7.1 Identifying Irrational Numbers
In previous years you studied rational numbers. Recall that rational numbers are any number that can be
expressed as a fraction where the numerator and denominator are both integers. Sometimes you will see this
B
fraction written as G where O and 5 are both integers.
You might also recall that this means that repeating decimals, meaning decimals that follow a repeating
mmmmmmmmmm are rational numbers
numeric pattern as some point are both rational numbers. For example, 0.083m and 0. 142857
mmmmmmmmmm = . Remember that even terminating
because they are a repeating pattern. In fact, 0.083m =
and 0. 142857
decimals, meaning decimals that stop, are really repeating decimals and therefore rational. For example, 0.75 =
0.75000000000000 … which shows that the zero is repeating meaning 0.75 is rational. In fact, it is equal to .
All of this helps us to define irrational numbers. Recall that the prefix ir- means “not” so that we can define
irrational numbers as numbers that are not rational. In other words, an irrational number cannot be written as a
fraction. An irrational number written as a decimal would go forever and have no repeating pattern.
The most common example of this is the number π which you may know is approximately 3.14 or
.
However, both of those values are only rational estimates of π. Other than a few special numbers like π or e (which
you’ll learn about in later math courses), irrational numbers come up most often when dealing with square roots.
Recall that a square root is the inverse operation of squaring a number. In other words, we are asking
ourselves, “What number multiplied by itself will equal the given number?” The symbol for square root is √ and
you should remember some basics such as √25 = 5 or √0.49 = 0.7 when we take the principal (or positive) square
root.
When square roots don’t have exact solutions such as the examples above, they are irrational. So all of the
following are irrational numbers because they don’t have an exact solution: √60, √11, √2, √77, √21. In particular,
note that √2 is irrational. This is something you should have memorized.
Using Technology
Most calculators have a square root button, but that will not necessarily tell you for sure whether a number
is rational or not as you may not be able to plug in the whole number you want to find the square root of due to
the display screen size. Also, the calculator may give you an answer such as 0.1344567927 which looks like it could
be irrational since we can’t see a pattern, but the number 0. mmmmmmmmmmmmmmmmm
1344567927 is rational and we don’t know for sure if
the pattern repeats because the calculator did not return enough numbers. Rely on your brain and not the
calculator to determine if a number is rational or irrational.
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Lesson 7.1
Identify which of the following numbers are rational or irrational and explain how you know.
1. √25
2. √24
3. −√36
4. −√27
6. 0.45
7. 0. 2m
8. √49
9. √18
10. −√10
11.
mmm
13. 0. m42
14. 0.39
15. −√16
5.
12.
16. −√43
17. If the number 0.77 is displayed on a calculator that can only display ten digits, do we know whether it is
rational or irrational? In one complete sentence explain why.
18. If the number 0.123456789 is displayed on a calculator that can only display ten digits, do we know whether
it is rational or irrational? In one complete sentence explain why.
19. If the number 0.987098709 is displayed on a calculator that can only display ten digits, do we know whether
it is rational or irrational? In one complete sentence explain why.
20. If the number 0.425364758 is displayed on a calculator that can only display ten digits, do we know whether
it is rational or irrational? In one complete sentence explain why.
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