Introduction to Rational numbers

Positive and Negative Rational Numbers

Consider the rational number 23. Both the numerator and denominator of this number are positive integers. Such a rational number is called a positivenegative rational number.

So, 34,57,29 etc. are positive rational numbers.

The numerator of −35 is a negative integer, whereas the denominator is a positive integer. Such a rational number is called a negativepositive rational number.

So, −57, −38, −95 etc.are negative rational numbers.

Is 8−3 a negative rational number?

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what about −3−5?

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Rational numbers

You began your study of numbers by counting objects around you. The numbers used for this purpose were called counting numbers or natural numbers.They are 1, 2, 3, 4, ... and so on.

By including 0 to natural numbers, we got the whole numbers, i.e., 0, 1, 2, 3, ... The negatives of natural numbers were then put together with whole numbers to make up integers.

Integers are ..., –3, –2, –1, 0, 1, 2, 3, .... We, thus, extended the number system, from natural numbers to whole numbers and from whole numbers to integers rational numbers.

You were also introduced to fractions.

These are numbers of the form

numeratordenominator

where the numerator is either 0 or a positive integer and the denominator, a positive integer. You compared two fractions, found their equivalent forms and studied all the four basic operations of addition, subtraction, multiplication and division on them.

In this Chapter, we shall extend the number system further. We shall introduce the concept of rational numbers along with their addition, subtraction, multiplication and division operations.

Let's know in detail? YesNo

Earlier, we have seen how integers could be used to denote opposite situations involving numbers.

For example, if the distance of 3 km to the right of a place was denoted by 3, then the distance of 5 km to the left of the same place could be denoted by -5-3.

If a profit of ₹ 150 was represented by 150 then a loss of ₹ 100 could be written as -100-150. There are many situations similar to the above situations that involve fractional numbers.

You can represent a distance of 750 m above sea level as 34 km. Can we represent 750m below sea level in km? Can we denote the distance of 34 km below sea level by −34? We can see −34 is neither an integer, nor a fractional number. We need to extend our number system to include such numbers.

Comparing Rational Numbers

We know how to compare two integers or two fractions and tell which is smaller or which is greater among them. Let us now see how we can compare two rational numbers.

Let's use the same method for rational numbers also. We can mark rational numbers on the number line. We can mark the points: −12 and −15 by divided the numberline into 1025 parts.

She marked −12 and −15 as follows: and found −12 >< −15

For example, to compare −75 and −53, we first compare 75 and 53.

We get 75 <> 53 and conclude that −75 >< −53.

Take five more such pairs and compare them.

Which is greater −38 or −27−27−38 ? And for: −43 or −32 −43−32?

Comparison of a negative and a positive rational number is obvious. A negative rational number is to the leftright of zero whereas a positive rational number is to the rightleft of zero on a number line. So, a negative rational number will always be lessgreater than a positive rational number.

Thus −27 <=> 12

To compare rational numbers −3−5 and −27 reduce them to their standard forms and then compare them.

Do 4−9 and −1636 represent the same rational number?

Solution:

YesNo

Because 4−9 = 4x−49x−4 = −16363616−1016

−1636 = −16+−435÷−4 = −4−994410

Reshma wanted to count the whole numbers between 3 and 10. From her earlier classes, she knew there would be exactly 6 whole numbers between 3 and 10. Similarly, she wanted to know the total number of integers between –3 and 3. The integers between –3 and 3 are –2, –1, 0, 1, 2. Thus, there are exactly integers between –3 and 3.

Thus, we find that number of integers between two integers are limited (finite). Will the same happen in the case of rational numbers also? Reshma took two rational numbers −35 and −13

So, we could find one more rational number between −35 and −13.

By using this method, you can insert as many rational numbers as you want between two different rational numbers.

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List three rational numbers between – 2 and – 1.

Solution:

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