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Unit 1: Rational numbers. Real numbers

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3

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f) =

−−⋅−

− 1

41

31

134

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73:

351

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REPRESENTATION OF RATIONAL NUMBERS The rational numbers are represented in a straight line with the integers.

To represent the rational numbers with precision:

1 Take a line segment with the length of a unit.

2 Set a segment assistant from the origin point and divide it into the parts required. In this example, it is divided into 4 parts.

3 Join the last point of the segment assistant with the end of the auxiliary segment and draw segments parallels to each of the points obtained in the partition of the segment assistant.

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In practice, the terms rational number and fraction are used as synonyms.

14. Represent these rational numbers in a straight line

f) 46 g) − 3

4 h) 96

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Types of decimal numbers

There are three different types of decimal number: exact, recurring and other decimals.

An exact or terminating decimal is one which does not go on forever, so you can write down all its digits. For example: 0.125

A recurring decimal is a decimal number which do not stop after a finite number of decimal places, but where some of the digits are repeated over and over again. For example: 0.1252525252525252525... is a recurring decimal, where '25' is repeated forever.

There exists two types of recurring decimals:

• Pure recurring decimal: It becomes periodic just after the decimal point. Ex. 1.3535… ( 35 is called the period)

• Eventually recurring decimal: When the period is not settled just after de decimal point. There is a not repeating number placed between the decimal point and the period.

Other decimals are those which go on forever and don't have digits which repeat. For example pi = 3.141592653589793238462643.. They are called irrational numbers.

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Number sets

All the numbers in the Number System are classified into different sets and those sets are called as Number Sets.

The set of real numbers is divided into natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

Examples of Number Sets

• N= {1, 2, 3…….} is the set of natural numbers. • (0, 1, 2, 3……} is the set of whole numbers. • I= {……-3, -2, -1, 0, 1, 2, 3….} is the set of integers.

• The numbers 10.3, 102.25, etc. are rational numbers but not integers.

• , , etc. are irrational numbers.

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15. Classify these numbers

𝟑𝟓 , −𝟖

𝟐, 0.75, -15, √𝟕 , 632, 0.14532…

How to convert a fraction into a decimal number

Divide the numerator by the denominator

How to convert a decimal number into a fraction

Examples

𝟏.𝟐𝟑���� = 𝟏𝟐𝟑 − 𝟏𝟗𝟗

=𝟏𝟐𝟐𝟗𝟗

𝟏.𝟒𝟔� = 𝟏𝟒𝟔 − 𝟏𝟒

𝟗𝟗=𝟏𝟑𝟐𝟗𝟗

=𝟔𝟏𝟒𝟓

16. Express the following numbers in a fraction format. Write down which type of decimal number is everyone.

a. 2.25 b. 0.2… c. 0.18… d. 8. 6� e. 12.36� f. 0.254�

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Estimating and rounding decimal numbers

Estimating is an important part of mathematics, since it supposes a very handy tool for everyday life. Get in the habit of estimating amounts of money, lengths of time or distances.

There are different ways to estimate a number:

a) Rounding up or down. To round up, just cut off the number at the place you want, and sum one to the last digit. To round down, just cut off the number at the place you want. See the table below.

b) Rounding is the most usual kind of estimating. To round a decimal number, find the place value you want and look at the digit just to the right. If it is 5 or more, then round up. If it is 4 or less, round down. Then, sometimes we round up, and sometimes we round down. It depends on the digit to the right.

15,2753 Rounding up Rounding down Rounding 15, 3 15,2 15,3 15,28 15,27 15,28 15,276 15,275 15,275

8,7451

Representing real numbers

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17. Represent these irrational numbers

a) √𝟖 b) √𝟐𝟔 c) √𝟒𝟗

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Intervals and Rays (half-line)

An interval is a set of real numbers between two given points: a and b, which are called ends of the interval.

Open Interval

An open interval, (a, b), is the set of all real numbers greater than a and smaller than b.

Closed Interval

A closed interval, [a, b], is the set of all real numbers greater than or equal to a and less than or equal to b.

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Half-Closed Intervals

Half-closed intervals are also called half-open intervals.

When there are a set of points formed by two or more of these intervals, the sign (Union) is used between them.

The half-line is determined by a number.

x > a

x ≥ a

x < a

x ≤ a

18. Plot these intervals and half-lines

a) (2,5) b) [2,5) c) [2,5]

d) (1,∞)

e) [1,∞)

f) (-∞,-3)

g) (-∞,-3]