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Rational Numbers

Rational Numbers on a Number Line

Every one of you knows what a number line is. You know that positive numbers are located on the right side of the number line, while the negative numbers are located on the left side. So let us now learn how to locate rational numbers on the number line. Let us study this in detail.

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Rational Numbers on Number Line

The rational number are the numbers which can be represented on the number line. In the figure below, we can see the number line. There are positive numbers, zero and negative numbers on the number line. Suppose you want to plot the number  \( \frac{2}{3} \) on the number line.

Number LineNow, first draw the number line and divide the line. To represent rationals, we divide the distance between two consecutive part into ‘n’ number of parts. Here in this example, we will divide the distance between 0 and 1, 1 and 2 and so on in three equal parts. How do you know that we are supposed to divide it into three divisions? The denominator tells us. The number of divisions will be equal to the denominator.

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Construction

  1.  Draw a line and first mark zero on the number line.
  2. Now from 0, divide the line into three equal parts on the R.H.S of 0. The third point will be marked as 1.
  3. Repeat the same steps to locate the number 2 on the number line.
  4. Now take the same length on L.H.S of the zero and divide the line into three equal parts so that we can locate -1 on the number line.
  5. The most important thing before locating  \( \frac{2}{3} \) on the number line is to write the divisions.
  6. Now the point after 0 will be denoted by \( \frac{1}{3} \),
  7. Now increase the numerator by 1 and so the next point that means the third point will be \( \frac{2}{3} \).
  8. Exactly, in the same way, denote the points on the L.HS of o.
  9. Now locate \( \frac{2}{3} \) and name it as point P.

Example

Now let us see another example. Represent  \( \frac{-3}{4} \) on the number line.

Construction

  1.   Again draw a line and mark zero on the number line.
  2. Now as the denominator is 4, divide the line into four equal parts on the l.H.S of 0. The fourth point will be  -1.
  3. Now repeat the same steps to locate the number -2.
  4. To write the divisions the point after 0 will be denoted by \( \frac{-1}{4} \).
  5. Now increase the numerator by 1 and so the next point that means the third point will be \( \frac{-2}{3} \).
  6. Now locate \( \frac{-1}{4} \) on the number line and name it as point P.
  7. Exactly, in the same way, denote the points on the R.HS of o.

In the same way, we can locate any number, may it be the positive number or negative number. In a rational number, the denominator tells the number of equal parts into which the first unit is to be divided.

Solved Examples for You

Question: Which of the following rational numbers lies between 0 and -1?

  1. 0
  2. -1
  3. \( \frac{-1}{4} \)
  4. \( \frac{1}{4} \)

Solution: C. It is very clear that o and -1 cannot lie between o and -1. Also, o = \( \frac{0}{4} \) and -1 = \( \frac{-4}{4} \). So it clear that \( \frac{-1}{4} \) lies between  o and -1.

Question: On the real number line below, numbers increase in value from left to right. If B > 0 then the value of A  must be:

Number Line

  1. Negative
  2. Positive
  3. Less than B
  4. Greater than B
  5. between 0 and B

Solution: C. Here the number increases from the value from left to right. Begin with zero the number increases in value from left to right and number decreases in value from right to left. So if we are going right to left, then the number decreases. So the value of A is less than B.

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