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

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PART ONE

Expressing

Quantities

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CHAPTER

Numbers and

Arithmetic

Mathematics is expressed in language alien to people unfamiliar with it.

Someone talking about mathematics can sound like a rocket scientist.

Written mathematical documents are often laden with symbology. Before

you proceed further, look over Table 1-1. It will help you remember symbols

used in basic mathematics, and might introduce you to a few symbols you’ve

never seen before!

If at ﬁrst some of this stuﬀ seems theoretical and far-removed from ‘‘the

everyday world,’’ think of it as basic training, a sort of math boot camp. Or

better yet, think of it as the classroom part of drivers’ education. It was good

to have that training so you’d know how to read the instrument panel, ﬁnd

the turn signal lever, adjust the mirrors, control the headlights, and read the

road signs. So get ready for a drill. Get ready to think logically. Get your

mind into math mode.

Table 1-1 Symbols used in basic mathematics.

Symbol Description

{ } Braces; objects between them are elements of a set

) Logical implication; read ‘‘implies’’

() Logical equivalence; read ‘‘if and only if ’’

8 Universal quantiﬁer; read ‘‘for all’’ or ‘‘for every’’

9 Existential quantiﬁer; read ‘‘for some’’

| Logical expression; read ‘‘such that’’

& Logical conjunction; read ‘‘and’’

N The set of natural numbers

Z The set of integers

Q The set of rational numbers

R The set of real numbers

1 The set with no elements; read ‘‘the empty set’’ or ‘‘the null set’’

\ Set intersection; read ‘‘intersect’’

[ Set union; read ‘‘union’’

 Proper subset; read ‘‘is a proper subset of ’’

 Subset; read ‘‘is a subset of ’’

2 Element; read ‘‘is an element of ’’ or ‘‘is a member of ’’

=2 Non-element; read ‘‘is not an element of ’’ or ‘‘is not a member of ’’

¼ Equality; read ‘‘equals’’ or ‘‘is equal to’’

6¼ Not-equality; read ‘‘does not equal’’ or ‘‘is not equal to’’

(Continued )

PART 1 Expressing Quantities

Table 1-1 Continued.

Symbol Description

 Approximate equality; read ‘‘is approximately equal to’’

< Inequality; read ‘‘is less than’’

 Weak inequality; read ‘‘is less than or equal to’’

> Inequality; read ‘‘is greater than’’

 Weak inequality; read ‘‘is greater than or equal to’’

þ Addition; read ‘‘plus’’

 Subtraction, read ‘‘minus’’



Multiplication; read ‘‘times’’ or ‘‘multiplied by’’





Quotient; read ‘‘over’’ or ‘‘divided by’’

Ratio or proportion; read ‘‘is to’’

Logical expression; read ‘‘such that’’

! Product of all natural numbers from 1 up to a certain value; read ‘‘factorial’’

( ) Quantiﬁcation; read ‘‘the quantity’’

[ ] Quantiﬁcation; used outside ( )

{ } Quantiﬁcation; used outside [ ]

CHAPTER 1 Numbers and Arithmetic 5

Sets

A set is a collection or group of deﬁnable elements or members. Set elements

commonly include:

points on a line

instants in time

coordinates in a plane

coordinates in space

coordinates on a display

curves on a graph or display

physical objects

chemical elements

locations in memory or storage

data bits, bytes, or characters

subscribers to a network

If an object or number (call it a) is an element of set A, this fact is

written as:

a 2 A

The 2 symbol means ‘‘is an element of.’’

SET INTERSECTION

The intersection of two sets A and B, written A \ B, is the set C such that

the following statement is true for every element x:

x 2 C if and only if x 2 A and x 2 B

The \ symbol is read ‘‘intersect.’’

SET UNION

The union of two sets A and B, written A [ B, is the set C such that the

following statement is true for every element x:

x 2 C if and only if x 2 A or x 2 B

The [ symbol is read ‘‘union.’’

PART 1 Expressing Quantities

SUBSETS

A set A is a subset of a set B, written A  B, if and only if the following

holds true:

x 2 A implies that x 2 B

The  symbol is read ‘‘is a subset of.’’ In this context, ‘‘implies that’’ is

meant in the strongest possible sense. The statement ‘‘This implies that’’ is

equivalent to ‘‘If this is true, then that is always true.’’

PROPER SUBSETS

A set A is a proper subset of a set B, written A  B, if and only if the following

both hold true:

x 2 A implies that x 2 B

as long as A 6¼ B

The  symbol is read ‘‘is a proper subset of.’’

DISJOINT SETS

Two sets A and B are disjoint if and only if all three of the following

conditions are met:

A 6¼ 1

B 6¼ 1

A \ B ¼ 1

where 1 denotes the empty set, also called the null set. It is a set that doesn’t

contain any elements, like a basket of apples without the apples.

COINCIDENT SETS

Two non-empty sets A and B are coincident if and only if, for all elements x,

both of the following are true:

x 2 A implies that x 2 B

x 2 B implies that x 2 A

CHAPTER 1 Numbers and Arithmetic 7

Numbering Systems

A number is an abstract expression of a quantity. Mathematicians deﬁne

numbers in terms of sets containing sets. All the known numbers can be

built up from a starting point of zero. Numerals are the written symbols

that are agreed-on to represent numbers.

NATURAL AND WHOLE NUMBERS

The natural numbers, also called whole numbers or counting numbers, are built

up from a starting point of 0 or 1, depending on which text you consult.

The set of natural numbers is denoted N. If we include 0, we have this:

N ¼f0ó 1ó 2ó 3ó ...ó nó ...g

In some instances, 0 is not included, so:

N ¼f1ó 2ó 3ó 4ó ...ó nó ...g

Natural numbers can be expressed as points along a geometric ray or

half-line, where quantity is directly proportional to displacement (Fig. 1-1).

DECIMAL NUMBERS

The decimal number system is also called base 10 or radix 10. Digits in this

system are the elements of the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Numerals are

written out as strings of digits to the left and/or right of a radix point,

which is sometimes called a ‘‘decimal point.’’

In the expression of a decimal number, the digit immediately to the left of

the radix point is multiplied by 1, and is called the ones digit. The next digit

to the left is multiplied by 10, and is called the tens digit. To the left of this

are digits representing hundreds, thousands, tens of thousands, and so on.

The ﬁrst digit to the right of the radix point is multiplied by a factor of

1/10, and is called the tenths digit. The next digit to the right is multiplied

by 1/100, and is called the hundredths digit. Then come digits representing

thousandths, ten-thousandths, and so on.

Fig. 1-1. The natural numbers can be depicted as points on a half-line or ray.

PART 1 Expressing Quantities

BINARY NUMBERS

The binary number system is a scheme for expressing numbers using only the

digits 0 and 1. It is sometimes called base 2. The digit immediately to the left

of the radix point is the ‘‘ones’’ digit. The next digit to the left is the ‘‘twos’’

digit; after that comes the ‘‘fours’’ digit. Moving further to the left, the digits

represent 8, 16, 32, 64, and so on, doubling every time. To the right of the

radix point, the value of each digit is cut in half again and again, that is,

1/2, 1/4, 1/8, 1/16, 1/32, 1/64, and so on.

When you work with a computer or calculator, you give it a decimal

number that is converted into binary form. The computer or calculator does

its operations with zeros and ones, also called digital low and high states.

When the process is complete, the machine converts the result back into

decimal form for display.

OCTAL AND HEXADECIMAL NUMBERS

The octal number system uses eight symbols. Every digit is an element of the

set {0, 1, 2, 3, 4, 5, 6, 7}. Starting with 1, counting in the octal system goes like

this: 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, ...16, 17, 20, 21, ...76, 77, 100, 101, ...776,

777, 1000, 1001, and so on.

Yet another scheme, commonly used in computer practice, is the hexa-

decimal number system, so named because it has 16 symbols. These digits are

the decimal 0 through 9 plus six more, represented by A through F, the ﬁrst

six letters of the alphabet. Starting with 1, counting in the hexadecimal

system goes like this: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 10, 11, ...1E,

1F, 20, 21, ...FE, FF, 100, 101, ...FFE, FFF, 1000, 1001, and so on.

PROBLEM 1-1

What is the value of the binary number 1001011 in the decimal system?

SOLUTION 1-1

Note the decimal values of the digits in each position, proceeding from right

to left, and add them all up as a running sum. The right-most digit, 1, is mul-

tiplied by 1. The next digit to the left, 1, is multiplied by 2, for a running sum

of 3. The digit to the left of that, 0, is multiplied by 4, so the running sum is

still 3. The next digit to the left, 1, is multiplied by 8, so the running sum is 11.

The next two digits to the left of that, both 0, are multiplied by 16 and 32,

respectively, so the running sum is still 11. The left-most digit, 1, is multiplied

by 64, for a ﬁnal sum of 75. Therefore, the decimal equivalent of 1001011

is 75.

CHAPTER 1 Numbers and Arithmetic 9

PROBLEM 1-2

What is the value of the binary number 1001.011 in the decimal system?

SOLUTION 1-2

To solve this problem, begin at the radix point. Proceeding to the left ﬁrst,

ﬁgure out the whole-number part of the expression. The right-most digit, 1,

is multiplied by 1. The next two digits to the left, both 0, are multiplied

by 2 and 4, respectively, so the running sum is still 1. The left-most

digit, 1, is multiplied by 8, so the ﬁnal sum for the whole-number part of

the expression is 9.

Return to the radix point and proceed to the right. The ﬁrst digit, 0, is

multiplied by 1/2, so the running sum is 0. The next digit to the right, 1,

is multiplied by 1/4, so the running sum is 1/4. The right-most digit, 1, is

multiplied by 1/8, so the ﬁnal sum for the fractional part of the expression

is 3/8.

Adding the fractional value to the whole-number value in decimal form

produces the decimal equivalent of 1001.011, which is 9-3/8.

Integers

The set of natural numbers can be duplicated and inverted to form an

identical, mirror-image set:

N ¼f0ó  1ó  2ó  3ó ...g

The union of this set with the set of natural numbers produces the set of

integers, commonly denoted Z:

Z ¼ N [N

¼f...ó  3ó  2 ó  1ó 0ó 1ó 2ó 3ó ...g

POINTS ALONG A LINE

Integers can be expressed as points along a line, where quantity is directly

proportional to displacement (Fig. 1-2). In the illustration, integers

Fig. 1-2. The integers can be depicted as points on a line.

PART 1 Expressing Quantities