Rational Numbers on a Number Line

You know how to represent integers on a number line. Let us draw one such number line.

Number Line.

The points to the right of 0 are denoted by + sign and are positive integers. The points to the left of 0 are denoted by - sign and are negative integers.

Representation of fractions on a number line is also known to you.

Let us see how the rational numbers can be represented on a number line.

Let us represent the number −12 on n the number line.

So if we have to show −12 on a number line, then, into how many equal parts should the length between 0 and -1 be divided? 256

We know how to mark the rational number 12. It is marked at a point which is half the distance between 0 and 1. So, −12 would be marked at a point half the distance between 0 and .

How far is the point: −12 from 0? −1221 units.

We know how to mark 32 on the number line. It is marked on the right of 0 and lies halfway between 1 and 2. Let us now mark −32 on the number line. It lies on the left of 0 and is at the same distance as 32 from 0.

In decreasing order, we have −12,−22 ( = -1), −32, −42( = -2). This shows that −32 lies between – 1 and – 2. Thus, −32 lies half-way between – 1 and .

How far is −32 from 0? -3/221 units.

Similarly, −13 is to the left of zero and at the same distance from zero as −13 is to the right. So as done above, −13 can be represented on the number line. Once we know how to represent −13 on the number line, we can go on representing −23,−43,−53 and so on All other rational numbers with different denominators can be represented in a similar way.

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Rational Numbers in Standard Form

The denominators of these rational numbers are positive integers and 1 is the only common factor between the numerators and denominators. Further, the negative sign occurs only in the numerator. Such rational numbers are said to be in standard form.

A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.

If a rational number is not in the standard form, then it can be reduced to the standard form.

Recall that for reducing fractions to their lowest forms, we divided the numerator and the denominator of the fraction by the same non zero positive integer. We shall use the same method for reducing rational numbers to their standard form.

Example 1

Reduce −4530 to the standard form.

We had to divide twice. First time by 3 and then by 5. This could also be done as

We have now understood how to reduce a given rational number into a standard form. Continue