Numbers and Operations - Easy Mathematics Step-by-Step

Easy Mathematics Step-by-Step (2012)

Chapter 1. Numbers and Operations

In this chapter, you learn about the various sets of numbers that make up the real numbers.

Natural Numbers and Whole Numbers

The natural numbers (or counting numbers) are the numbers

1, 2, 3, 4, 5, 6, 7, 8, …

The three dots indicate that the pattern continues without end.

You can represent the natural numbers as equally spaced points on a number line, increasing endlessly in the direction of the arrow, as shown in Figure 1.1.

Figure 1.1 Natural numbers

When you include the number 0 with the set of natural numbers, you have the whole numbers:

0, 1, 2, 3, 4, 5, 6, 7, 8, …

The number 0 is a whole number, but not a natural number.

Like the natural numbers, you can represent the whole numbers as equally spaced points on a number line, increasing endlessly in the direction of the arrow, as shown in Figure 1.2.

Figure 1.2 Whole numbers

The graph of a number is the point on the number line that corresponds to the number, and the number is the coordinate of the point. You graph a set of numbers by marking a large dot at each point corresponding to one of the numbers. The graph of the numbers 2, 3, and 7 is shown in Figure 1.3.

Figure 1.3 Graph of 2, 3, and 7

Integers

On the number line shown in Figure 1.4, the point 1 unit to the left of 0 corresponds to the number –1 (read as “negative one”), the point 2 units to the left of 0 corresponds to the number –2, the point 3 units to the left of 0 corresponds to the number –3, and so on. The number –1 is the opposite of 1, –2 is the opposite of 2, –3 is the opposite of 3, and so on. The number 0 is its own opposite.

Figure 1.4 Whole numbers and their opposites

A number and its opposite are exactly the same distance from 0. For instance, 3 and –3 are opposites, and each is 3 units from 0.

The number 0 is neither positive nor negative.

The whole numbers and their opposites make up the integers:

…, –3, –2, –1, 0, 1, 2, 3, …

The integers are either positive (1, 2, 3, …), negative (…, –3, –2, –1), or 0. Positive numbers are located to the right of 0 on the number line, and negative numbers are to the left of 0, as shown in Figure 1.5.

Figure 1.5 Integers

It is not necessary to write a + sign on positive numbers (although it’s not wrong to do so). If no sign is written, then you know the number is positive.

Problem Find the opposite of the given number.

a. 8

b. –4

Solution

a. 8

Step 1. Describe the location of 8 and its opposite on a number line.
8 is 8 units to the right of 0. The opposite of 8 is 8 units to the left of 0.

Step 2. State the opposite of 8.
The number that is 8 units to the left of 0 is –8.

b. –4

Step 1. Describe the location of –4 and its opposite on a number line.
–4 is 4 units to the left of 0. The opposite of –4 is 4 units to the right of 0.

Step 2. State the opposite of –4.
The number that is 4 units to the right of 0 is 4.

Problem Graph the integers –5, –2, 3, and 7.

Solution

Step 1. Draw a number line.

Step 2. Mark a large dot at each of the points corresponding to –5, –2, 3, and 7.

Rational, Irrational, and Real Numbers

The number is an example of a rational number. A rational number is a number that can be expressed as a quotient of an integer divided by an integer other than 0. That is, the rational numbers are all the numbers that can be expressed as

The number 0 is excluded as a denominator for because division by 0 is undefined, so has no meaning no matter what number you put in the place of p.

Fractions, decimals, and percents are rational numbers. All of the natural numbers, whole numbers, and integers are rational numbers as well because you can express each of these numbers, as shown here.

The decimal representations of rational numbers terminate or repeat. For instance, is a rational number whose decimal representation terminates, and is a rational number whose decimal representation repeats. You can show a repeating decimal by placing a line over the block of digits that repeats, like this: . You also might find it convenient to round the repeating decimal to a certain number of decimal places. For instance, rounded to two decimal places, .

The symbol ≈ means “is approximately equal to.”

Note: Fractions, decimals, and percents are discussed at length in Chapters 5–7.

The irrational numbers are the real numbers whose decimal representations neither terminate nor repeat. These numbers cannot be expressed as ratios of two integers. For instance, the positive number that multiplies by itself to give 2 is an irrational number called the positive square root of 2. You use the square root symbol to show the positive square root of 2 like this: . Every positive number has two square roots: a positive square root and a negative square root. The other square root of 2 is . It also is an irrational number. (See Chapter 10 for an additional discussion of square roots.)

You cannot express as the ratio of two integers, nor can you express it precisely in decimal form. Its decimal equivalent continues on and on without a pattern of any kind, so no matter how far you go with decimal places, you can only approximate . For instance, rounded to three decimal places, . Do not be misled, however, because even though you cannot determine an exact value for , it is a number that occurs frequently in the real world. For instance, designers and builders encounter as the length of the diagonal of a square that has sides with length of 1 unit, as shown in Figure 1.6.

Figure 1.6 Diagonal of unit square

Not all roots are irrational. For instance, .

There are infinitely many square roots and other roots as well that are irrational.

Be careful: Square roots of negative numbers are not real numbers.

Two famous irrational numbers are π and e. The number π is the ratio of the circumference of a circle to its diameter, and the number e is used extensively in calculus. Most scientific and graphing calculators have π and e keys. To nine decimal place accuracy, π ≈ 3.141592654 and e ≈ 2.718281828.

Although, in the past, you might have used 3.14 or for π, π does not equal either of these numbers. The numbers 3.14 and are rational numbers, but π is irrational.

The real numbers are all the rational and irrational numbers put together. They are all the numbers on the number line (see Figure 1.7). Every point on the number line corresponds to a real number, and every real number corresponds to a point on the number line.

Figure 1.7 Real number line

The relationship among the various sets of numbers included in the real numbers is shown in Figure 1.8.

Figure 1.8 Real numbers

Note: Hereafter in this book, all numbers are understood to be real numbers.

Problem Categorize the given number according to the various sets of the real numbers to which it belongs. (State all that apply.)

a. 0

b. 0.75

c. –25

d. 36

e. –0.35

Solution

Step 1. Recall the descriptions of the natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Step 2. Categorize the number.

a. 0 is a whole number, an integer, a rational number, and a real number.

b. 0.75 is a rational number and a real number.

c. –25 is an integer, a rational number, and a real number.

d. 36 is a natural number, a whole number, an integer, a rational number, and a real number.

e. –0.35 is a rational number and a real number.

Problem Graph the numbers –4, –2.5, 0, , and 3.6.

Solution

Step 1. Draw a number line.

Step 2. Mark a large dot at each of the points corresponding to –4, –2.5, 0, , and 3.6.

Terminology for the Four Basic Operations

For the math you do in your everyday world, you work with the real numbers. Addition, subtraction, multiplication, and division are the four basic operations you use. Each of the operations has special symbolism and terminology associated with it. Table 1.1 shows the terminology and symbolism for the operations.

Table 1.1 The Four Basic Operations

As you can see from the examples in Table 1.1, addition and subtraction “undo” each other. Similarly, multiplication and division undo each other, as long as no division by 0 occurs.

Division Involving Zero

You must be very careful when you have zero in a division problem. The number 0 can be the dividend, provided the divisor is not 0—the quotient will be 0. But 0 can never be the divisor. The quotient of any number divided by 0 has no meaning; that is, division by 0 is undefined.

Problem State whether the quotient is 0 or undefined.

Solution

Step 1. Recall that division by 0 is undefined, so state that the quotient is undefined.

Step 1. Recall that the quotient is 0 when 0 is divided by a nonzero number, so state that the quotient is 0.

Step 1. Recall that division by 0 is undefined, so state that the quotient is undefined.

Properties of Real Numbers

The real numbers have the following 11 properties under the operations of addition and multiplication.

For all real numbers a, b, and c, you have:

1. Closure Property of Addition. is a real number. This property guarantees that the sum of any two real numbers is always a real number.

Examples

2. Closure Property of Multiplication. is a real number. This property guarantees that the product of any two real numbers is always a real number.

Examples

3. Commutative Property of Addition. . This property allows you to reverse the order of the numbers when you add, without changing the sum.

Examples

4. Commutative Property of Multiplication. . This property allows you to reverse the order of the numbers when you multiply, without changing the product.

Examples

5. Associative Property of Addition. . This property says that when you have three numbers to add together, the final sum will be the same regardless of the way you group the numbers (two at a time in the same order) to perform the addition.

Example

Suppose you want to compute . In the order given, you have two ways to group the numbers for addition:

Either way, 16 is the final sum.

6. Associative Property of Multiplication. . This property says that when you have three numbers to multiply together, the final product will be the same regardless of the way you group the numbers (two at a time in the same order) to perform the multiplication.

Example

Suppose you want to compute . In the order given, you have two ways to group the numbers for multiplication:

Either way, 7 is the final product.

The associative property is needed when you have to add or multiply more than two numbers because you can do addition or multiplication on only two numbers at a time. Thus, when you have three numbers, you must decide which two numbers you want to start with—the first two or the last two (assuming you keep the same order). Either way, your final answer is the same.

7. Additive Identity Property. There is a real number 0, called the additive identity, such that and . This property guarantees that you have the number 0 for which its sum with any real number is the number itself.

Examples

8. Multiplicative Identity Property. There is a real number 1, called the multiplicative identity, such that and . This property guarantees that you have the number 1 for which its product with any real number is the number itself.

Examples

9. Additive Inverse Property. Every real number a has an additive inverse, –a (its opposite), such that and . This property guarantees that every real number has an opposite whose sum with the number is 0.

Examples

10. Multiplicative Inverse Property. Every nonzero real number a has a multiplicative inverse, (its reciprocal), such that and . This property guarantees that every real number, except 0, has a reciprocal whose product with the number is 1.

Example

11. Distributive Property. and . This property says that when you have a number times a sum, you can either add first and then multiply or multiply first and then add. Either way, the final answer is the same.

Example

can be computed two ways:

add first to obtain or multiply first to obtain

The distributive property is the only property that involves both addition and multiplication at the same time. Another way to express the distributive property is to say that multiplication distributes over addition.

Either way, the answer is 45.

Subtraction and division are not mentioned in the properties listed because you can always turn subtraction into addition by “adding the opposite,” and you can turn division into multiplication by “multiplying by the reciprocal.” That is,

When you subtract a number, you get the same answer as you do when you add its opposite.

When you divide by a nonzero number, you get the same answer as you do when you multiply by its reciprocal.

Problem Identify the property illustrated.

Solution

Step 1. Recall the 11 properties: closure property of addition, closure property of multiplication, commutative property of addition, commutative property of multiplication, associative property of addition, associative property of multiplication, additive identity property, multiplicative identity property, additive inverse property, multiplicative inverse property, and distributive property.