1.8 The real numbers  (Page 2/13)

Identify integers, rational numbers, irrational numbers, and real numbers

We have already described numbers as counting number s , whole number s , and integers    . What is the difference between these types of numbers?

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number    is a number that can be written as a ratio of two integers.

All signed fractions, such as $\frac{4}{5},-\frac{7}{8},\frac{13}{4},-\frac{20}{3}$ are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to $\frac{3}{1},\frac{6}{2},\frac{9}{3},\frac{12}{4},\frac{15}{5}\text{…}$

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers ! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer $\mathrm{-8}$ could be written as the decimal $\mathrm{-8.0}.$ So, clearly, some decimals are rational.

Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means $7\frac{3}{10},$ we can write it as an improper fraction, $\frac{73}{10}.$ So 7.3 is the ratio of the integers 73 and 10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 or $\mathrm{-1.2684})$ is a rational number. We can use the place value of the last digit as the denominator when writing the decimal as a fraction.

Let’s look at the decimal form of the numbers we know are rational.

We have seen that every integer is a rational number , since $a=\frac{a}{1}$ for any integer, a . We can also change any integer to a decimal by adding a decimal point and a zero.

We have also seen that every fraction is a rational number . Look at the decimal form of the fractions we considered above.