Revision Notes on Number Systems
Decimal Expration of Rational Number (2)
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Rational Numbers on the basis of sign (2)
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Properties of Rational Numbers (4)
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Additive Inverse योग प्रतिलोम– If sum of two rational number is zero than they are called additive inverse.
Ex- 14 + (-14) = 0,
Note- To gate additive inverse of any no change the sign.
Multiplicative Inverse गुणज प्रतिलोम– If product of two rational number is one than they are called multiplicative inverse.
Ex- 14 × = 1,
Note- Reciprocal of number is multiplicative inverse of itself.
Real Numbers- The combination of rational and irrational numbers in number system is called natural numbers.
* It is denoted by R.
Natural Numbers- The counting numbers are called natural numbers.
i.e. 5, 6, 7, 8, ……….
* It is denoted by N.
Whole numbers- All non-negative counting numbers including zero are called whole numbers.
i.e. 0, 1, 2, 3, 4, 5, …………….
* It is denoted by W.
Integers- All negative and non-negative numbers including zero altogether known as integers.
i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, …………..
* It is denoted by Z.
* The sum, product, and difference of any two integers is also an integer. But this is not true for division just try 1 ÷ 2.
* The sum their additive inverse is zero. Ex {1+ (-1) = 0, 12 + (-12) = 0}
Prime Numbers (रूढ़ संख्या): – Numbers which have only two factors namely 1 and itself are called prime numbers.
Ex- 2, 3, 5, 7, 11, 19, 37 etc are prime numbers.
Composite Numbers (भाज्य संख्या/यौगिक संख्या)- Numbers having more than two factors are called composite numbers.
Ex- 4, 6, 8, 10 etc are composite numbers.
Notes:
(a) 1 is neither prime nor composite.
(b) 2 is the lowest and the only even prime number.
(c) 4 is the lowest even composite number.
(d) 9 is the lowest odd composite number.
Co-prime Numbers (सह अभाज्य संख्या)– Two numbers which have no common factor except 1called co-prime numbers.
* Two numbers whose HCF is 1 called co-prime numbers.
* Co-prime numbers need not be prime numbers.
Ex- (i) 7 and 10 are co-prime.
(ii) 15 and 17 are co-prime.
Twin Prime Numbers (जुड़वाँ अभाज्य संख्या)- Two prime numbers whose difference is 2 are called twin prime numbers.
Ex- (i) 3 and 5, (ii) 17 and 19
(iii) 41 and 43, (iv) 29 and 31,
(v) 71 and 73.
Rational numbers परिमेय संख्या – Those numbers which can be written in the form where p and q are integers but q≠ 0 are called rational numbers.
* Every integer p is also a rational number, can be written as
* It is denoted by Q.
Irrational numbers अपरिमेय संख्या – Those numbers which cannot be written in the form where p and q are integers but q≠ 0 are called irrational numbers.
Ex- √2, √3 etc.
* The decimal expansion of a rational number is either terminating or non-terminating recurring.
* If p is a prime number and p divides a2 , then p is one of the prime factors of a2 which divides a, where a is a positive integer.
* If p is a positive number and not a perfect square, then √n is definitely an irrational number.
* If p is a prime number, then √p is also an irrational number.
* The decimal form of irrational numbers is non-terminating and non-repeating.
How to find a rational number between two given numbers?
To find the rational number between two given numbers ‘a’ and ‘b’.
Example:
Find 2 rational numbers between 4 and 5.
Solution:
To find the rational number between 4 and 5
To find another number we will follow the same process again.
Hence the two rational numbers between 4 and 5 are 9/2 and 17/4.
Remark: There could be unlimited rational numbers between any two rational numbers.
Real Numbers and their Decimal Expansions
1. Rational Numbers
If the rational number is in the form of a/b then by dividing a by b we can get two situations.
a. If the remainder becomes zero
While dividing if we get zero as the remainder after some steps then the decimal expansion of such number is called terminating.
Example:
7/8 = 0.875
b. If the remainder does not become zero
While dividing if the decimal expansion continues and not becomes zero then it is called non-terminating or repeating expansion.
Example:
1/3 = 0.3333….
Hence, the decimal expansion of rational numbers could be terminating or non-terminating recurring and vice-versa.
2. Irrational Numbers
If we do the decimal expansion of an irrational number then it would be non –terminating non-recurring and vice-versa. i. e. the remainder does not become zero and also not repeated.
Example:
π = 3.141592653589793238……
Representing Real Numbers on the Number Line
To represent the real numbers on the number line we use the process of successive magnification in which we visualize the numbers through a magnifying glass on the number line.
Example:
Step 1: The number lies between 4 and 5, so we divide it into 10 equal parts. Now for the first decimal place, we will mark the number between 4.2 and 4.3.
Step 2: Now we will divide it into 10 equal parts again. The second decimal place will be between 4.26 and 4.27.
Step 3: Now we will again divide it into 10 equal parts. The third decimal place will be between 4.262 and 4.263.
Step 4: By doing the same process again we will mark the point at 4.2626.
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.
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:
Finding Roots of a Positive Real Number ‘x’ geometrically and mark it on the Number Line
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.
Identities Related to Square Roots
If p and q are two positive real numbers
Examples:
1. Simplify
We will use the identity
2. Simplify
We will use the identity
Rationalizing the Denominator
हर का प्रमेयकरण
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 of 7/(7- √3).
Solution:
We will use the identityhere.
Laws of Exponents for Real Numbers
वास्तविक संख्या के घातांक का नियम
If we have a and b as the base and m and n as the exponents, then
1. am × an =am+n
2. (am)n = amn
4. am bm = (ab)m
5. a0 = 1
6. a1 = a
7. 1/an = a-n
* Let a > 0 be a real number and n a positive integer.
* Let a > 0 be a real number. Let m and n be integers such that m and n have no common factors other than 1, and n > 0. Then,
Example:
Simplify the expression (2x3y4) (3xy5)2.
Solution:
Here we will use the law of exponents
am × an =am+n and (am)n = amn
= (2x3y4)(3xy5)2
= (2x3y4)(3 2 x 2 y10)
= 18. x3. x2. y4. y10
= 18. x3+2. y4+10
= 18x5y14