What is a Number Line and How to represent a Rational Number on a Number Line ?
- A number line is a line which represent all the number. A number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point
- We most shows the integers as specially-marked points evenly spaced on the line. but the line includes all real numbers, continuing forever in each direction, and also numbers not marked that are between the integers.
- It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.
- The number on the right side are greater than number on the left side
Note :
- Rational numbers can be represented on a number line as shown in the video.
- For decimal expression, we need to use the process of Successive Magnification
- Number like can be represent on number like using Pythagorus Theorem
What is process of Successive Magnification
Suppose we need to locate the decimal 2.665 on the Number line.
- We know that the number is between 2 and 3 on the number line.
- Divide the portion between 2 and 3 into 10 equal part.Then it will represent 2.1,2.2...2.9
- We know that 2.66 lies between 2.6 and 2.7
- Now lets divide the portion between 2.6 and 2.7 into 10 equal parts. Then these will represent 2.61,2.62,2.63,2.64,2.65,2.66...2.69
- Also 2.665 lies between 2.66 and 2.67
- Divide the portion between 2.66 and 2.67 into 10 equal parts. Then these will represent 2.661,2.662,2.663,2.664,2.665,2.666...2.669
- So we have located the desired number on the Number line.
This process is called the Process of successive Magnification
Note
- Pythagoras Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Using this theorem we can represent the irrational numbers on the number line.
- They have non terminating and non repeating decimal expression. If a number is non terminating and non repeating decimal expression,then it is irrational number.
- The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division
Pythagoras theorem to locate an irrational number √n on the real number line
Steps to locate irrational number:
(i) Step 1: Find the Pythagorean triplet for given √n. Let x and y be
the two other Pythagorean triplets than √n (Assume x > y).
(ii) Step 2: Out of x and y, locate from origin (O) the point which is larger x in this case on the real number line.
(iii) Step 3: Draw from x a perpendicular line segment of length y units.
(iv) Step 4: Draw an arc of radius Oy on the number line. The point where this arc will intersect represents √n.
In other Words
To find √x geometrically
1. First of all, mark the distance x unit from point A on the line so that AB = x unit.
2. From B mark a point C with the distance of 1 unit, so that BC = 1 unit.
3. Take the midpoint of AC and mark it as O. Then take OC as the radius and draw a semicircle.
4. From the point B draw a perpendicular BD which intersects the semicircle at point D.
The length of BD = √x.
To mark the position of √x on the number line, we will take AC as the number line, with B as zero. So C is point 1 on the number line.
Now we will take B as the centre and BD as the radius, and draw the arc on the number line at point E.
Now E is √x on the number line.
For Example:
Locate √5 on the number line.
(i) Firstly, we will find the other two numbers whose result will be √5 satisfying the Pythagoras theorem.
(ii) In this case √(2)2 + √(1)2 = √5.
(iii) Now, draw a number line. Mark point A which will be 2 units from
origin. Then draw perpendicular line segment AB of unit length. Take
origin as center and OB as radius; draw an arc intersecting number line
at C.
Locating √n point on number line for already drawn √n-1:
For Example:
Locate √3 on the number line.
(i) In this case, we will locate √2 on number as shown in the above example.
(ii) For already drawn √2, draw unit perpendicular length BD to OB. Now, keeping O as center draw an arc from point D which will intersect the number line at Q.
(iii) In the figure, OQ represents √3.
Example:
If 5 is a rational number and √7 is an irrational number then 5 + √7 and 5 - √7 are irrational numbers.
3. If we multiply or divide a non-zero rational number with an irrational number then also the outcome will be irrational.
Example:
If 7 is a rational number and √5 is an irrational number then 7√7 and 7/√5 are irrational numbers.
4. The sum, difference, product and quotient of two irrational numbers could be rational or irrational.
Example:
Surds
If 'a' is a positive rational number which cannot be expressed as the nth
The symbol n√ is called the radical sign, n is called the order of the surd and 'a' is called the radicand.
- a is a rational number
- n√ a is an irrational number
Identities Related to Square Roots
If p and q are two positive real numbers
Examples:
1. Simplify
2. Simplify
Rationalizing the Denominator
Rationalize the denominator means to convert the denominator containing square root term into a rational number by finding the equivalent fraction of the given fraction.
For which we can use the identities of the real numbers.
Example:
Rationalize the denominator
Example :
