Which topics will you learn here ?
- What are Rational Numbers ?
- Find Equivalent Rational Numbers
- Operation on Rational Numbers - Addition , Subtraction, Multiplication , Division.
- How to represent Rational Numbers on Number Line ?
- Properties of Rational Number
- Additive Inverse and Multiplicative Inverse
- How to find infinite Rational Numbers between 2 given Rational Numbers.
- How to represent Irrational Numbers on Number Line ?
- Process of Magnification
- Properties of Irrational Numbers
What are Rational Numbers ?
In Number system, a number is called rational number if it can be expressed in the form where p and q are integers ( q> 0).
Example : etc.
- In Number system; every integers, natural and whole number is a rational number as they can be expressed in terms of p/q.
- Rational Numbers are denoted by the alphabet 'Q'.
- ‘Rational’ comes from the word ‘ratio’, and Q comes from the word ‘quotient’.
- There are infinite rational number between two rational numbers.
- They either have termination decimal expression or repeating non terminating decimal expression.So if a number whose decimal expansion is terminating or non-terminating recurring then it is rational.
- The sum, difference and the product of two rational numbers is always a rational number.
- The quotient of a division of one rational number by a non-zero rational number is a rational number.
- Rational numbers satisfy the closure property under addition, subtraction, multiplication and division.
For more information about Terminating and Non-Terminating Rational Numbers , click here.
Equivalent Rational Numbers
- By multiplying or dividing the numerator and denominator of a rational number by the same integer, we can obtain another rational number equivalent to the given rational number.
- Numbers are said to be equivalent if they are proportionate to each other.
- We can find infinite number of Equivalent Rational Numbers for a given Rational Number.
Example
Therefore 1/2, 2/4, 4/8 are equivalent to each other as they are equal to each other.
Positive and Negative Rational Numbers
1. Positive Rational Numbers are the numbers whose both the numerator and denominator are positive.
Example: 3/4, 12/24 etc.
2. Negative Rational Numbers are the numbers whose one of the numerator or denominator is negative.
Example: (-2)/6, 36/(-3) etc.
Remark: The number 0 is neither a positive nor a negative rational number.
Rational Numbers on the Number Line
Representation of whole numbers, natural numbers and integers on a number line is done as follows
Rational Numbers can also be represented on a number line like integers i.e. positive rational numbers are on the right to 0 and negative rational numbers are on the left of 0.
Representation of rational numbers can be done on a number line as follows
Rational Numbers in Standard Form
- A rational number is in the standard form if its denominator is a positive integer and there is no common factor between the numerator and denominator other than 1.
- If any given rational number is not in the standard form then we can reduce it to its standard form or the lowest form by dividing its numerator and denominator by their HCF ignoring its negative sign.
Comparison of Rational Numbers
- To compare the two positive rational numbers we need to make their denominator same, then we can easily compare them.
- To make their denominator same, we need to take the LCM of the denominator of both the numbers.
case 2
- To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.
- To compare, we need to compare them as normal numbers.
- If we have to compare one negative and one positive rational number then it is clear that the positive rational number will always be greater as the positive rational number is on the right to 0 and the negative rational numbers are on the left of 0.
Rational Numbers between Rational Numbers
To find the rational numbers between two rational numbers, we have to make their denominator same then we can find the rational numbers.
Example: Find Rational Numbers between 1 and 2.
There are “infinite” numbers of rational numbers between any two rational numbers.
Operations on Rational Numbers - Addition
a. Addition of two rational numbers with the same denominator
i. We can add it using a number line.
Example:
Add 1/5 and 2/5
Solution:
On
the number line we have to move right from 0 to 1/5 units and then move
2/5 units more to the right. add your sum would be 3/5
ii. If we have to add two rational numbers whose denominators are same then we simply add their numerators and the denominator remains the same.
Example
b. Addition of two rational numbers with the different denominator
Operations on Rational Numbers - Subtraction
Additive Inverse
a + (-a) = 0, where a is a whole number
b +(-b) = 0, where b is an integer
(a/b) + (-a/b) = 0, where a/b is a rational number
So we say that (-a/b) is the additive inverse of a/b and a/b is the additive inverse for (-a/b)
If we have to subtract two rational numbers then we have to add the additive inverse of the rational number that is being subtracted to the other rational number
a - b = a + (-b)
Perform Subtraction on Rational Numbers with same denominators
In this method
- we will simply subtract the numerator
- the denominator remains the same.
Example
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Perform Subtraction on Rational Numbers with different denominators
In this method
- We will first make the denominators same ( By Cross Multiplication method or LCM Method- explained in the video)
- we will add the additive inverse of the second number to the first number.
Operations on Rational Numbers - Multiplication
a. Multiplication of a Rational Number with a Positive Integer.
To multiply a rational number with a positive integer
- we simply multiply the integer with the numerator and
- the denominator remains the same.
To multiply a rational number with a negative integer
- we simply multiply the integer with the numerator and
- the denominator remains the same
- the resultant rational number will be a negative rational number.
Example
To multiply a rational number with another rational number
- we have to multiply the numerator of two rational numbers and
- multiply the denominator of the two rational numbers.
Operations on Rational Numbers - Division.
Reciprocal or multiplicative inverse
- The multiplicative inverse of any rational number a/b is defined as b/a so that
(a/b) x (b/a) =1
- Zero does not have any reciprocal or multiplicative inverse
Product of Reciprocal
- If we multiply the reciprocal of the rational number with that rational number then the product will always be 1.
Division of Rational Numbers
- Write the first Rational number as it is
- Change the sign of division to multiplication
- Write the reciprocal of the second Rational Number
- Solve
- Bring the answer in Standard Form
Properties on Rational Numbers
(i) Closure Property
Rational numbers are closed under addition, subtraction and multiplication.
Division :
(-3)/5 ÷ 2/3 = (-9)/10, which is also a rational number.
but, For any rational number a, a ÷ 0 is not defined. So, rational number are not closed under division.
However, if we exclude zero then the rational numbers are closed under
division.
(ii) Commutativity
Addition:
- Two rational numbers can be added in any order, i.e.,
commutativity holds for rational numbers under addition. - For any two rational number a and b, a + b = b + a.
Subtraction:
- For any two rational number a and b, a - b ≠ b - a.
- Subtraction is not associative for rational numbers.
Multiplication:
- Multiplication is commutative for rational
numbers. - In general, a × b = b × a, for any two rational numbers a and b.
Division:
- In general, a ÷ b ≠ b ÷ a , for any two rational numbers a and b.
- Hence, division is not Cumulative for rational numbers.
(iii) Associativity
Addition:
- Addition is associative for rational numbers,
- For any three rational numbers a, b and c, a + (b + c) = (a + b) + c.
Subtraction:
- Subtraction is not associative for rational numbers.
Multiplication:
- Multiplication is associative for rational number
- For any three rational numbers a, b and c, a × (b × c) = (a × b) × c.
Division:
- Division is not associative for rational numbers.
(iv)Distributivity of multiplication over addition for rational number
- For all rational numbers a, b and c, a(b + c) = ab + ac
(v)Distributivity of multiplication over subtraction for rational number:
- For any three rational numbers a, b and c, a (b – c) = ab – ac
Additive Identity/Role of Zero
We observea + 0 = 0 + a = a, where a is a whole number
b + 0 = 0 + b = b, where b is an integer
c + 0 = 0 + c = c, where c is a rational number
Zero is called the identity for the addition of rational numbers. It is the additive identity for integers and whole numbers as well.
Multiplicative identity/Role of one
We observea × 1 = = a, where a is a whole number
b × 1 = b, where b is an integer
c × 1 = c, where c is a rational number
1 is the multiplicative identity for rational numbers. It is the multiplicative identity for integers and whole numbers as well
Additive Inverse
We observea + (-a) = 0, where a is a whole number
b +(-b) = 0, where b is an integer
(a/b) + (-a/b) = 0, where a/b is a rational number
So we say that (-a/b) is the additive inverse of a/b and a/b is the additive inverse for (-a/b)
What is the additive inverse of following numbers
a) 1/6
b) -8/9
Solution:
Additive Inverse of any rational number a/b is defined as -a/b so that (a/b) +(-a/b)=0
So for 1/6, additive inverse will be -1/6
Similarly for -8/9 ,it will be 8/9
Reciprocal or multiplicative inverse
The multiplicative inverse of any rational number a/b is defined as b/aZero does not have any reciprocal or multiplicative inverse
-54
-11/12
Solution
The multiplicative inverse of any rational number a/b is defined as b/a so that (a/b) x (b/a) =1
So answer would be
-54=-54/1 multiplicative inverse will 1/-54 =-1/54
Similarly for -11/12, the multiplicative inverse will be -12/11
Representation of Rational Number on a number line
- We draw a line.
- We mark a point O on it and name it 0. Mark a point to the right of
0. Name it 1. The distance between these two points is called unit
distance. - Mark a point to the right of 1 at unit distance and name it 2.
- Proceeding in this manner, we can mark points 3, 4, 5,
- Similarly we can mark – 1, – 2, – 3, – 4, – 5, ……… to the left of 0.
This line is called the number line. - This line extends indefinitely on both sides.
Note :
- The positive rational numbers are represented by points on the number
line to the right of O . - the negative rational numbers are represented by points on the number line to the left of O.
- Any rational number can be represented on this line. The denominator of the rational number indicates the number of equal parts into which the first unit has been divided whereas the numerator indicates as to how many of these parts are to be taken into consideration.
Rational Numbers Between Two Rational Numbers
- We can find infinitely many rational numbers between any two given
rational numbers. - We can take the help of the idea of the mean for this
purpose. Between two rational numbers x and y, there exists a rational number = (x+y)/2
Make the denominator Same and find Rational numbers accordingly.
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 :
Operations on Real Numbers
1. The sum, difference, product and quotient of two rational numbers will be rational.
Example:
2. If we add or subtract a rational number with an irrational number then the outcome will be irrational.
