Page 10 - 新思维数学学生用书9 样章

1.1 Irrational numbers

Integers are whole numbers. For example, 13, −26 and 100 004 are integers.
Tip
3
4
You can write rational numbers as fractions. For example, 9 , −3 and
4 15 The set of
5
18 are rational numbers.
11 rational numbers
You can write any fraction as a decimal. includes integers.
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9 = 975 −3 4 = −3 26666666... 18 5 = 18 4545454...
.
.
.
4 15 11
The fraction either terminates (for example, 9.75) or it has recurring Tip
digits (for example, −3.266666666666… continues with 6s and Square roots
18.45454545454… continues with the digits 4 and 5 repeating). of negative
There are many square roots and cube roots that you cannot write as numbers do not
fractions. When you write these fractions as decimals, they do not belong to the
terminate and there is no recurring pattern. For example, a calculator set of rational
gives the answer 7 = 2 645751... The calculator answer is not exact. The
.
decimal does not terminate and there is no recurring pattern. Therefore, or irrational
7 is not a rational number. numbers. You
Numbers that are not rational are called irrational numbers. 7, 23, will learn more
3
3 10 and 45 are irrational numbers. Irrational numbers that are square about these
roots or cube roots are called surds. numbers if you
There are also numbers that are irrational but are not square roots or continue to study

cube roots. One of these irrational numbers is called pi, which is the mathematics to a
Greek letter π. Your calculator will tell you that π = 3.141592… You will higher level.
meet π later in the course.

Worked example 1.1

Do not use a calculator for this question.

a Show that 90 is between 9 and 10.
b N is an integer and 90 is between N and N + 1. Find the value of N.
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Answer

a 9 = 81 and 10 = 100
2
2
81 < 90 < 100 This means 90 is between 81 and 100.
So 81 < 90 < 100
And so 9 < 90 10
<
b 4 = 64 and 5 = 125
3
3
64 < 90 < 125 and so
3 64 < 3 90 < 3 125
So 4 < 3 90 < 5 and N = 4

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