Number Systems
Types of Numbers
Real numbers: Real numbers comprise
the full spectrum of numbers. They can take on any form – fractions or whole
numbers, decimal points or no decimal points. The full range of real numbers
includes decimals that can go on forever and ever without end.
For Example: 8, 6, 2 +
,
3/5 etc.
Natural numbers: A natural number is a
number that comes naturally. Natural Numbers are counting numbers from 1, 2, 3,
4, 5, ……..
Whole numbers: hole numbers are just
all the natural numbers plus zero.
For Example: 0, 1, 2, 3, 4, 5, and so on upto infinity.
Integers: Integers incorporate all
the qualities of whole numbers and their opposites (or additive inverses of the
whole numbers) . Integers can be described as being positive and negative whole
numbers.
For Example: … –3, –2, –1, 0, 1, 2, 3, . . .
Rational Numbers: All numbers of the form p/q where p and q are integers (q ≠ 0)
called Rational numbers. http://facebook.com/groups/nextgencareers
Examples: 1/12, 42, 0, −8/11 etc.
All integers, fractions and
terminating or recurring decimals are rational numbers.How to find rational numbers between two given rational numbers?
If m and n be two rational numbers
such that m < n then 1/2 (m + n) is
a rational number between m and n.
Question: Find out a rational number
lying halfway between 2/7 and 3/4.
Solution:
http://facebook.com/groups/nextgencareers
Required number = 1/2 (2/7 + 3/4)
= 1/2 ((8 + 21)/28)
= {1/2 × 29/28)
= 29/56
Hence, 29/56 is a rational number
lying halfway between 2/7 and 3/4.
Question: Find out ten rational
numbers lying between -3/11 and 8/11.
Solution:
http://facebook.com/groups/nextgencareers
We know that -3 < -2 < -1 <
0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8
Therefore, -3 /11< -2/11 <
-1/11 < 0/11 < 1/11 < 2/11 < 3/11 < 4/11 < 5/11 < 6/11
< 7/11 < 8/11
Hence, -2/11, -1/11, 0/11, 1/11,
2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the ten rational numbers lying
between -3/11 and 8/11.
Irrational Numbers: Irrational numbers are the opposite of rational numbers. An
irrational number cannot be written as a fraction, and decimal values for
irrational numbers never end and do not have a repeating pattern in them. ‘pi’
with its never-ending decimal places, is irrational.
Example: π, √2, (3+√5), 4√3
(meaning 4×√3), 6√3 etc
Please note that the value of π = 3.14159 26535 89793 23846 26433 83279 50288
41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679…
We cannot write π as a simple
fraction (The fraction 22/7 = 3.14…. is just an approximate value of π)
Number Systems
Number Systems
http://facebook.com/groups/nextgencareers
Even Numbers: An even number is one
that can be divided evenly by two leaving no remainder, such as 2, 4, 6, and 8.
Odd Numbers: An odd number is one
that does not divide evenly by two, such as 1, 3, 5, and 7.
Prime Number: A prime
number is a number which can be divided only by 1 and itself. The prime number
has only two factors, 1 and itself or in other words, A number
greater than 1 is called a prime number, if it has only two factors, namely 1
and the number itself.
http://facebook.com/groups/nextgencareers
For example: 2, 3, 7, 11, 13, 17, …. are prime numbers.
Prime Numbers:
Prime numbers up to 100 are: 2, 3, 5,
7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83,
89, 97
Procedure to find out the prime number
Suppose A is the given number.
Step 1: Find a whole number nearly
greater than the square root of A.
Let K is nearby square root of A
Step 2: Test whether A is divisible by any prime number less than K. If yes A is not a prime number. If not, A is a prime number.
Let K is nearby square root of A
Step 2: Test whether A is divisible by any prime number less than K. If yes A is not a prime number. If not, A is a prime number.
Example:
Find out whether 337 is a prime
number or not?
Step 1: 19 is the nearby square root
(337) Prime numbers less than 19 are 2, 3, 5, 7, 11, 13, 17
Step 2: 337 is not divisible by any of them
Step 2: 337 is not divisible by any of them
Therefore, 337 is a prime number
Composite Number: A Composite Number is a number which can be divided evenly. Any
composite number has additional factors than 1 and itself.
For example: 4, 6, 8, 9, 10 …..
Co-primes or Relatively prime numbers: A pair of numbers not having any common factors other than 1 or
–1. (Or alternatively their greatest common factor is 1 or –1)
Or in other words, In number
theory, two integers a and b are said to be co-prime if the only
positive integer that evenly divides both of them is 1. That is, the only
common positive factor of the two numbers is 1. This is equivalent to their
H.C.F. being 1. e.g. (2,3), (6,13), (10,11), (25,36) etc
For Example: 15 and 28 are co-prime, because the factors of 15
(1,3,5,15) , and the factors of 28 (1,2,4,7,14,28) are not in common (except
for 1) .
In number theory, two integers a and
b are said to be co-prime if the only positive integer that evenly
divides both of them is 1. That is, the only common positive factor of the two
numbers is 1. This is equivalent to their H.C.F. being 1. e.g. (2,3), (6,13),
(10,11), (25,36) etc
http://facebook.com/groups/nextgencareers
Twin Primes: A pair of prime numbers
that differ by 2 (successive odd numbers that are both Prime numbers) .
For Example: (3,5) , (5,7) , (11,13) , …
Perfect Numbers:
A perfect number is a positive
integer that is equal to the sum of its positive divisors excluding the number
itself (proper positive divisors).
The first perfect number is 6, because 1, 2, and 3 are its proper positive
divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the
sum of all its positive divisors: ( 1 + 2 + 3 + 6 )/2= 6. The next perfect number is 28 = 1 + 2 + 4 + 7 +
14.
There are total 27 perfect numbers.
Surds :
Let a be any rational number and n be any positive integer such that a√n is irrational. Then a√n is a surd.
Example: √3, 10√6, √43 etc
Every surd is an irrational
number. But every irrational number is not a surd. (E.g.: Π, e etc are
not surds though they are irrational numbers.)
Fractions
A fraction is known as a rational number and written in the form
of p/q where p and q are integers and q ≠ 0. The lower number ‘q’ is known as
denominator and the upper number ‘p’ is known as numerator.
Type of Fractions
Proper Fraction: The fraction in which
numerator is less the denominator is called a proper fraction.
For Example:
|
2
|
,
|
5
|
,
|
10
|
3
|
6
|
11
|
Improper fraction: The fraction in which numerator is greater than the denominator is
called improper fraction.
For Example:
|
3
|
,
|
6
|
,
|
8
|
32
|
5
|
7
|
Mixed fraction: Mixed fraction is a
composition of fraction and whole number.
For Example: 2
|
1
|
, 3
|
3
|
, 5
|
6
|
2
|
4
|
7
|
Complex fraction: A complex fraction is that fraction in which numerator or
denominator or both are fractions.
For Example:
|
2
|
,
|
2
|
,
|
3
|
3
|
5
|
7
|
|||
4
|
6
|
5
|
|||
7
|
6
|
Decimal fraction: The fraction whose denominator is 10 or its higher power, is
called a decimal fraction.
For Example:
|
7
|
,
|
11
|
,
|
12
|
10
|
100
|
1000
|
Continued fraction: Fractions which contain addition or subtraction of fractions or a
series of fractions generally in denominator (sometimes in numerator also) are
called continued fractions.
It is also defined as a fraction whose numerator is an integer and
whose denominator is an integer plus a fraction.
For Example: 2 -
|
2
|
|
2
|
||
2+
|
||
3
|
||
4
|
NEXTGEN CAREER COACHING-www.facebook.com/nextgencareer
Comparison of Fractions
If the
denominators of all the given fractions are equal then the fraction of greater
numerator will be the greater fraction.
For Example:
|
4
|
,
|
2
|
,
|
8
|
,
|
9
|
then,
|
9
|
>
|
8
|
>
|
4
|
>
|
2
|
7
|
7
|
7
|
7
|
7
|
7
|
7
|
7
|
For Example:
|
7
|
,
|
7
|
,
|
7
|
,
|
7
|
then,
|
7
|
>
|
7
|
>
|
7
|
>
|
7
|
4
|
2
|
8
|
9
|
2
|
4
|
8
|
9
|
For Example:
|
5
|
,
|
7
|
,
|
11
|
,
|
8
|
then,
|
5
|
>
|
7
|
>
|
8
|
>
|
11
|
2
|
4
|
8
|
5
|
2
|
4
|
5
|
8
|
NEXTGEN CAREER
COACHING-www.facebook.com/nextgencareer
Quicker Method (Cross Multiplication)
This is a
short-cut method to compare fractions. Using this method we can compare all
types of fractions.5×7=35 9×4=36
5/9 ? 4/7
The fraction whose numerator is in the greater product is greater.
NEXTGEN CAREER
COACHING-www.facebook.com/nextgencareer
Since 36 is
greater than 35, hence,
Divisibility
Rules
- Divisibility by 2: A number is divisible by 2 if its
unit’s digit is even or 0.
- Divisibility by 3: A number is divisible by 3 if the sum
of its digits are divisible by 3.
- Divisibility by 4: A number is divisible by 4 if the last
2 digits are divisible by 4, or if the last two digits are 0’s.
- Divisibility by 5: A number is divisible by 5 if its unit’s
digit is 5 or 0.
Divisibility by 6: A number is divisible by 6 if it is simultaneously divisible by 2
and 3. NEXTGEN CAREER
COACHING-www.facebook.com/nextgencareer
- Divisibility by 7: A number is divisible by 7 if unit’s
place digit is multiplied by 2 and subtracted from the remaining digits
and the number obtained is divisible by 7.
- Divisibility by 8: A number is divisible by 8 if the last
3 digits of the number are divisible by 8, or if the last three digits of
a number are zeros.
- Divisibility by 9: A number is divisible by 9 if the sum
of its digits is divisible by 9.
- Divisibility by 10: A number is divisible by 10 if its
unit’s digit is 0.
- Divisibility by 11: A number is divisible by 11 if the sum
of digits at odd and even places are equal or differ by a number divisible
by 11.
- Divisibility by 12: A number is divisible by 12 if the
number is divisible by both 4 and 3.
- Divisibility by 13: A number is divisible by 13 if its
unit’s place digit is multiplied by 4 and added to the remaining digits
and the number obtained is divisible by 13.
- Divisibility by 14: A number is divisible by 14 if the
number is divisible by both 2 and 7.
- Divisibility by 15: A number is divisible by 15 if the
number is divisible by both 3 and 5.
- Divisibility by 16: A number is divisible by 16 if its
last 4 digits is divisible by 16 or if the last four digits are zeros.
- Divisibility by 17: A number is divisible by 17 if its
unit’s place digit is multiplied by 5 and subtracted from the remaining
digits and the number obtained is divisible by 17.
- Divisibility by 18: A number is divisible by 18 if the
number is divisible by both 2 and 9.
- Divisibility by 19: A number is divisible by 19 if its
unit’s place digit is multiplied by 2 and added to the remaining digits
and the number obtained is divisible by 19.
PRACTICE TEST ON NUMBER SYSTEM:
NEXTGEN CAREER
COACHING-www.facebook.com/nextgencareer
Q1. 1/5 of a number exceeds 1/7 of
the same number by 10. The number is:
a)
125 b)
150 c)
175 d)
200
Q2. Two numbers differ by 5. If their
product is 336, the sum of the two numbers is:
a)
21 b)
28 c)
37 d) 51
Q3. Sum of two numbers is 40 and
their product is 375. What will be the sum of their reciprocals?
a) 8/75 b)
1/40 c) 75/8 d) 75/4
Q4. Which of the following fractions
is the smallest?
a) 9/13 b)
17/26 c)
28/29 d)
33/52
Q5. A number when divided by 899
gives a remainder 63. If the same number is divided by 29. The remainder will
be:
a)
10 b)
5 c)
4 d) 2
Q6. The smallest number to be added
to 1000, so that 45 divides the sum exactly is:
a)
35 b)
80 c)
20 d) 10
NEXTGEN CAREER
COACHING-www.facebook.com/nextgencareer
Q7. A six digit number is formed by
repeating a three digit number; for example, 256, 256 or 678, 678 etc. Any
number of this form is always exactly divisible by:
a) 7
only b)
11
only c)
13 only d)
1001
Q8. The sum of three consecutive odd
natural numbers is 147, then, the middle number is:
a)
47 b)
48 c)
49 d) 51
Q9. The sum of all natural numbers
between 100 and 200, which are multiples of 3 is:
a) 5000 b)
4950 c)
4980 d) 4900
Q10. How many digits in all are
required to write numbers from 1 to 50?
a) 100 b)
92 c)
91 d) 50
Q11. A number of friends decided to
go on picnic and planned to spend Rs 108 on eatables. Three of them however did
not turn up. As a consequence each one of the remaining had to contribute Rs 3
extra. The number of them who attended the picnic was:
a)
15 b)
12 c)
9 d) 6
Q12. The divisor is 25 times the
quotient and 5 times the remainder. If the quotient is 16, the dividend is:
a) 6400 b) 6480 c)
400 d) 480
Q13. In a test, 1
mark is awarded for each correct answer and one mark is deducted for each wrong
answer. If a boy answers all 20 items of the test and gets 8 marks, the number
of questions answered correct by him was:
a)
16 b)
14 c)
12 d) 8
Q14. Of the three numbers, the second
is twice the first and it is also thrice the third. If the average of three
numbers is 44, the difference of the first number and the third number is:
a)
24 b)
18 c)
12 d) 6
Q15. A number when divided
successively by 4 and 5 leaves remainders 1 and 4 respectively. When it is
successively divided by 5 and 4 the respective remainders will be
a)
4,1 b) 3,2 c)
2,3 d) 1,2
Q16: The digit in unit’s place of the
product 81 X 82 X 83………X 89 is
a)
0 b)
2 c) 6 d)
8
Q 17. The sum of first sixty numbers
from one to sixty is divisible by
a)
13 b)
59 c)
60 d) 61
Q18. The digit in the unit’s place of
the product
(2464)1793 X (615)317 X (131)491 is
a)
0 b) 2
c) 3 d) 5
Q19. (719 + 2) is divided by 6, the remainder is:
a)
5 b)
3 c)
2 d) 1
Q20. 0.39393939………. is
equal to
a)
39/100 b)
13/33 c)
93/100 d)
39/990
NEXTGEN CAREER
COACHING-www.facebook.com/nextgencareer
ANSWER KEY:
1c 2c 3a 4a 5b 6a 7d 8c 9b 10c
11c 12b 13b 14c 15c 16a 17d 18a 19c 20b
11c 12b 13b 14c 15c 16a 17d 18a 19c 20b
Now, you can solve the following Questions out of Test Paper No161A-E20 and submit answers in the form of 1a,2b,3c,4d,5e,6b in the Comments Section. You will get your scorecard, answer key, and detailed explanation of each question. Please submit your email address, WhatsApp# to get full Practice Test Paper No161A-E20 and similar other practice test papers. You can also download Lessons and Test Papers from https://www.facebook.com/groups/NextGenCareers/
No comments:
Post a Comment