Representation of Rational Numbers on the Number Line
In representation of rational numbers on the number line are discussed here. We know how to represent integers on the number line.To represent the integers on the number line, we need to draw a line and take a point O on it. Call it 0 (zero).
Set of equal distances on the right as well as on the left of O. Such a distance is known as a unit length. Let A, B, C, D, etc. be the points of division on the right of 'O' and A',B', C', D', etc. be the points of division on the left of 'O'. If we take OA = 1 unit, then clearly, the point A, B, C, D, etc. represent the integers 1, 2, 3, 4, etc. respectively and the point A', B', C', D', etc. represent the integers -1, -2, -3, -4, etc. respectively.
Note: The point O represents integer 0.
Thus, we may represent any integer by a point on the number line. Clearly, every positive integer lies to the right of O and every negative integer lies to the left of O.
We can represent rational numbers on the number line in the same way as we have learnt to represent integers on the number line.
In order to represent rational numbers on the number line, first we need to draw a straight line and mark a point O on it to represent the rational number zero. The positive (+ve) rational numbers will be represented by points on the number line lying to the right side of O and negative (-ve) rational numbers.
If we mark a point A on the line to the right of O to represent 1, then OA = 1 unit. Similarly, if we choose a point A' on the line to the left of O to represent -1, then OA' = 1 unit.
Consider the following examples on representation of rational numbers on the number line;
1. Represent \(\frac{1}{2}\) and \(\frac{-1}{2}\) on the number line.
Solution:
Draw
a line. Take a point O on it. Let the point O represent 0. Set off unit lengths
OA to the right side of O and OA' to the left side of O.
Then, A represents the integer 1 and A' represents the integer -1.
Now, divide the segment OA into two equal parts. Let P be the mid-point of segment OA and OP be the first part out of
these two parts. Thus, OP = PA = \(\frac{1}{2}\). Since, O represents 0 and A represents 1, therefore P represents the rational number \(\frac{1}{2}\).
Again,
divide OA' into two equal parts. Let OP' be the first part out of these two parts. Thus, OP' = PA' = \(\frac{-1}{2}\). Since, O represents 0 and A' represents -1, therefore P' represents the rational number \(\frac{-1}{2}\).
2. Represent \(\frac{2}{3}\) and \(\frac{-2}{3}\) on the number line.
Solution:
Draw
a line. Take a point O on it. Let it represent 0. From the point O set
off unit distances OA to the right side of O and OA' to the left side of O respectively.
Divide OA into three equal parts. Let OP be the segment showing 2 parts out of 3. Then the point P represents the rational number \(\frac{2}{3}\).
Again, divide OA' into three equal parts. Let OP' be the segment consisting
of 2 parts out of these 3 parts. Then, the point P' represents the
rational number \(\frac{-2}{3}\).
3. Represent \(\frac{13}{5}\) and \(\frac{-13}{5}\) on the number line.
Solution:
Draw a line. Take a point O on it. Let it represent 0.
Now, \(\frac{13}{5}\) = 2\(\frac{3}{5}\) = 2 + \(\frac{3}{5}\)
From
O, set off unit distances OA, AB and BC to the right of O. Clearly, the
points A, B and C represent the integers 1, 2 and 3 respectively. Now,
take 2 units OA and AB, and divide the third unit BC into 5 equal parts.
Take 3 parts out of these 5 parts to reach at a point P. Then the point
P represents the rational number \(\frac{13}{5}\).
Again, from the point O, set off unit distances to the left. Let these
segments be OA', A' B', B’ C’, etc. Then, clearly the points A’, B’ and
C’ represent the integers -1, -2, -3 respectively.
Now, = -\(\frac{13}{5}\) = -(2 + \(\frac{3}{5}\))
Take
2 full unit lengths to the left of O. Divide the third unit B’ C’ into 5
equal parts. Take 3 parts out of these 5 parts to reach a point P’.
Then, the point P’ represents the rational number -\(\frac{13}{5}\).
Thus, we can represent every rational number by a point on the number line.
● Rational Numbers
Introduction of Rational Numbers
Is Every Rational Number a Natural Number?
Is Every Rational Number an Integer?
Is Every Rational Number a Fraction?
Equivalent form of Rational Numbers
Rational Number in Different Forms
Properties of Rational Numbers
Lowest form of a Rational Number
Standard form of a Rational Number
Equality of Rational Numbers using Standard Form
Equality of Rational Numbers with Common Denominator
Equality of Rational Numbers using Cross Multiplication
Comparison of Rational Numbers
Rational Numbers in Ascending Order
Rational Numbers in Descending Order
Representation of Rational Numbers on the Number Line
Rational Numbers on the Number Line
Addition of Rational Number with Same Denominator
Addition of Rational Number with Different Denominator
Properties of Addition of Rational Numbers
Subtraction of Rational Number with Same Denominator
Subtraction of Rational Number with Different Denominator
Subtraction of Rational Numbers
Properties of Subtraction of Rational Numbers
Rational Expressions Involving Addition and Subtraction
Simplify Rational Expressions Involving the Sum or Difference
Multiplication of Rational Numbers
Properties of Multiplication of Rational Numbers
Rational Expressions Involving Addition, Subtraction and Multiplication
Reciprocal of a Rational Number
Rational Expressions Involving Division
Properties of Division of Rational Numbers
Rational Numbers between Two Rational Numbers
8th Grade Math Practice
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