2.10: Properties of Rational Numbers versus Irrational Numbers
What if you wanted to identify a number like ? Would you classify it as rational or irrational? After completing this Concept, you'll be able to decide which category numbers like this one fall into.
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CK12 Foundation: 0210S Irrational Numbers (H264)
Guidance
Not all square roots are irrational, but any square root that can’t be reduced to a form with no radical signs in it is irrational. For example, is rational because it equals 7, but can’t be reduced farther than . That factor of is irrational, making the whole expression irrational.
Example A
Identify which of the following are rational numbers and which are irrational numbers.
a) 23.7
b) 2.8956
c)
d)
Solution
a) 23.7 can be written as , so it is rational.
b) 2.8956 can be written as , so it is rational.
c) We know from the definition of that the decimals do not terminate or repeat, so is irrational.
d) . We can’t reduce it to a form without radicals in it, so it is irrational.
Repeating Decimals
Any number whose decimal representation has a finite number of digits is rational, since each decimal place can be expressed as a fraction. For example, This decimal goes on forever, but it’s not random; it repeats in a predictable pattern. Repeating decimals are always rational; this one can actually be expressed as .
Example B
Express the following decimals as fractions.
a.) 0.439
b.)
Solution:
a.) 0.439 can be expressed as , or just . Also, any decimal that repeats is rational, and can be expressed as a fraction.
b.) can be expressed as , which is equivalent to .
Classify Real Numbers
We can now see how real numbers fall into one of several categories.
If a real number can be expressed as a rational number, it falls into one of two categories. If the denominator of its simplest form is one, then it is an integer . If not, it is a fraction (this term also includes decimals, since they can be written as fractions.)
If the number cannot be expressed as the ratio of two integers (i.e. as a fraction), it is irrational .
Example C
Classify the following real numbers.
a) 0
b) 1
c)
d)
e)
Solution
a) Integer
b) Integer
c) Irrational (Although it’s written as a fraction, is irrational, so any fraction with in it is also irrational.)
d) Irrational
e) Rational (It simplifies to , or .)
Watch this video for help with the Examples above.
CK12 Foundation: Irrational Numbers
Vocabulary
 The square root of a number is a number which gives the original number when multiplied by itself. In algebraic terms, for two numbers and , if , then .
 A square root can have two possible values: a positive value called the principal square root , and a negative value (the opposite of the positive value).
 A perfect square is a number whose square root is an integer.

Some mathematical properties of square roots are:
 Square roots of numbers that are not perfect squares (or ratios of perfect squares) are irrational numbers . They cannot be written as rational numbers (the ratio of two integers). In decimal form, they have an unending, seemingly random, string of numbers after the decimal point.
 Computing a square root on a calculator will produce an approximate solution since the calculator only shows a finite number of digits after the decimal point.
Guided Practice
Place the following numbers in numerical order, from lowest to highest.
Solution:
Since is the only negative number, it is the smallest.
Since , .
Since the , then .
Since , then
This means that the ordering is:
Practice
For questions 17, classify the following numbers as an integer, a rational number or an irrational number.
 Write 0.6278 as a fraction.
 Place the following numbers in numerical order, from lowest to highest.
 Use the marked points on the number line and identify each proper fraction.
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Description
Learning Objectives
Here you'll learn how to differentiate between rational and irrational numbers. You'll also learn how to classify and order real numbers.