Grade 9 Number Systems
Natural numbers The counting numbers 1, 2, 3 … are called natural numbers. The set of natural numbers is denoted by N. N = {1, 2, 3, …} Whole numbers If we include zero to the set of natural numbers, then we get the set of whole numbers. The set of whole numbers is denoted by W. W = {0, 1, 2, …} Integers The collection of numbers … –3, –2, –1, 0, 1, 2, 3 … is called integers. This collection is denoted by Z, or I. Z = {…, –3, –2, –1, 0, 1, 2, 3, …}
Rational numbers Rational numbers are those which can be expressed in the form integers and q Note: 1.
12 12 3 4 , where the HCF of 4 and 5 is 1 15 15 3 5 12 4 and are equivalent rational numbers (or fractions) 15 5 a Thus, every rational number ‘x ’can be expressed as x , where a, b are integers b
p , where p, q are q
0.
Example: , , , etc.
1 3 6 2 4 9
such that the HCF of a and b = 1 and b
0.
2. Every natural number is a rational number. 3. Every whole number is a rational number. [Since every whole number W can be expressed as 4. Every integer is a rational number.
There are infinitely many rational numbers between any two given rational numbers.
W ]. 1
Example:
The Term Paper on Conversion Of Number Systems
The number is a symbol or a word used to represent a numeral, while a system is a functionally related groups of elements, so as whole, a number is set or group of symbols to represent numbers or numerals. In other words, any system that is used for naming or representing numbers is a number system. We are quite familiar with decimal number system using ten digits. However digital devices and ...
Find 5 rational numbers Solution:
3 5 and . 8 12
3 3 3 9 9 6 54 8 8 3 24 24 6 144 5 5 2 10 10 6 60 12 12 2 24 24 6 144
It can be observed that:
54 55 56 57 58 59 60 144 144 144 144 144 144 144 3 55 7 19 29 59 5 8 144 18 48 72 144 12 55 7 19 29 59 3 5 . Thus, , , , and are 5 rational numbers between and 144 18 48 72 144 8 12
Irrational Numbers Irrational numbers are those which cannot be expressed in the form are integers and q Example: 0.
p , where p, q q
2, 7, 14, 0.0202202220…….
There are infinitely many irrational numbers. Real Numbers The collection of all rational numbers and irrational numbers is called real numbers. So, a real number is either rational or irrational. Note: Every real number is represented by a unique point on the number line (and vice versa).
So, the number line is also called the real number line. Example: Locate 6 on the number line. Solution: It is seen that:
6 5
2
12
To locate 6 on the number line, we first need to construct a length of 5 .
5 22 1
By Pythagoras Theorem:
OB2 OB OA 2 AB2 5 22 12 4 1 5
Steps: (a) Mark O at 0 and A at 2 on the number line, and then draw AB of unit length 5 perpendicular to OA. Then, by Pythagoras Theorem, OB (b) Construct BD of unit length perpendicular to OB. Thus, by Pythagoras Theorem, OD
5
2
12
6
(c) Using a compass with centre O and radius OD, draw an arc intersecting the number line at point P. Thus, P corresponds to the number 6 .
Real numbers and their decimal expansions: The decimal expansion of a rational number is either terminating or non-terminating recurring (repeating).
Moreover, a number whose decimal expansion is terminating or non-terminating repeating is rational. Example:
3 2 15 8 4 3 24 13 1.5 1.875 Terminating Terminating Non – terminating recurring Non-terminating recurring
1.333……. 1.3
1.846153846153 1.846153
Example:
The Term Paper on Real Estate Market Evaluation
Real Estate Market Evaluation According to the company, health and beauty products have been available only through independent operators. Because of the growing worldwide interest in well-being, the company is expanding to meet the needs of customers by strategically placing the new store on High Street. The company reports that its health and beauty services markets are growing at roughly 10 ...
Show that 1.23434 …. can be written in the form and q 0. Solution:
Let x 1.23434….. 1.234 1
p , where p and q are integers q
Here, two digits are repeating. Multiplying (1) by 100, we get: 100x = 123.43434……… =122.2 + 1.23434 …….. Subtracting (1) from (2), we get:
99 x 122.2 x 122.2 99 1222 990 661 495 611 495
(2)
Thus,1.234
The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non- recurring is irrational. Example: 2.645751311064……. is an irrational number Representation of real numbers on the number line Example: Visualize 3.32 on the number line, upto 4 decimal places. Solution:
3.32 3.3232…… 3.3232 approximate upto 4 decimal place
Now, it is seen that 3
Operation on real numbers Some facts (a) The sum or difference of a rational number and an irrational number is always irrational. (b) The product or quotient of a non-zero rational number with an irrational number is always irrational. (c) If we add, subtract, multiply or divide two irrational numbers, then the result may be rational or irrational. Illustrations 2 3 is irrational 2 2 0 is rational 3 5 15 is irrational 2 2 2 is rational
6 2 2 2 3 is irrational 1 is rational
Identities If a and b are positive real numbers, then ab a b a. b.
a b a b
c. d. e. f.
a
a b b a a b
2
a b b c
a b2 a2 b d ac ad bc bd
a
b
a 2 ab b
The denominator of
a x
b can be rationalised by multiplying both the y
numerator and the denominator by x Laws of exponents 1) a p .a q 2)
ap
q
y , where a, b, x, y are integers.
ap
a pq
q
ap 3) q a
ap
p
q
4) ab Note:
x
a pb p , where a > 0 is a real number and p, q are rational numbers.
a
a
1 x
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