Table of contents | |
Introduction | |
Irrational number | |
Real Numbers and their Decimal Expansion | |
Operations on Real Numbers | |
Laws of Exponent for Real Numbers | |
Summary |
Number system is a writing system used for expressing the numbers. A number is a mathematical object used to count, label and measure.
Example: N = {1, 2, 3, 4 ... .}
Natural numbers are represented on the number line as follows:
Whole numbers: The group of natural numbers including zero is called whole numbers. It is denoted by “W”. Zero is a very powerful number because if we multiply any number with zero it becomes zero. All natural numbers are called whole numbers.
Example: W = {0, 1, 2, 3, 4 ... .}
Whole numbers are represented on number line as follows:
Integers: The collection of all whole numbers and negatives of all natural numbers or counting numbers are called integers. They are denoted by “Z” or “I”. All whole numbers are integers, but all integers are not whole numbers
Example: Z or I = {... − 3, −2, −1, 0, 1, 2, 3 ... . .}
Integers represented on number line as follow:
Real numbers
Example: √2 , √3, −5, 0 , 1/5 5 etc.... All rational numbers are real numbers but all the real numbers are not rational numbers. Also, all irrational numbers are real numbers, but the reverse is not true
Rational numbers: Numbers that can be represented in the form of p/q where p & q are integers & q ≠ 0 are called rational numbers. The word rational came from the word ‘ratio’. It is denoted by letter Q and Q is taken from the word quotient. All integers are rational numbers.
Example: Q ={1/2, 3, -4, 3/2 etc ....}
Rational numbers also include natural numbers, whole numbers and integers. This can be explained using following example:
Example: -16 can also be written as -16/1 Here p = – 16 and q = 1. Therefore, the rational numbers also include the natural numbers, whole numbers and integers.
Equivalent rational numbers: The equivalent rational numbers are numbers that have same value but are represented differently.
Example: If a/b is equivalent to c/d and a/b = x then c/d = x
Also, if a/b = c/d, then a x d = b x c.
Irrational numbers: A number which can’t be expressed in the form of p/q and its decimal representation is non-terminating and non-repeating is known as irrational numbers. It is denoted by “S”.
Example: S = √2 = 1.4142135... . , √3 = 1.73205 ... . , etc
(i) When one rational number is to be determined:
Let a and b be two rational numbers, such that b > a. Then, is a rational number lying between a and b
Example: Find a rational number between 4 and 5 Here, 5 > 4
We know that, if a and b are two rational numbers, such that b > a. Then, is a rational number lying between a and b.
So, a rational number between 4 and 5 =
(ii) When more than one rational number are to be determined:
Let a and b be two rational numbers, such that b > a and we want to find n rational numbers between a and b. Then, n rational numbers lying between a and b are
(a + b), (a + 2d ), (a + 3d), ..........(a + nd), where, d = ( b -a ) / (n + 1)
Here, a and b are two rational numbers n is the number of rational numbers between a and b
Example (i): Find six rational numbers between 3 and 4
Here, 4 > 3
So, let a = 3 and b = 4 and n = 6
Since, d = b -a / n + 1
Now,
Hence, the required six rational numbers lying between 3 and 4 are
Example (ii): Find four rational numbers between − 6 and– 7.
Here, −6 > −7
Let a = − 7, b = − 6 and n = 4
Now, d = b -a / n + 1
=
So, four rational numbers between – 6 and − 7 are (a + d), (a + 2d), (a + 3d) and (a + 4d)
i.e., and
= and
= and
The above rational numbers are the rational numbers which lie between – 6 and – 7.
A number which can’t be expressed in the form of p/q and its decimal representation is non-terminating and non-repeating is known as irrational numbers. The set of irrational numbers is denoted by “S”.
Example: S = √2 ,√3 , π, etc. ..
Locate an irrational number on the number line:
We see how to locate an irrational number on number line with the help of following example:
Example: Locate √17 on the number line
Here, 17 = 16 + 1 = (4)2 + (1)2 (Sum of squares of two natural numbers)
So, we take a = 4 and b = 1
Now, draw OA = 4 units on the number line and then draw AB = 1 join OB.
By using Pythagoras theorem, in ∆OAB
Taking O as centre and radius equal to OB, draw an arc, which cuts the number line at C. Hence, OC represents √17.
Example: √2, √3, −5, 0, 1/5, 5 etc.... All rational numbers are real number but all real numbers are not rational numbers. Also, all irrational numbers are real number, but the reverse case is not true.
Real numbers and their decimal expansion: The decimal expansion of real numbers can be either terminating or non – terminating, repeating or non – terminating non – repeating. With the help of decimal expansion of real numbers, we can check whether it is rational or irrational.
(i) Decimal expansion of rational numbers:
Rational numbers are present in the form of p/q, where q ≠ 0, on dividing p by q, two main cases occur,
(a) Either the remainder becomes zero after few steps
(b) The remainder never becomes zero and gets repeating numbers.
Case I: Remainder becomes zero
On dividing p by q, if remainder becomes zero after few steps, and then the decimal expansion terminates or ends after few steps. Such decimal expansion is called terminating decimal expansion.
Example:
On dividing we get exact value 0.625 and remainder is zero.
So, we say that is a terminating decimal expansion.
On dividing we get exact value 0.625 and remainder is zero. So, we say that is a terminating decimal expansion.
Case II: Remainder never becomes zero
On diving p by q, if remainder never becomes zero and the sets of digits repeats periodically or in the same interval, then the decimal expansion is called non – terminating repeating decimal expansion. It is also called non – terminating recurring decimal expansion.
Example (i):
= 0.333... . . or = 0. 3= [The block of repeated digits is denoted by bar ‘– ‘over it]
On dividing we get the repeated number 3 and remainder never becomes zero. Hence, 1 by 3 has a non – terminating repeating decimal expansion.
Example (ii):
Hence, = 0.On dividing 4 by 13 we get the repeated numbers 0.30769230 again and again, and remainder never becomes zero. Hence, 4 by 13 has a non – terminating repeating decimal expansion.
Step III: Subtract equation (1) from equation (2) we get,
Example (i): Express in the form of p by q
Assume the given decimal expansion as x
Let,
x =
x = 0.666 ... ... . . (i)
Here, only 1 digit is repeating. Hence, multiplying both side of equation (i) by 10 we get,
10x = 6.66... ... ... (ii)
Subtracting equation (i) from (ii) we get,
10x – x = 6.66 – 0.66
9x = 6.66 – 0.66
9x = 6
x = 6/9 = 2/3
Hence,
Example (ii): Express 0.4 in the form, where p and q are integers and q ≠ 0
Let, x = 0.43535 ......(i)
Here, we see that one digit exit between decimal point and recurring number
So, we multiply both sides of equation (i) by 10, we get
10x = 4.3535 ...... (ii)
Here we see that two digits are repeated in the recurring number
So, we multiply equation (ii) by 100, we get
1000x = 435.3535 ...... (iii)
Subtracting equation (ii) from equation (iii), we get
1000x − 10x = 435.3535 − 4.3535
990x = 431
x =
Hence,
Example (iii): Express 0.00232323.... in the form, where p and q are integers and q ≠ 0
Let, x = 0.00232323 = 0.00 ......(i)
Here, we see that two digits exist between decimal point and recurring number
So, we multiply both sides of equation (i) by 100,
100x = 0.232323...... (ii)
Here we see that two digits are repeated in the recurring number
So, we multiply equation (ii) by 100, we get
10000x = 23.2323 ...... (iii)
Subtracting equation (ii) from equation (iii), we get
10000x − 100x = 23.2323 − 0.232329
990x = 23
x =
Hence, 0.002323 =
Example (i): Find the irrational number between and
∴
Now,
Thus,
It means that the required rational numbers will lie between and . Also, we know that the irrational numbers have non-terminating non-recurring decimals. Hence, one irrational number between and is 0.20101001000... . .
Example (ii): Find the two irrational numbers between and
If = 0.333 (Given)
We have, = 0.333 (Given)
Hence, = 2 × = 2 × 0.333 = 0.666
So, the two rational numbers between and may be 0.357643... and 0.43216 (In this solution we can write infinite number of such irrational numbers)
Example (iii): Find two irrational numbers between √2 and √3 .
We know that, the value of
√2 = 1. 41421356237606 and
√3 = 1.7320508075688772
From the above value we clearly say that √2 and √3 are two irrational numbers because the decimal representations are non-terminating non-recurring. Also, √3 > √2
Hence, the two irrational numbers may be 1.501001612 and 1.602019
Example: Visualise the representation of 4.36% on the number line up to 4 decimal places.
Here, we can understand representation of 4.36% on the number line up to 4 decimal places with the help of following steps:
Step I: Here, we know that, the number 4.36% lies between 4 and 5. Hence, first draw the number line and look at the portion between 4 and 5 by a magnifying glass.
Step II: Divide the above part into ten equal parts and mark first point to the right of 4 as 4.1, the second as 4.2 and so on.
Step III: Now 4.36 lies between 4.3 and 4.4. So, divide this portion again into ten equal parts and mark first point to the right of 4.3 as 4.31, second 4.32 and so on.
Step IV: Now, 4.366 lies between 4.36 and 4.37. So, divide this portion again into ten equal parts and mark first point to the right of 4.36 as 4.361, second 4.362 and so on.
Step V: To visualize 4.36 more accurately, again divide the portion between 4.366 and 4.367 into 10 equal parts and visualize the representation of 4.36% as in the figure given below:
We can proceed endlessly in this manner. Thus, 4.3666 is the 6th mark in this subdivision.
Example: Visualise 2.565 on the number line, using successive magnification.
We know that, 2.565 lies between 2 and 3. So, divide the part of the number line between 2 and 3 into 10 equal parts and look at the portion between 2.5 and 2.6 through a magnifying glass.
Now, 2.565 lies 2.5 and 2.6 hence first draw the number line and look at the portion between 2.5 and 2.6 by a magnifying glass.
Now, we imagine and divide this again into 10 equal parts. The first mark will represent 2.51, the next 2.52 and so on. To see this clearly we magnify this as shown in the following figure,
Again 2.565 lies between 2.56 and 2.57 so, now focus on this portion of the number line and imagine to divide it again into 10 equal parts as shown in the following figure
This process is called visualization of representation of number on the number line through a magnifying glass.
Thus we can visualise that 2.561 is the first mark and 2.565 is the fifth mark in these subdivision.
Both Rational & irrational numbers satisfy commutative law, associative law, and distributive law for addition and multiplication. However, the sum, difference, quotients and products of irrational numbers are not always irrational. If we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number. But this statement is not true for irrational numbers. We can see the example of this one by one
Rational Number + Rational Number = Rational Number
Let, a = (rational) and b = (rational),
=
=
=
= (rational number)
Rational Number - Rational Number = Rational Number
Let, a = (rational) and b = (rational)
=
=
= (rational number)
Rational Number X Rational Number = Rational Number
Example:
Let, a = (rational) and b = (rational)
Hence, × = (rational)
Rational Number / Rational Number = Rational Number
Let, a = (rational) and b = (rational)
divided by
i.e.,
Hence, ÷ = x = x 3
The sum and difference of a rational number and an irrational number is an irrational number.
Example:
Let, a = (rational) and b = √3 (irrational) then,
a + b = + √3 = (irrational)
a − b = − √3 = (irrational)
The multiplication or division of a non-zero rational number with an irrational number is an irrational number.
Example:
Let, a = (rational) and b = √2 (irrational) then,
ab = × √2 = (irrational)
= = × = (irrational)
If we add, subtract, multiply or divide two irrational numbers, we may get an irrational number or rational number.
Example:
Let two irrational numbers be
a = 3 + √2 and b = 3 − √2 then
a + b = ( 3 + √2 ) + ( 3 − √2 )
= 3 + √2 + 3 − √2
= 3 + 3
= 6 (rational)
Let two irrational numbers be
a = √3 + 1 and b = √3 − 1 then
A + b = (√3 + 1 ) + (√3 − 1)
= √3 + 1 + √3 − 1
= 2√3 (irrational)
Examples: Write which of the following numbers are rational or irrational.
(a) π − 2
(b) (3 + √27 ) − (√12 + √3)
(c)
(a) π − 2
We know that the value of the π = 3.1415
Hence, 3.1415 – 2 = 1.1415
This number is non-terminating non-recurring decimals.
(b) (3 + √27 ) − (√12 + √3)
On simplification, we get
( 3+ ) - ( + )
= 3 + 3√3 − 2√3 − √3
= 3 + √3 − √3
= 3, which is a rational number.
(c)
Here, 4 is a rational number and √5 is an irrational number. Now, we know that division of rational number and irrational number is always an irrational number.
Example: Add: 3 √2 + 6 √3 and √2 - 3√3
= (3 √2 + 6 √3 ) + (√2 – 3 √3 )
= (3 √2 + √2 ) + (6 √3 – 3 √3 )
= (3 + 1) √2 + (6 − 3) √3
= 4√2 + 3√3
Example: Multiply: 5√3 x 3√3
5√3 x 3√3
= 5 x 3 x √3 x √3
= 15 x 3 = 45
Step I: Firstly mark the distance x from fixed point on the number line i.e. PQ = x
Step II: Mark a point R at a distance 1 cm from point Q and take the mid-point of PR.
Step III: Draw a semicircle, taking O as centre and OP as a radius.
Step IV: Draw a perpendicular line from Q to cut the semi-circle to find √x
Step V: Take the line QR as a number line with Q as zero.
Step VI: Draw an arc having centre Q and radius QS to represent √x on number line.
We can see this method with the help of example
Example: let us find it for x = 4.5, i.e., we find √4.5
(i) Firstly, draw a line segment AB = 4.5 units and then extend it to C such that BC = 1 unit.
(ii) Let O be the Centre of AC. Now draw the semi- circle with centre O and radius OA.
(iii) Let us draw BD from point B, perpendicular to AC which intersects semi-circle at point D.
Hence, the distance BD represents √4.5 ≈ 2.121 geometrically. Now take BC as a number line, draw an arc with centre B and radius BD from point BD, meeting AC produced at E. So, point E represents √4.5 on the number line.
Radical Sign: Let a > 0 be a real number and n be a positive integer, such that
(a) = is a real number, then n is called exponent, and a is called radical and “√ ” is called radical sign.
The expression is called surd.
Example: If n = 2 then (4) = is called square root of 2.
Let’s solve some examples on the basis some of identities:
Examples: Simplify each of the following
(a) x
(b)
(c) ( √2 + √3 ) (√2 - √3)
(d) (5 + √5 ) (5 - √5 )
(a) x
We know that,
x =
=
= = 2
= (25)
= 21
= 2
b)
We know that,
=
=
=(34)
= 3
(c) (√2 + √3 ) (√2 - √3)
(d) (5 + √5 ) (5 - √5 )
We know that,
Rationalising the Denominators
Looking at the value can you tell where this value will lie on the number line? It is a little bit difficult. Because the value containing square roots in their denominators and division is not easy as addition, subtraction, multiplication and division are convenient if their denominators are free from square roots. To make the denominators free from square roots i.e. they are whole numbers, we multiply the numerator and denominators by an irrational number. Such a number is called a rationalizing factor.
Note: Conjugate of is , and conjugate of ,
Let’s solve some examples on rationalizing the denominators:
Examples: Rationalise the denominator of the following
(a)
(b)
(c)
(d)
(a)
Rationalization factor for
Here, we need to rationalise the denominator i.e., remove root from the denominator. Hence, multiplying and dividing by
∴
(b)
We know that the conjugate of 4 + √2 = 4 - √2
∴
(c)
We know that the conjugate of √3 - √5 = √3 + √5
∴
(d)
We know that the conjugate of 5 + 3√2 = 5 - 3√2
Now we will list some laws of exponents, out of these some you have learnt in your earlier classes. Let a (> 0) be a real number and m, n be rational numbers.
(i) am X an = am+n
(ii) (am)n = amn
(iii) = am-n
(iv) am X bm = (ab)m
(v) a-m =
(vi) () -m = ()m
(vii) (a) = where m and n ∈ N
Note: The value of zero exponent i.e. a° =1
Let us now discuss the application of these laws in simplifying expression involving rational exponents of real numbers.
Examples: Simplify each of the following
(i) (2)5 x (2)3
(ii) (43)2
(iii)
(iv) 72 × 62
(v) 6-2
(vi)
(vii) 33/2
(i) (2)5 x (2)3
We know that,
am x an = am+n
Hence,
(2)5 × (2)3 = (2)5+3 = (2)8
(ii) (43)2
We know that,
(am)n = amn
(43)2 = (4)3 ×2 = (4)6
(iii)
We know that,
am/an = am-n
= = 53-2 = 51
(iv) 72 × 62
We know that,
am x bm = (ab)m
(7)2 × (6)2 = (7 × 6)2 = (42)2
(v) 6-2
We know that,
a-m =1/am
6-2 = 1/62 = 1/36
(vi)
We know that
(vii) 33/2
We know that
1. Rational Numbers:
2. Irrational Numbers:
3. Real Numbers:
4. Operations with Rational and Irrational Numbers:
5. Identities for Positive Real Numbers:
For any positive real numbers a and b:
6. Rationalizing the Denominator:
To rationalize the denominator in terms like 1 / a + b multiply by the conjugate a - b / a - b.
7. Exponential Properties:
Let a > 0 be a real number and p and q be rational numbers. Then:
44 videos|412 docs|55 tests
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1. What is an irrational number and how is it different from a rational number? |
2. How are real numbers classified based on their decimal expansion? |
3. What are the basic operations that can be performed on real numbers? |
4. What are the laws of exponents for real numbers? |
5. How can I summarize the key concepts of the number system for easy revision? |
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