + (+5) – (+1) = +1.

Example 3 cont’d


Multiplication and Division of Signed Numbers2.3

• Multiplying/Dividing Two Signed Numbers

• 1. If the two numbers have the same sign, multiply or

divide their absolute values. This product is always

positive.

• 2. If the two numbers have different signs, multiply or

divide their absolute values and place a negative sign

before the product.

Multiplication and Division of Signed Numbers

• Multiply:

• a. (+2)(+3) = +6

• b. (–4)(–7) = +28

• c. (–2)(+4) = –8

• d. (–6)(+5) = –30

Example 1

Multiply the absolute values of the signed numbers; the product is positive because the two numbers have the same sign.

Multiply the absolute values of the signed numbers; the product is negative because the two numbers have different signs.

Multiply the absolute values of the signed numbers; the product is negative because the two numbers have different signs.

Multiply the absolute values of the signed numbers; the product is positive because the two numbers have the same sign.

• e. (+3)(+4) = +12

• f. (–6)(–9) = +54

• g. (–5)(+7) = –35

• h. (+4)(–9) = –36

Example 1 cont’d

• Multiplying More Than Two Signed Numbers

• 1. If the number of negative factors is even (divisible by

2) multiply the absolute values of the numbers.

This

product is positive.

• 2. If the number of negative factors is odd, multiply the

absolute values of the numbers and place a negative

sign before the product.

Multiplication and Division of Signed Numbers

• Multiply: (–11)(+3)(–6) = +198

Example 2

The number of negative factors is 2, which is even; therefore, the product is positive.

• Divide:

Example 4

Divide the absolute values of the signed numbers; the quotient is positive because the two numbers have the same sign.

Divide the absolute values of the signed numbers; the quotient is positive because the two numbers have the same sign.

Divide the absolute values of the signed numbers; the quotient is negative because the two numbers have different signs.

Divide the absolute values of the signed numbers; the quotient is negative because the two numbers have different signs.

• e. (+30) (+5) = +6

• f. (–42) (–2) = +21

• g. (+16) (–4) = –4

• h. (–45) (+9) = –5

Example 4 cont’d


Signed Fractions2.4

Example 1The LCD is 16.

Combine thenumerators.

• Add:

• Equivalent Signed Fractions

• That is, a negative fraction may be written in three different but equivalent forms.

• However, the form is the customary form.

• For example,

• Note • using the rules for dividing signed numbers.

Signed Fractions

Example 12

The LCD is 12.

Combine the numerators.

Change to customary form.• Add:


1.15 Powers and Roots

• The square of a number is the product of

that number times itself. The square of 3 is

3 3 or 32 or 9.

The square of a number may be found

with a calculator as follows.

Powers and Roots

• The square root of a number is that

positive number which, when multiplied by

itself, gives the original number.

The square root of 25 is 5 and is written as

• The symbol is called a radical.

Powers and Roots

• Find the square roots of a. 16, b. 64, c. 100, and d. 144.

• a. = 4 because 4 4 = 16

• b. = 8 because 8 8 = 64

• c. = 10 because 10 10 = 100

• d. = 12 because 12 12 = 144

• Numbers whose square roots are whole numbers are called perfect squares. For example,1, 4, 9, 16, 25, 36, 49, and 64 are perfect squares.

Example 3

• The cube of a number is the product of

that number times itself three times. The

cube of 5 is 5 5 5 or 53 or 125.

Powers and Roots

• Find the cubes of a. 2, b. 3, c. 4, and d. 10.

• a. 23 = 2 2 2

• b. 33 = 3 3 3

• c. 43 = 4 4 4

• d. 103 = 10 10 10

Example 6

= 8

= 27

= 64

= 1000

• The cube root of a number is that number

which, when multiplied by itself three

times, gives the original number.

The cube root of 8 is 2 and is written as

• (Note: 2 2 2 = 8. The small 3 in the

radical is called the index.)

Powers and Roots

• Find the cube roots of a. 8, b. 27, and c.

125.

• a. = 2 because 2 2 2 = 8

• b. = 3 because 3 3 3 = 27

• c. = 5 because 5 5 5 = 125

Example 9

• Numbers whose cube roots are whole numbers are

called perfect cubes. For example, 1, 8, 27, 64, 125,

and 216 are perfect cubes.

• In general, in a power of a number, the exponent

indicates the number of times the base is used as a

factor.

For example, the 4th power of 3 is written 34, which

means that 3 is used as a factor 4 times (34 = 3 3 3 3 = 81).

Powers and Roots


Powers of 102.5

• Multiplying Powers of 10

• To multiply two powers of 10, add the

exponents as follows:

• 10a 10b = 10a + b

Powers of 10

• Multiply: (102)(103)

• Method 1: (102)(103) =

• (10 10)(10 10 10)

• = 105

• Method 2: (102)(103) = 102 + 3

• = 105

Example 1

Add the exponents.

• Dividing Powers of 10

• To divide two powers of 10, subtract the

exponents as follows:

• 10a 10b = 10a – b

Powers of 10

• Divide:

• Method 1:

• Method 2:

Example 4

Subtract the exponents.

• Raising a Power of 10 to a Power

• To raise a power of 10 to a power, multiply

the exponents as follows:

• (10a)b = 10ab

Powers of 10

• Find the power (102)3.

• Method 1: (102)3 = 102 102 102

• = 102 + 2 + 2

• = 106

• Method 2: (102)3 = 10(2)(3)

• = 106

Example 7

Multiply the exponents.

Use the product of powers rule.

•

• For example, and

• In a similar manner, we can also show that

Powers of 10

• For example, and

• Combinations of multiplications and

divisions of powers of 10 can also be done

easily using the rules of exponents.

Powers of 10


Scientific Notation2.6

• Scientific Notation

Scientific notation is a method that is

especially useful for writing very large or

very small numbers.

To write a number in scientific notation,

write it as a product of a number between

1 and 10 and a power of 10.

Scientific Notation

• Write 226 in scientific notation.

• 226 = 2.26 102

• Remember that 102 is a short way of writing 10 10 =

100.

• Note that multiplying 2.26 by 100 gives 226.

Example 1

• Writing a Decimal Number in Scientific Notation• To write a decimal number in scientific notation,

• 1. Reading from left to right, place a decimal point after the first nonzero digit.

• 2. Place a caret (^) at the position of the original decimal point.

• 3. If the decimal point is to the left of the caret, the exponent of the power of 10 is the same as the number of decimal places from the caret to the decimal point.

Scientific Notation

• 4. If the decimal point is to the right of the caret, the

exponent of the power of 10 is the same as the negative

of the number of places from the caret to the decimal

point.

• 5. If the decimal point is already after the first nonzero digit,

the exponent of 10 is zero.

• 2.15 = 2.15 100

Scientific Notation

• Write 2738 in scientific notation.

Example 3

• Writing a Number in Scientific Notation in Decimal Form

• To change a number in scientific notation to decimal form,

•1. Multiply the decimal part by the given positive power of 10 by moving the decimal point to the right the same number of decimal places as indicated by the exponent of 10. Supply zeros when needed.

•2. Multiply the decimal part by the given negative power of 10 by moving the decimal point to the left the same number of decimal places as indicated by the exponent of 10. Supply zeros when needed.

Scientific Notation

• Write 2.67 102 as a decimal.

• 2.67 102 = 267

Example 5

Move the decimal point two places to the right, since the exponent of 10 is +2.

• You may find it useful to note that a number in scientific notation with

• a. a positive exponent greater than 1 is greater than 10, and

b. a negative exponent is between 0 and 1.

That is, a number in scientific notation with a positive exponent represents a relatively large number.

A number in scientific notation with a negative exponent represents a relatively small number.

Scientific Notation

• Scientific notation may be used to compare two positive numbers expressed as decimals.

First, write both numbers in scientific notation.

• The number having the greater power of 10 is the larger.

If the powers of 10 are equal, compare the parts of the numbers that are between 1 and 10.

Scientific notation is especially helpful for multiplying and dividing

very large and very small numbers.

Scientific Notation

• To perform these operations, you must first know some rules for exponents.

• Multiplying Numbers in Scientific Notation

• To multiply numbers in scientific notation, multiply the decimals between 1 and 10.

• Then add the exponents of the powers of 10.

Scientific Notation

• Multiply (4.5 108)(5.2 10–14). Write the result in scientific notation.

• (4.5 108)(5.2 10–14) = (4.5)(5.2) (108)(10–14)

• = 23.4 10–6

• = (2.34 101) 10–6

• = 2.34 10–5

• Note that 23.4 10–6 is not in scientific notation, because 23.4 is not between 1 and 10.

Example 10

• Dividing Numbers in Scientific Notation

• To divide numbers in scientific notation,

divide the decimals between 1 and 10.

• Then subtract the exponents of the powers

of 10.

Scientific Notation

• Powers of Numbers in Scientific

Notation

• To find the power of a number in scientific

notation, find the power of the decimal

between 1 and 10. Then multiply the

exponent of the power of 10 by this same

power.

Scientific Notation

• Find the power (4.5 106)2. Write the result in scientific notation.

• (4.5 106)2 = (4.5)2 (106)2

• = 20.25 1012

• = (2.025 101) 1012

• = 2.025 1013

Example 13

Note that 20.25 is not between 1 and 10.


Engineering Notation2.7

• Numbers may also be written in engineering notation,

similar to scientific notation, as follows:

•

Engineering Notation

• Engineering notation is used to write a number with its

decimal part between 1 and 1000 and a power of 10

whose exponent is divisible by 3.


• Writing a Decimal Number in Engineering Notation

• To write a decimal number in engineering notation,

• 1. Move the decimal point in groups of three digits until the

decimal point indicates a number between 1 and 1000.

2. If the decimal point has been moved to the left, the

exponent of the power of 10 in engineering notation is

the same as the number of places the decimal point was

moved.


• 3. If the decimal point has been moved to the right, the

exponent of the power of 10 in engineering notation is

the same as the negative of the number of places the

decimal point was moved.

In any case, the exponent will be divisible by 3.


• Write 48,500 in engineering notation.

• 48,500 = 48.5 103

• Check

The exponent of the power of 10 must be

divisible by 3.

Example 1

Move the decimal point in groups of three decimal places until the decimal part is between 1 and 1000.

• Writing a number in engineering notation in decimal form is similar to writing a number in scientific notation in decimal form.

Operations with numbers in engineering notation using a calculator are very similar to operations with numbers inscientific notation.

If your calculator has an engineering notation mode, set it in this mode.

If not, use scientific notation and convert the result toengineering notation.


• For comparison purposes, the following table shows six numbers written in both scientific notation and engineering notation:


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Signed Numbers, Powers, & Roots Chapter 2 Sections 2.1-2.6 Chapter 1C Section 1.15