ESL Mathematics for Standardized Tests

Chapter 1

Real Numbers

Taking a standardized test in mathematics is an anxiety-provoking experience for many people. The best way to accomplish your goal of passing whatever test you must take is to learn math. Sometimes, this involves learning formulae and rules, but studying mathematics itself will give you the solid foundation that you need to feel confident on the day of your test.

1.0 A Brief Look at Numbers

Let’s begin by reviewing several different types of numbers:

Vllhole numbers include the counting numbers: 0, 1, 2, 3, 4, 5,...

Integers include whole numbers, as well as negative numbers such as -1, -2.

Prime numbers are numbers that can be divided evenly by only themselves and 1. Examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. The number 2 is the only even prime number. The rest are odd, such as number 7. Zero (0) is not a prime number.

Rational numbers include fractions (but not fractions whose denominator is zero) and their decimal equivalents. Examples are e9780738665016_i0003.jpg 1.574, and 7.231. The fraction e9780738665016_i0004.jpg is not a rational number, because its denominator is zero. The denominator of any whole number is 1, so e9780738665016_i0005.jpg for example.

Irrational numbers are numbers that cannot be written as fractions. They can be written as decimals that never end and have no pattern, such as 1.743968... (The three dots at the end mean the decimals keep going on forever.)

Real numbers are those that can be placed on a number line, and include both rational and irrational numbers, whole numbers, integers, and prime numbers.

Figure 1.1: A Number Line

1.1 The Number Line

Numbers can be represented on a number line (see fig. 1.1). Notice that the numbers to the right of 0 (zero) are positive (+), and those to the left are negative (-). Zero itself is neither a positive nor a negative number and is always less than or smaller than any positive number.

For any two unequal numbers, the larger of them will always be located on the number line to the right of the smaller one. For example, is -2 larger than -3, or is -2 less than -3? We can answer this question quickly if we draw a number line like the one shown in figure 1.1. We can see that -2 is to the right of -3, so -2 is larger than -3. We can write this in the following way: -2 > -3 (negative two is greater than negative three). The symbol > (or <) represents greater than (or less than). The larger number is always at the open end, so 7 > 3, and 3 < 7.

Example 1

Fill in the box with one of these three symbols: >, =, or <.

Step 1 Draw a number line similar to the one shown in figure 1.1. Your number line should also show the following negative integers placed to the left of zero: -1, -2, -3, -4, -5, -6, -7, -8.... Be sure -1 is next to 0, as in figure 1.1.

Step 2 Notice that -8 is to the left of -2 on your number line. So -8 must be less than -2.

Step 3 Place the less than (or smaller than) symbol in the box:

Ordering Integers in a Sequence

Now that we understand how to construct a number line, we can use this knowledge to position several integers in a sequence.

Example 2

Working from left to right, place the numbers +1, 0, -2, +3, and -3 in a sequence from the largest to the smallest.

Step 1 Draw a number line (see fig. 1.2).

Step 2 Mark the location of each number on the number line with a dot. Remember that numbers become larger as we move from left to right.

Figure 1.2: Ordering Numbers

Step 3 Looking at our number line above, we can see that the largest number is +3, and the next largest is +1. The smallest is -3. Thus, the correct order from the largest to the smallest is:

1.2 Working with Real Numbers

We begin this section by learning about numbers and place value. Each number can be made up of one or more digits. For example, the number 2 can also be written as 2.0 (pronounced two point zero).

Place Value

Each digit has a place value, depending on its position in the number (see table 1.1).

As we move down the table, we can see that each number is multiplied by 10. Consider the following examples:

Table 1.1: Numbers and Place Values

Table 1.2: Digits and Place Value

What is the place value of 0 in the number 1,023?

Reading from left to right, enter each digit in a separate box like the one shown in table 1.2.

The digit 0 is in the hundreds box. Thus, the place value of 0 in the number 1,023 is hundreds.

Example 3

What is the place value of 5 in the number 1,451?

Step 1 Draw a box similar to the one shown in table 1.2.

Step 2 Note which place value the digit 5 is in.

The answer is tens.

A number such as 25.321 is called a decimal. Notice that there are digits to the right of the decimal point (.). This rational number is pronounced twenty-five point three, two, one. The digits 3, 2, and 1 after the decimal point in the number 25.321 also have a place value.

As we move down the table, we can see that each number is divided by 10. Consider the following examples:

Note that we add a th to the place value to indicate decimal values. That is, we say tenths rather than tens.

Example 4

How do we read the number 47.005?

Step 1 Draw a box showing decimals and place values.

Step 2 Insert the digits to the right of the decimal point in the box.

Thus, we read the number 47.005 as forty-seven and five thousandths. We can also say forty-seven point zero, zero, five.

Table 1.3: Decimals and Place Value

Rounding Numbers Up and Down

Rounding a number is a straightforward method of expressing an approximate value for the number. Numbers can be either rounded up to any given place value, or rounded down.

Example 5

Round 38,731 to the nearest thousand (place value).

Step 1 Note the given place value (in this example, 8), and then look at the digit immediately to the right (7).

Step 2 If the digit to the right of the given place value is equal to or greater than 5, add 1 to the number in the place value position, and replace all the digits to the right of the number with zeros. This is called rounding up. Add 1 to the 8 in the thousands’ place value, and replace all the digits to the right with zeros.

The solution is 39,000.

Example 6

Round 27,331 to the nearest thousand (place value).

Step 1 Note the given place value (in this example, 7), and then look at the digit immediately to the right (3).

Step 2 If the digit to the right of the given place value is less than 5, the number in the given place value remains the same, but all the digits to the right of this number are replaced with zeros. This is called rounding down.

The answer is 27,000.

Changing Decimals to Fractions

Example 7 shows us how to change a decimal number to a fraction.

Example 7

How do we change 19.54 to fractional notation?

Step 1 How many numbers are to the right of the decimal point? Two—the digits 5 and 4.

19.54

Step 2 Move the decimal point two places to the right. Begin writing a fraction with 1954 as the numerator (the number on the top of a fraction).

Remember that the denominator of any number is 1.

Since the decimal point in the numerator was moved two places, the decimal point in the denominator (the number on the bottom) is also moved two places. So it is 1 followed by two zeros.

The answer is e9780738665016_i0020.jpg

Example 8

Express 0.753 in fractional notation.

Step 1 How many numbers are to the right of the decimal point? Three—the numbers 7, 5, and 3.

Step 2 Move the decimal point three places to the right.

Step 3 Write a fraction with 753 as the numerator.

Step 4 Since the decimal point in the numeratorwas moved three places, the denominator is 1 followed by three zeros.

The result: e9780738665016_i0023.jpg

Changing Fractions to Decimals

To change a fraction to a decimal, we divide the fraction’s numerator by its denominator.

Example 9

change e9780738665016_i0024.jpg to decimal notation.

Step 1 Divide the numerator 4 by the denominator 5.

Remember that 4 is the same as 4.0.

Divide until the remainder is 0 (zero) or, as we will see in the next example, the division may repeat itself. (Often an exercise will tell you how many places to go, such as Change e9780738665016_i0026.jpg to decimal notation to two places.)

The answer is 0.8.

Since the remainder equals zero, the result is called a terminating decimal.

Example 10

Change e9780738665016_i0028.jpg to decimal notation.

Step 1 Divide the numerator 2 by the denominator 3.

The remainder is 2, and this pattern keeps repeating itself.

The solution is e9780738665016_i0031.jpg

Since the remainder repeats itself, the decimal is called a repeating decimal. The bar sign over the last 6 indicates that the decimal is a repeating one.

Ordering Real Numbers in a Sequence

To order real numbers in a sequence, we build upon the knowledge we learned about number lines, place value, fractions, and decimals. All real numbers can be positioned on a number line. These include, but are not limited to, integers, fractions, and decimals (see fig. 1.3).

Figure 1.3: Ordering Real Numbers

Example 11

Place the following numbers from smallest to largest on a number line: 0, 3.5, -1.5, 2.75, 1

Step 1 Draw a number line similar to the one shown in figure 1.3.

Step 2 Mark the numbers on the line.

Step 3 Write down the numbers in order from smallest to largest.

The answer: -1.5, 0,1, 2.75, 3.5

Notice that for positive numbers, we do not have to place the plus sign (+) in front of the number. For example, +3.5 is usually written as 3.5.

Example 12

Jack is working in his parents’ hardware store during the summer vacation. If Jack is asked to stock packets of nails on the shelf from the left to the right, from the smallest size in inches (in.) to the largest, in which order would he arrange the nails whose sizes are e9780738665016_i0033.jpg in., e9780738665016_i0034.jpg in., e9780738665016_i0035.jpg in., and e9780738665016_i0036.jpg in.?

Step 1 Express each fraction as a decimal:

Step 2 Compare the decimals for each fraction. The smallest fraction is the one with the smallest decimal. Thus, we order the fractions from left to right, from smallest to largest as follows:

.1875, .28125, .625, .75, or

Stocking the shelf from left to right, Jack begins with the e9780738665016_i0039.jpg in. nails, then continues with the e9780738665016_i0040.jpg in., the e9780738665016_i0041.jpg in., and ends with the largest nails, e9780738665016_i0042.jpg in.

Comparing Fractions

You can compare fractions by converting to their decimals, as shown in example 12. Or, for a quicker answer, you can use a method called cross multiplication to find out which of two fractions is the larger or the smaller.

Example 13

Is e9780738665016_i0043.jpg 7 larger than e9780738665016_i0044.jpg

Step 1 Begin at the bottom (the denominator position of the fraction) and multiply diagonally (follow the arrows shown in fig. 1.4). The result obtained from multiplication is called the product.

Figure 1.4: Cross Multiplying

Step 2 Write down the products next to each of the two fractions.

Step 3 Write down which of the two products is the larger. The larger is 28.

Step 4 The fraction located next to the larger product is the larger fraction of the two.

Thus, e9780738665016_i0046.jpg is larger than e9780738665016_i0047.jpg

Example 14

IS e9780738665016_i0048.jpg larger than e9780738665016_i0049.jpg

Step 1 Use the cross multiplication method shown in figure 1.4.

Step 2 Write down the product next to each one of the two fractions.

Step 3 The two products are 30 and 45. Notice that 30 is less than 45.

Thus, e9780738665016_i0051.jpg is smaller than e9780738665016_i0052.jpg

1.3 Properties of Real Numbers

In this section, we study several properties of real numbers that help us to understand mathematics better.

The Associative Property of Addition

The associative property of addition means that it doesn’t matter in which order we group numbers; we always obtain the same result. Thus, for any real numbers a, b, and c:

Let’s test this property by assigning values to the letters a, b, and c.

Example 15

Let a = 2, b = 1, and c= 3.

a + (b + c) =

2 + (1 + 3) =

2 + (4) =

2+4 =6

and

(a + b) + c =

(2 + 1) + 3 =

(3) + 3 =

Notice we obtained the same answer—the number 6.

The Commutative Property of Addition

For any real numbers a and b,

The commutative property of addition means that the order in which we add numbers is not significant, because a + b will give us the same result as b + a.

Example 16

Let a = 5 and b = 4.

a + b =

5 + 4 = 9

and

b + a =

4 + 5 = 9

The Additive Identity Property

When any real number a is added to zero (which is called the additive identity), the result is the original number a. Thus,

Example 17

If a = 7, then

a + 0 =

7+0=7

MATH TIPS

Multiplication & Division Symbols

Several ways are used to indicate that two or more numbers are to be multiplied or divided.

The following symbols all denote multiplication as shown by the examples:

3 · 3

3 × 3

3(3)

(3)(3)

Here are two symbols used in examples that denote division:

3 ÷ 3

The Associative Property of Multiplication

For any real numbers a, b, and c:

The associative property of multiplication states that the way in which we group numbers when we multiply them is not significant, because we obtain the same product regardless of how a, b, and c are grouped.

Example 18

Let a = 1, b = 2, and c = 3.

a(bc) =

1(2 × 3) =

1(6) = 6

and

(ab)c =

(1 × 2)3 =

(2)3 = 6

The Commutative Property of Multiplication

For any real numbers a and b,

The commutative property of multiplication states that no matter how we order a and b, we obtain the same product.

Example 19

If a = 4 and b = 1, then

ab = 4 × 1 = 4 and

ba = 1 × 4 = 4

The Multiplicative Identity Property

When any real number a is multiplied by 1 (called the multiplicative identity), the result is the original number a. Thus,

Example 20

Let a = 5.

a × 1 =

5 × 1 = 5

Subtract

Subtract means to take away. For example, when we are required to

subtract 5 from 9, we write this mathematically as 9 - 5.

The Distributive Property

Unlike other properties, the distributive property involves multiplication and either addition or subtraction. This property states that for any real numbers a, b, and c:

Example 21

Let a = 1, b = 2, and c = 3.

a(b + c) =

1(2 + 3) =

1(5) = 5

and

ab + ac =

1(2) + 1(3) =

2+3 =5

Thus, a(b + c) = ab + ac.

Example 221

Let a = 1, b = 3, and c = 2.

a(b - c) =

1(3 - 2) =

1(1) = 1

and

ab - ac =

1(3) - 1(2)

3 -2 = 1

Thus, a(b - c) = ab - ac.

1.4 Absolute Value

This section on absolute value serves as the foundation for further work on the addition, subtraction, multiplication, and division of real numbers. We can define absolute value as the distance of a number from zero (its point of origin) on a number line. The symbol |x| is used to indicate the absolute value of a number x. Figure 1.5 shows that the number 3 is three units from zero on the number line. Thus, we say that the absolute value of 3 is 3. However, -3 is also three units from zero, so the absolute value of -3 is also 3. We can say, then, that the absolute value of a number, with the exception of 0 (zero), is always positive.

Figure 1.5: Absolute Value

Example 23

What is the absolute value of 4?

There are 4 units between 0 and 4 on the number line, so the absolute value of 4 is 4.

Example 24

Find: |-3|

There are 3 units between zero and -3, so the absolute value of -3 is 3, which is the numerical part of the number.

Example 25

Step 1 Evaluate |-8|, and then add 6.

Example 26

Evaluate: 8 + |-8| -2

Step 1 Evaluate |-8|.

Step 2 Add and subtract.

1.5 Addition with Negatives

Adding Negative Numbers

Figure 1.6 illustrates the addition of negative numbers. For example, when we add -2 and