Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

Подождите немного. Документ загружается.

If a ¼ 3 and b ¼ 52, however, it’s necessary to use the ‘‘times sign’’ and

write this:

3  52 ¼ 52  3

The reason becomes obvious if the above expression is written without using

the ‘‘times signs.’’ This results in a false statement:

352 ¼ 523

ASSOCIATIVE LAW FOR ADDITION

When adding any three real numbers, it does not matter how the addends are

grouped. The operation of addition is associative over the set of real numbers.

For all real numbers a

, a

, and a

, the following equation holds true:

ða

þ a

Þþa

¼ a

þða

þ a

ASSOCIATIVE LAW FOR MULTIPLICATION

When multiplying any three real numbers, it does not matter how the

multiplicands are grouped. Multiplication, like addition, is associative

over the set of real numbers. For all real numbers a, b, and c, the following

equation holds:

ðabÞc ¼ aðbcÞ

DISTRIBUTIVE LAWS

For all real numbers a, b, and c, the following equation holds. The operation

of multiplication is distributive with respect to addition:

aðb þ cÞ¼ab þ ac

The above statement logically implies that multiplication is distributive with

respect to subtraction, as well:

aðb  cÞ¼ab  ac

The distributive law can also be extended to division as long as there aren’t

any denominators that end up being equal to zero. For all real numbers a, b,

and c, where a 6¼ 0, the following equations are valid:

ðab þ acÞ=a ¼ ab=a þ ac=a ¼ b þ c

ðab  acÞ=a ¼ ab=a  ac=a ¼ b  c

CHAPTER 1 Numbers and Arithmetic 21

PRECEDENCE OF OPERATIONS

When addition, subtraction, multiplication, division, and exponentiation

(raising to a power) appear in an expression and that expression

must be simpliﬁed, the operations should be performed in the following

sequence:

Simplify all expressions within parentheses, brackets, and braces from

the inside out.

Perform all exponential operations, proceeding from left to right.

Perform all products and quotients, proceeding from left to right.

Perform all sums and diﬀerences, proceeding from left to right.

The following are examples of this process, in which the order of the numer-

als and operations is the same in each case, but the groupings diﬀer:

½ð2 þ 3Þð3  1Þ



¼½5 ð4Þ



¼ð5  16Þ

¼ 80

¼ 6400

½ð2 þ 3 ð3Þ1Þ



¼½ð2 þð9Þ1Þ



¼ð8

¼ 64

¼ 4096

A note of caution is in order here: This rule doesn’t apply to exponents

of exponents. For example, 3

raised to the power of 3 is equal to

or 19,683. But 3 raised to the power of 3

is equal to 3

7,625,597,484,987.

PROBLEM 1-7

What is the value of the expression 5  6 þ 8  1?

SOLUTION 1-7

First, multiply 5 times 6, getting the product 30. Then add 8 to this,

getting 22. Finally subtract 1, getting 23. Thus:

5  6 þ 8  1 ¼23

PART 1 Expressing Quantities

PROBLEM 1-8

What is the value of the expression 5  (6 þ 8  1)?

SOLUTION 1-8

First ﬁnd the sum 6 þ 8, which is 14. Then subtract 1 from this, getting 13.

Finally multiply 13 by 5, getting 65. Thus:

5 ð6 þ 8  1Þ¼65

More Principles Worth Memorizing

The following rules and deﬁnitions apply to arithmetic operations for real

numbers, with the constraint that no denominator can ever be equal to

zero. It’s good to memorize these, because an ‘‘automatic’’ knowledge of

them can save you some time as you hack your way through the occasional

forest of calculations that life is bound to confront you with. After a while,

these laws will seem like nothing more or less than common sense.

ZERO NUMERATOR

For all nonzero real numbers a, if 0 is divided by a, then the result is equal

to 0. This equation is always true:

0=a ¼ 0

ZERO DENOMINATOR

For all real numbers a,ifa is divided by 0, then the result a/0 is undeﬁned.

Perhaps a better way of saying this is that, whatever such an ‘‘animal’’

might happen to be, it is not a real number! Within the set of real numbers,

there’s no such thing as the ratio of a number or variable to 0.

If you’re into playing around with numbers, you can have lots of fun

trying to deﬁne ‘‘division by 0’’ and explore the behavior of ratios with zero

denominators. If you plan to go on such a safari, however, take note: The

‘‘animals’’ can be unpredictable, and you had better fortify your mind with

lots of ammunition and heavy armor.

CHAPTER 1 Numbers and Arithmetic 23

MULTIPLICATION BY ZERO

Whenever a real number is multiplied by 0, the product is equal to 0. The

following equations are true for all real numbers a:

a  0 ¼ 0

0  a ¼ 0

ZEROTH POWER

Whenever a nonzero real number is taken to the zeroth power, the result is

equal to 1. The following equation is true for all real numbers a except 0:

¼ 1

ARITHMETIC MEAN

Suppose that a

, a

, ..., a

be real numbers. The arithmetic mean, denoted

(also known as the average)ofa

, a

, ..., a

is equal to the sum of the

numbers, divided by the number of numbers. Mathematically:

¼ða

þ a

þ ...þ a

Þ=n

PRODUCT OF SIGNS

When two real numbers with plus signs (meaning positive, or larger than 0)

or minus signs (meaning negative, or less than 0) are multiplied by each

other, the following rules apply:

ðþÞðþÞ ¼ ðþÞ

ðþÞðÞ ¼ ðÞ

ðÞðþÞ ¼ ðÞ

ðÞðÞ ¼ ðþÞ

QUOTIENT OF SIGNS

When two real numbers with plus or minus signs are divided by each other,

the following rules apply:

ðþÞ=ðþÞ ¼ ðþÞ

ðþÞ=ðÞ ¼ ðÞ

ðÞ=ðþÞ ¼ ðÞ

ðÞ=ðÞ ¼ ðþÞ

PART 1 Expressing Quantities

POWER OF SIGNS

When a real number with a plus sign or a minus sign is raised to a positive

integer power n, the following rules apply:

ðþÞ

¼ ðþÞ

ðÞ

¼ ðÞ if n is odd

ðÞ

¼ ðþÞ if n is even

RECIPROCAL OF RECIPROCAL

For all nonzero real numbers, the reciprocal of the reciprocal is equal to

the original number. The following equation holds for all real numbers a

provided that a 6¼ 0:

1=ð1=aÞ¼a

PRODUCT OF SUMS

For all real numbers a, b, c, and d, the product of (a þ b) with (c þ d) is given

by the following formula:

ða þ bÞðc þ dÞ¼ac þ ad þ bc þ bd

CROSS MULTIPLICATION

Given two quotients or ratios expressed as fractions, the numerator of the ﬁrst

times the denominator of the second is equal to the denominator of the ﬁrst

times the numerator of the second. Mathematically, for all real numbers a,

b, c, and d where neither a nor b is equal to 0, the following statement is valid:

a=b ¼ c=d , ad ¼ bc

RECIPROCAL OF PRODUCT

For any two nonzero real numbers, the reciprocal (or minus-one power)

of their product is equal to the product of their reciprocals. If a and b are

both nonzero real numbers:

1=ðabÞ¼ð1=aÞð1=bÞ

ðabÞ

1

¼ a

1

CHAPTER 1 Numbers and Arithmetic 25

PRODUCT OF QUOTIENTS

The product of two quotients or ratios, expressed as fractions, is equal to the

product of their numerators divided by the product of their denominators.

For all real numbers a, b, c, and d, where b 6¼ 0 and d 6¼ 0:

ða=bÞðc=dÞ¼ðacÞ=ðbdÞ

RECIPROCAL OF QUOTIENT

For any two nonzero real numbers, the reciprocal (or minus-one power) of

their quotient or ratio is equal to the quotient inverted or the ratio reversed.

If a and b are real numbers where a 6¼ 0 and b 6¼ 0:

1=ða=bÞ¼b=a

ða=bÞ

1

¼ b=a

Still More Principles

The following general laws of arithmetic can be useful from time to time. You

don’t really have to memorize them, as long as you can look them up when

you need them. (But if you want to memorize them, go right ahead.)

QUOTIENT OF PRODUCTS

The quotient or ratio of two products can be rewritten as a product of two

quotients or ratios. For all real numbers a, b, c, and d, where cd 6¼ 0:

ðabÞ=ðcdÞ¼ða=cÞðb=dÞ

¼ða=dÞðb=cÞ

QUOTIENT OF QUOTIENTS

The quotient or ratio of two quotients or ratios can be rewritten either as

a product of two quotients or ratios, or as the quotient or ratio of two

products. For all real numbers a, b, c, and d, the following equations hold

PART 1 Expressing Quantities

as long as b 6¼ 0, c 6¼ 0, and d 6¼ 0:

ða=bÞ=ðc=dÞ¼ða =bÞð d=cÞ

¼ða=cÞðd=bÞ

¼ðadÞ=ðbcÞ

SUM OF QUOTIENTS (COMMON DENOMINATOR)

If two quotients or ratios have the same nonzero denominator, their sum is

equal to the sum of the numerators, divided by the denominator. For all

real numbers a, b, and c, where c 6¼ 0:

a=c þ b=c ¼ða þ bÞ=c

SUM OF QUOTIENTS (GENERAL)

If two quotients or ratios have diﬀerent nonzero denominators, their sum

must be found using a speciﬁc formula. For all real numbers a, b, c, and d,

where b 6¼ 0 and d 6¼ 0:

a=b þ c=d ¼ðad þ bcÞ=ðbdÞ

INTEGER ROOTS

Suppose that a is a positive real number. Also suppose that n is a positive

integer. Then the nth root of a can also be expressed as the 1/n power of a.

Thus, the second root (or square root) is the same thing as the 1/2 power;

the third root (or cube root) is the same thing as the 1/3 power; the fourth

root is the same thing as the 1/4 power; and so on.

RATIONAL-NUMBER POWERS

Suppose that a is a real number. Also suppose that b is a rational number

such that b ¼ m/n, where m and n are integers and n 6¼ 0. Then the following

formula holds:

¼ a

m=n

¼ða

1=n

¼ða

1=n

CHAPTER 1 Numbers and Arithmetic 27

NEGATIVE POWERS

Suppose that a is a real number. Also suppose that b is a rational number.

Then the following formula holds:

b

¼ð1=aÞ

¼ 1=ða

SUM OF POWERS

Suppose that a is a real number. Also suppose that b and c are rational num-

bers. Then the following formula holds:

ðbþcÞ

¼ a

DIFFERENCE OF POWERS

Suppose a is a nonzero real number. Also suppose that b and c are rational

numbers. Then the following formula holds:

ðbcÞ

¼ a

LET a 5 x, b 5 y, AND c 5 z

In mathematical literature, the expression ‘‘Let such-and-such be the case’’

is often used instead of ‘‘Suppose that such-and-such is true’’ or ‘‘Imagine

such-and-such.’’ When you are admonished to ‘‘let’’ things be a certain

way, you are in eﬀect being asked to imagine, or suppose, that they are so.

That’s the way the next few principles are stated. Also, diﬀerent variables

are used in the next few statements: x replaces a, y replaces b, and z replaces

c. It’s a good idea to get used to this sort of variability (along with repetitive-

ness, ironically) to arm your mind against the assaults of mathematical

discourse.

PRODUCT OF POWERS

Let x be a real number. Let y and z be rational numbers. Then the following

formula holds:

¼ðx

PART 1 Expressing Quantities

QUOTIENT OF POWERS

Let x be a real number. Let y and z be rational numbers, with the constraint

that z 6¼ 0. Then the following formula holds:

y=z

¼ðx

1=z

POWER OF RECIPROCAL

Let x be a nonzero real number. Let n be a natural number. Then the nth

power of the reciprocal of x is equal to the reciprocal of x to the nth

power. That is:

ð1=xÞ

¼ 1=ðx

POWERS OF SUM

Let x and y be real numbers. Then the following formulas hold:

ðx þ yÞ

¼ x

þ 2xy þ y

ðx þ yÞ

¼ x

þ 3x

y þ 3xy

þ y

ðx þ yÞ

¼ x

þ 4x

y þ 6x

þ 4xy

þ y

POWERS OF DIFFERENCE

Let x and y be real numbers. Then the following formulas hold:

ðx  yÞ

¼ x

 2xy þ y

ðx  yÞ

¼ x

 3x

y þ 3xy

 y

ðx  yÞ

¼ x

 4x

y þ 6x

 4xy

þ y

PROBLEM 1-9

What is the value of (1/27)

1/3

SOLUTION 1-9

We are seeking the 1/3 power (or the cube root) of 1/27. This means we are

looking for some number which, when multiplied by itself and then multi-

plied by itself again, will produce 1/27.

Imagine some number y such that (1/y)

¼ (1/y)(1/y)(1/y) ¼ 1/27. Then y

must be the cube root of 27. Using a calculator, the cube root of 27 is found

CHAPTER 1 Numbers and Arithmetic 29

to be 3. From this we can conclude that 1/y ¼ 1/3, and thus that (1/3)

¼ 1/27.

The value we seek, the cube root (or 1/3 power) of 1/27, is therefore 1/3.

PROBLEM 1-10

Find the sum of 2/7 and 5/8.

SOLUTION 1-10

Use the formula for the sum of quotients in general, stated earlier in this

chapter. As long as b and d are nonzero, the following holds for all real

numbers a, b, c, and d:

a=b þ c=d ¼ðad þ bcÞ=ðbdÞ

Assign a ¼ 2, b ¼ 7, c ¼5, and d ¼ 8. Then plug the numbers into the formula:

2=7 þð5=8Þ¼½ð2  8Þþð7 5Þ=ð7  8Þ

¼½16 þð35Þ=56

¼19=56

Quiz

Refer to the text in this chapter if necessary. A good score is eight correct.

Answers are in the back of the book.

1. The reciprocal of a number is the same thing as the

(a) additive inverse

(b) exponent

(d) cube root

2. What is the value of 2

 3 þ 2?

(a) 46

(b) 50

(d) This expression is ambiguous without parentheses to tell us the

precedence of the operations.

3. The sum of 3/x and 5/x, where x represents any real number other

than 0, is

(a) 8/(x

)

(b) 8/(2x)

(d) 15/x

PART 1 Expressing Quantities