1.1 Real numbers: algebra essentials  (Page 2/35)

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The set of rational numbers    is written as Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

1. a terminating decimal: $\text{\hspace{0.17em}}\frac{15}{8}=1.875,$ or
2. a repeating decimal: $\text{\hspace{0.17em}}\frac{4}{11}=0.36363636\dots =0.\overline{36}$

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing integers as rational numbers

Write each of the following as a rational number.

1. 7
2. 0
3. –8

Write a fraction with the integer in the numerator and 1 in the denominator.

1. $7=\frac{7}{1}$
2. $0=\frac{0}{1}$
3. $-8=-\frac{8}{1}$

Write each of the following as a rational number.

1. 11
2. 3
3. –4
1. $\frac{11}{1}$
2. $\frac{3}{1}$
3. $-\frac{4}{1}$

Identifying rational numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

1. $-\frac{5}{7}$
2. $\frac{15}{5}$
3. $\frac{13}{25}$

Write each fraction as a decimal by dividing the numerator by the denominator.

1. $-\frac{5}{7}=-0.\stackrel{\text{———}}{714285},$ a repeating decimal
2. $\frac{15}{5}=3\text{\hspace{0.17em}}$ (or 3.0), a terminating decimal
3. $\frac{13}{25}=0.52,$ a terminating decimal

Write each of the following rational numbers as either a terminating or repeating decimal.

1. $\frac{68}{17}$
2. $\frac{8}{13}$
3. $-\frac{17}{20}$
1. 4 (or 4.0), terminating;
2. $0.\overline{615384},$ repeating;
3. –0.85, terminating

Irrational numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even $\text{\hspace{0.17em}}\frac{3}{2},$ but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers    . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating rational and irrational numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

1. $\sqrt{25}$
2. $\frac{33}{9}$
3. $\sqrt{11}$
4. $\frac{17}{34}$
5. $0.3033033303333\dots$
1. $\sqrt{25}:\text{\hspace{0.17em}}$ This can be simplified as $\text{\hspace{0.17em}}\sqrt{25}=5.\text{\hspace{0.17em}}$ Therefore, $\sqrt{25}\text{\hspace{0.17em}}$ is rational.
2. $\frac{33}{9}:\text{\hspace{0.17em}}$ Because it is a fraction, $\frac{33}{9}\text{\hspace{0.17em}}$ is a rational number. Next, simplify and divide.
$\frac{33}{9}=\frac{\stackrel{11}{\overline{)33}}}{\underset{3}{\overline{)9}}}=\frac{11}{3}=3.\overline{6}$

So, $\text{\hspace{0.17em}}\frac{33}{9}\text{\hspace{0.17em}}$ is rational and a repeating decimal.

3. $\sqrt{11}:\text{\hspace{0.17em}}$ This cannot be simplified any further. Therefore, $\text{\hspace{0.17em}}\sqrt{11}\text{\hspace{0.17em}}$ is an irrational number.
4. $\frac{17}{34}:\text{\hspace{0.17em}}$ Because it is a fraction, $\text{\hspace{0.17em}}\frac{17}{34}\text{\hspace{0.17em}}$ is a rational number. Simplify and divide.
$\frac{17}{34}=\frac{\stackrel{1}{\overline{)17}}}{\underset{2}{\overline{)34}}}=\frac{1}{2}=0.5$

So, $\text{\hspace{0.17em}}\frac{17}{34}\text{\hspace{0.17em}}$ is rational and a terminating decimal.

5. $0.3033033303333\dots \text{\hspace{0.17em}}$ is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.

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