NUMBER
SYSTEM 1
All the numbers can be expressed as
Complex Numbers .A complex number is an expression of the form a + bi, where a and b are real numbers and i = √-1. In the complex number a + bi, a is called
the real part and bi is the imaginary part. When .a = 0, the complex number is called a
pure imaginary number. If b = 0, the
complex number reduces to the real number a thus, complex numbers
include all real numbers and all pure
imaginary numbers.
Classification of Numbers
All the numbers that we see or use on a regular
basis can be classified a given
in the chart
be below:
Real numbers: Real numbers represent actual
physicaI quantities in a
meaningful way e.g. length, height, density etc. Real numbers can be further divided into sub groups. For
example, rational/irrational, odd/even,
prime/composite etc.
Natural
or Counting Numbers
To count objects we
use counting numbers like 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. The set of positive counting numbers is
called natural numbers .These are also at times called positive integers.
N={1,2,3,4....}
Whole Numbers
The set of natural
numbers taken along with 0, gives us the set of Whole Numbers. Thus, the numbers 0, 1, 2, 3, 4, 5, 6,
7, 8, ..,.. represent the set of whole numbers.
W = {0, 1, 2, 3, 4 .....} `-
Integers
All the counting numbers (positive and negative)
including zero are called the integers. For example
-100,-99,-50, -40, 0, 13, 17 are all
integers.
I/Z
= {0,±1, ±2, ±3 ...}
Rational Numbers
This is the set of real numbers that can be written in the form,),. where a and b are integers and b is not
equal to zero (b # 0). All integers and all fractions are rational numbers
including the finite decimal numbers (i.e.
terminating) numbers -4,2/3 ,50/2,-10/3.-1/4,0,145
(also represented as 145/1) and 15/1 are examples of rational numbers.
Q = {a/b : a,b є I
& b ≠ 0}
Irrational
Numbers
The
numbers which are io rational are ca d irrational numbers, such as 42, n. These numbers give an approximate answer in terms of decimals. Also the
digits after the decimal are non-terminating and non-recurri g. Thus,
√2 =
1.4142135..., π= 3.141592... etc.
Even
Numbers
The set of Even Numbers is
the set of integers which are divisible by 2. e.g. 2, 4, 6, 8, 10 ... Even numbers are expressed
in the form 2n, where n is an integer. Thus, 0,-2, -6
etc. are also even numbers.
Odd Numbers
The set of Odd Numbers is the set of integers which are
not divisible by 2. e.g. 1, 3, 5, 7, 9 ... Odd numbers are expressed in the form (2n.+ 1), where n
is an integer (not necessarily prime). Thus, -1, -3, -9 etc. are
all odd numbers.
The smallest natural number is 1. The
smallest whole number is 0.
Ø The smallest natural number is 1.
Ø The smallest whole number is 0.
Prime Numbers
A natural number which does not have any other factors besides itself and unity, is a prime
number. For example 2, 3, 5, 7, 11, 13 etc. The set of such numbers is the set of prime numbers.
Imp.
Ø 1 is neither prime nor composite. `
Ø The
only even prime number is 2.
Ø
Two numbers are said to
be relatively prime to each other or co-prime
when their HCF is 1. e.g. (i) 9 and 28, (ii) 3 and 5, (iii) 14 and
29 etc.
Ø If number has no prime factor equal to or less than its square root, then the number is prime. This is a test to judge whether a number is
prime
or not.
Composite
Numbers
The set of Composite Numbers is the set of natural numbers which have other factors also, besides itself and unity. e.g. 8, 72,
39 etc . Alternatively, we might say that a natural number except 1 which is
not prime is a composite number.
System
of Real Numbers
The
system of real numbers as we know it today, is a result of gradual progress, as
the following indicates.
Ø Natural
numbers 1, 2, 3, 4, ... used in counting are also known as positive integers. If two such numbers are added or multiplied, the result is always
a natural number.
Ø Numbers
that can be expressed in the form p/q, where p and q are co-prime
integers, are known as rational numbers. For example 2/3, 8/5, 121/17, -3/2 etc.
Ø Irrational
numbers are numbers which cannot be expressed in the above form. Such as √2, π etc.
Ø Zero
helps us enlarge the number system so as to permit such operations as 6 - 6 or
10 - 10. Zero has the properties that
(i) Any number multiplied by zero is zero.
(ii) Zero divided by any number ( 0) is zero.
(iii) Any number divided by zero is undefined.
Ø When
no sign is placed before a number, a plus sign is understood. Thus, 5 is +5,
'12 is +12.
Important to Remember that..
Ø Zero is a rational number without any sign.
Ø The real number system
consists of the collection of Positive & Negative, Rational &
Irrational Numbers and Zero.
Graphical
Representation of Real Numbers
It is often useful to represent real numbers by points on
a line. To
do this, we choose a point on the line to represent the real number zero and
call this point the origin. The positive integers +1, +2, +3, ... are then associated with points on the line at distance 1, 2, 3, ... units respectively to the
right of the origin (see the figure), while the negative integers -1,
-2, -3, ... are associated with points on the line at distances 1, 2, 3, ...
units respectively to the left of the origin.
-3/2
-5 -4 -3 -2 -1 0 +1/2 +1 +2 +3 +4 +5
The rational number 1/2 is represented on this scale by
a point P halfway between 0 and +1. The negative number -3/2 or -1/2 is represented
by a point R, 1½ units to the left of the origin.
There is one and only
one point on the number line corresponding to each real number and
conversely, to every point on the line, there corresponds one and only one real
number.
The position of real numbers on a
line establishes an order to the real number system. If a point A
lies to the right of another point B on
the line we say that the number corresponding to A is greater than
the number corresponding to B or that the number corresponding
to B is less or greater than the number corresponding
to A. The symbols for "greater than" and "less than" are > and
< respectively. These symbols are called
"inequality
signs."
Thus
since 5 is to the right of 3, 5 is greater than 3 (5 > 3), we may also say 3 is less than 5 (3 < 5).
Similarly, since -6 is to the left of - 4, -6 is smaller than - 4, (-6 < - 4), we may also write (- 4
> - 6).
Absolute Value
The absolute value or numerical value of a
number means the distance of the number from the origin on the number line.
Thus |-6| = 6, |+4|; = 4, |-3/4|
= 3/4. In other words, absolute value of a number means the
value of the number without its sign i.e. the magnitude only.
El.
Write the absolute values of the following
(1) —51 (2) 15
(3) 237 (4) —354
(5) 0
Sol. The
absolute values of the given integers are
(1) 51 (2)
15
(3) 237 (4) 354
(5) 0
Properties of Numbers
Closure Property
A set has closure under an operation
if the result of performing the operation on
elements of the set is also an element of the set. The set X is closed under the operation * if for all elements a
and b in set X, the result a * b is also in set X.
For example,
integers possess this property under the operations +, x, — etc. As we can see, 5 + 8 = 13 and 5, 8, 13 are
all integers.
Identity
Property
A set has
an identity under an operation if there is an element in the set that when
combined with each element in the set leaves that element unchanged. The set X
has an identity under the operation * if
there is an element j in set X such that j * a = a * j = a for all elements a in set X. For example 0
and 1 are the identity elements for addition and multiplication
respectively. As we can see, 15 + 0 = 15 and 7 x 1 = 7.
Inverse
A
set has inverses under an operation if for each element of the set, there is an
element of the set such that when these two elements are combined using the
operation, the result is the identity
element for the set under the operation. If a set does not have an identity j under an operation, it
cannot have the inverse property for
the operation. If X is a set that has identity j under operation *, then it has inverse if for each element a in set
X there is an element a' in set X such that a * a' = j and a'
a = j. For example, 2 is the additive inverse of —2
and 1/2 is the multiplicative inverse of 2. As we can see, 2 + (-2) = 0 and 2 x 1/2=1,
Associative
& Commutative Property
Sets under an operation may also follow the associative property
and the commutative property. Associative and commutative properties are based on the order of the numbers
in any operation.
If there are two
operations on the set, then the set could have the
distributive property.
A→ (a+b)+c=a+(b+c)
axb=bxa.
E2. Which
properties are true for the sets of natural numbers, whole numbers, integers, rational numbers,
irrational numbers and er al numbers under the operation of addition?
Sol.
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Natural
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Whole
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Integers
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Rational
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Irrational
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Real
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Closure
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Yes
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Yes
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Yes
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Yes
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No
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Yes
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Identity
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No
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Yes
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Yes
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Yes
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No
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Yes
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Inverse
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No
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No
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No
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Yes
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No
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Yes
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Associativity
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Yes
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Yes
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Yes
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Yes
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Yes
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Yes
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Commutativity
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Yes
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Yes
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Yes
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Yes
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Yes
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Yes
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There are some properties that sets of numbers have, that
do not depend on an operation to be true.
Three such properties are order, density and completeness.
Order
A set of
numbers has order if, given two distinct elements in the set, one element is
greater than the other.
Density
A set of numbers has density if, between any two elements
of the set, there is another element of the set.
Completeness
A set of numbers has completeness, if the points using
its elements as coordinates completely fill a line or plane.
Rounding
off a number
To round off a number,
following rules are observed
Ø If the digit to be dropped is smaller than 5, the
preceding digit should be left unchanged but if the digit to be
dropped is greater than 5, the preceding digit should be raised by 1.
Ø
If the digit to be
dropped is 5, the preceding digit if odd should
be raised by 1 but if even it should be left unchanged.
Ø For
example, by rounding off the number 37.4867, we can obtain the numbers 37.487,
37.49 and 37.5 as per
our requirement.
Ø As
another example, we consider the number 4735.25. If would be rounded off to
4735.2 If we further round off this
number, we get 4735.
Four Basic Mathematical Operations
The four basic
mathematical operations are Addition, Subtraction, Multiplication
and Division.
Addition
When two numbers 'a' and 'b' are added, their sum is indicated by a + b.
Thus, the sum of 273 and 127 is 400 etc.
Subtraction
When a number
"b" is subtracted from a number "a
” the difference is
indicated by a — b. Thus, the difference between 1532 and 382 is 1150 etc.
Subtraction may be
defined in terms of addition. We may define a — b
to represent a number X such that X added to b yields a or X +
b = a. e.g. 8 — 3 is the number X which when added to 3 yields 8, i.e., X + 3 =
8; thus 8 — 3 = 5.
Multiplication
The product of two numbers "a" and
"b" is a. number "c" such that,
a x b = c. The operation of multiplication may be indicated by a cross,
a dot or parentheses.
Thus
5 x 3 = 5 . 3 = 5(3) = (5)(3) = 15, where the factors are 5 and
3 and the product is 15. When alphabets are used, the notation p x q is
usually avoided since x may be confused with the alphabet x.
Division
When
a number "a" is divided by the other number "b'
the quotient obtained is written as a = b or a/b, where
"a" is called the dividend and "b" the divisor. The
expression a/b is also called a fraction, having
numerator "a" and denominator "b". Division may be defined
in terms
of multiplication. We may consider a/b as a number X which on multiplication by b yields a or bX = a. For example, 6/3 is the number X such that 3 multiplied by X yields 6
or 3X = 6; thus 6/3 = 2.
Imp.
Ø Division
by zero is not defined.
Ø Multiplication or division of any number by 1 gives the
same result
i.e. the number itself.
Ø Product of any number with zero is zero.
Ø Division
of the number by itself is equal to 1.
Properties of addition and
multiplication of real numbers
Commutative
property for addition
The
order of addition of two numbers
does not affect the result.
a
+ b = b +
a,
For
example, 5 + 3 = 3 +
5
= 8.
Associative
property for addition
The terms of a sum can
be grouped in any manner without affecting the result.
a+ b+c=a + (b +
c) = (a
+ b)
+ c, For
example,3+4+1=3+(4+1)=(3+4)+1.
Commutative
property for multiplication
The order of the factors of
a product does not affect the result. a x b = b x a,
For example, 2x5=5x2=10.
Associative
property for multiplication
The
factors of a product can be grouped in any manner without affecting the result.
abc = a(bc) = (ab)c,
For
example, 3 x4x 6 = 3 x (4 x 6)
=(3 x 4) x 6 = 72.
Distributive
property for multiplication over addition
The
product of a number 'a' with the sum of two numbers (b &
c) is equal to the sum, of the products ab and ac. ax
(b+c)=axb+axc, e.g.4x(3+2)=4x3+4x2=20.
Imp.
Extensions
of these laws may be made. Thus we may add the numbers a, b, c, d, e by grouping
in any order, as (a + b) + c + (d + e), a + (b + c) + (d + e), etc. Similarly, in
multiplication, we may write (ab)c(de) or a(bc)(de), the result being
independent of order or
grouping.
Rules of Signs
Ø To add two
numbers with similar signs, add their absolute values and prefix the common sign.
Thus 3+4=7,(-3)+(-4)=-7.
Ø To add two
numbers with opposite signs, find the difference between their
absolute values and prefix the sign of the number with greater absolute value. For
example
17+(-8)=9,(-6)+4=-2,(-18)+15=-3.
Ø To subtract one number b from
another number a, change the
operation to addition and replace b by its opposite,-b.
Thus, 12-(7)=
12+(-7)=5, (-9) - (4) =
-9 + (-4) = -13, 2-(-8)=2+8= 10.
Ø To multiply (or
divide) two numbers having similar signs, multiply (or divide) their absolute
values and prefix a plus sign
(or no sign). For example,
(5)(3) = 15, (-5)(-3) = 15, -6/-3 =
2.
Ø To multiply (or
divide) two numbers having opposite signs, multiply (or divide) their absolute
values and prefix a minus sign. For example,
(-3)(6)= -18, (3)(-6)
= -18, -12/4 = -3.
Other
special numbers
Perfect Numbers
If the sum of the
divisors of N excluding N itself but including units
equal to N, then N is called a perfect number. e.g. 6, 28 etc.
6 = 1 + 2 + 3, where 1, 2 and 3 are divisors of 6
28 = 1 + 2 + 4 +
7
+ 14.
Fibonacci Numbers ,
Fibonacci numbers
form a sequence in which each term is the sum of the two terms immediately
preceding it. It is named for its discoverer, Leonardo Fibonacci (Leonardo Pisano). The
Fibonacci sequence that has 1 as its first
term is 1, 1, 2, 3, 5, 8, 13, 21, 34,
55, ... These numbers are referred to as Fibonacci numbers. The defining
property can be symbolically represented as
Tr+2 = tr + tr+1
If X1 = 1 and Xn+1 =
2Xn + 5, where n = 1, 2.., then what is the value of X100?
(1) (5 x 299
- 6) (2) (5 x 299 + 6)
(3) (6 x 299 + 5) (4)
(6 x 299 - 5)
Sol.X1=1=6x2°-5=X2=7=6x21-5
X3=19=6x22-5=
Xn=6x2n-1-5 •
X100 = ( 6 x 299 -
5).
Hence Ans.(4)
Classify each of the following numbers
according to the categories: real number, positive integer, negative
integer, rational number, irrational number, imaginary.
-5, 3/5,
3π, 2, -1/4, 6.3, 0, √5, √-1,
0.3782, √4, -18/7.
Sol. If
the number belongs to a category it is indicated by a tick (ü) mark.
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Real number
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Positive
integer
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Negative
integer
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Rational
number
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Irrational
number
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Imaginary
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-5
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ü
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ü
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ü
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3/5
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ü
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ü
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3π
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ü
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ü
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2
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ü
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ü
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ü
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-1/4
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ü
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ü
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6.3
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ü
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ü
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0
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ü
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ü
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√5
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ü
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ü
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√(-1)
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ü
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0.3782
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ü
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ü
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√4
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ü
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ü
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ü
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-18/7
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ü
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ü
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Some
Important Results on Numbers
Ø If the sum of two positive quantities is given, their
product is
greatest when they are equal.
Ex. Given X + Y = 30 Possible (X, Y) are (1, 29), (2, 28),
(3,
27) ... and so on. Out of all these, the pair that gives the maximum product will be (15, 15).
Ø If the product of two positive quantities is given, their
sum is
least when they are equal.
Ex. Given X Y = 100 Possible (X, Y) are (1, 100), (2, 50),
(4,
25) ... and so on. Out of all these, the pair that gives the minimum sum will be (10, 10).
Ø The sum of a positive number and its reciprocal is
always greater
than or equal to 2, i.e., (a/b + b/a) ≥ 2,
(a/b
+ b/c + c/d + d/a) ≥ 4, [(X) + (1/X)] ≥ 2.
Base System
The numerals we
use today are 0, 1, 2, 3, 4, 5 , 6, 7, 8 and 9. These numbers are a part of
decimal system, because there are only 10 basic symbols.
Expressing
the decimal number 63472 in the expanded form, we get
In the decimal number 63472,
Ø 2 is
at the unit's digit and has a place value of 1 ... 2 ones
Ø 7 is at the ten's digit
and has a place value of 10 ... 7 tens
Ø
4 is at the hundred's
digit and has a place value of 100 ... 4
hundreds
Ø 3
is at the thousand's digit and has a place value of 1000 ... 3
thousands
Ø
6 is at the ten thousand's digit
and has a place value of 10000 ... 6 ten thousands
Thus, the
number can be represented as
63472=6x
104+3x 103+4x 102+7x 101+2x 100
= 60000 + 3000 + 400 + 70 + 2
Thus, since
there are 10 symbols, this system of representation of numbers is known as the Decimal System (base 10). In a similar
way, a system in which only 0 and 1 exist, is known as Binary System (base 2).
Similarly, other systems can also be established
like Hexadecimal (base 16), Octal
(base 8) etc. In each of these systems, the number of symbols used is
restricted to the base number. Thus an octal system has only 8 symbols (0 to 7)
and hexadecimal system has 16 symbols (0 to 9, A to F).
Conversion
Decimal to Binary Conversion
To convert the decimal number to
binary we begin by dividing the decimal number by 2 and then dividing each resulting
quotient by
2 until there is a 1
quotient.
E6. Convert the decimal number 50 to a binary number.
Remainder
Sol.
2 50
E7. Convert the
binary number 110010 to a decimal number
Sol. 1 x 25 + 1 x 24 + 0 x 23 +
0 x 22 + 1 x 21 +
0 x 2° = 32 + 16 + 0 + 0
+ 2 + 0 = 50.
Decimal
to Octal Conversion
The
method of converting a decimal number into an octal number is repeated division by 8, similar to the method
used in conversion of decimal to binary. Each successive division by 8
yields a remainder that becomes a digit in the equivalent octal number.
E8. Convert the decimal number 20579 to an octal number.
the octal number is 50143.
Decimal
to Hexadecimal Conversion
Repeated division of a decimal number by 16 will
give the equivalent hexadecimal
number which is formed by the remainders of each division. It is similar
to the method used in conversion of decimal to binary.
E9. Convert (650)10 to a hexadecimal number.
Octal
to Binary Conversion
To convert an octal number
to a binary number, simply replace each octal digit by the appropriate three
digits binary equivalent.
E10.Change octal (3574)$ to its binary equivalent.
Sol. 357 4
011 101 111 100
Hence (3574)8 = (011 101 111 100)2 =
(011101111100)2.
Binary
to Octal Conversion
Conversion of a binary to an octal number is also
a straight forward process. Beginning from the least significant
position, simply break the binary number into groups of three digits and
convert each group into its equivalent octal digit.
E11. Convert
(110101)2 to an octal number.
Sol. 110 101
6
5
(110101)2 = (65)
Binary
to Hexadecimal Conversion
Converting a binary number to
hexadecimal is a straightforward procedure. Simply break the binary number into
four bit groups starting from the least
significant position and replace each group with the equivalent
hexadecimal number.
E12.Convert (1100101001010111)2 to a hexadecimal
number.
1100 1010 0101 0111
C A 5 7
(1100101001010111)2
= (CA57)16
Hexadecimal
to Binary Conversion
To convert from a hexadecimal number to a binary number, reverse the process and replace each hexadecimal number with their
equivalent four digits binary number.
E13.
Convert (10A4)16 to a binary number.
Sol.
1 0 A
4
0001 0000 1010 0111
Or
(10A4)16 =
(0001000010100100)2
Addition
of Binary Numbers
The method of addition of
binary numbers is also similar to that of decimal numbers. The three basic rules that will be
used in binary addition are
(1)
0+0=0
(2)
1+0= 1
(3)
1 + 1 = 10
(1 is carry over)
The process of addition will be explained in following example
E14.Add
the binary numbers 101010 and 10101.
Sol. 101010
10101
111111
Addition
of Octal Numbers
The method of addition of octal numbers is similar to that of decimal numbers.
The basic rules that will be used in octal addition are
(1)
1+7=2+6=3+5etc.=0(1
is carry over)
(2)
2+7=3+6=4+5etc.=1(1
is carry over)
(3)
3+7=4+6=5+5etc.=2(1
is carry over)
(4) 4+7=5+6=6+5etc.=3(1 is carry over)
(5)
5+7=6+6=7+5
etc.= 4(1 is carry over)
(6)
6+7=7+6=8+5etc.=5(1
is carry over)
(7) 7 + 7 = 6 (1
is carry over)
The
process of addition will be explained in following example.
E15.(3232)8 + (1256)8 =?
Sol.
3232
+1256
4510
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The unit's digit should be even or
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2
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0(i.e.in the given number atthe
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26, 48 etc.
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6=2=3,8-2=4
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units place we should have 2, 4,
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6, 8, 0)
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3
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The sum of the digits of the
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12729
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(1+2+7+2+9 = 21),
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number should be divisible by 3.
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21 + 3 = 7.
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The number formed by the last
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4
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two digits (units' and tens') of the
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21964
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64-4=16
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given number should be divisible
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by 4.
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5
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The unit's digit should be 0 or 5.
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1835, 15440
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Last digits are 5 and 0
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respectively
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The sum of the digits of the
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(1+2+7+2=12),
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6
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number should be divisible by 3
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1272
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12 / 3 = 4, Number is
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and the number should be even.
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even
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The number formed by the last
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8
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three digits (units', tens' and
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52672
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672 + 8 = 84
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hundreds') of the given number
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should be divisible by 8.
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9
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The sum of the digits of the
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127296
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(1+2+7+2+9+
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number should be divisible by 9.
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6 = 27), 27 = 9 = 3
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10
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The unit's digit should be 0.
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3220
|
Unit's digit is
zero.
|
|
|
The difference between the sums
|
|
5
+ 0 + 3 = 4 + 4 = 8
Hence difference is zero
|
|
11
|
of the digits in the even and odd
|
54043
|
|
|
|
places should be zero or a
|
|
|
|
|
multiple of 11.
|
|
|
|
|
The sum of the digits of the
|
|
(1+7+2+8=18),
18/3 = 6, also 28/4=7
|
|
12
|
number should be divisible by 3
|
1728
|
|
|
|
and the number should also be
|
|
|
|
|
divisible by 4.
|
|
|
|
|
The sum of the digits of the
|
|
(8 + 1 + 0 + 6 + 4 +
|
|
15
|
number should be divisible by 3
|
810645
|
5=24),24+3=8,
|
|
|
and unit's digit of the number
|
|
also last digit is 5.
|
|
|
should be 0 or 5.
|
|
|
|
|
The number formed by the last
|
|
|
|
|
four digits (units', tens',
|
|
|
|
16
|
hundreds' and thousands') of the
|
12978320
|
8320 / 16 = 520
|
|
|
given number should be divisible
|
|
|
|
|
by 16.
|
|
|
|
25
|
The last two digits of the number
|
1125, 975,
|
The last two digits are
|
|
|
should be 25, 50, 75 or 00.
|
15500, 50
|
as required.
|
|
|
|
1125, 1875,
|
|
|
|
|
15500,
|
|
|
|
The last three digits of the
|
35625,
|
The last three digits
|
|
125
|
number should be 125, 250, 375,
|
76375,
|
are as required.
|
|
|
500, 625, 750, 875 or 000.
|
22250,
|
|
|
|
|
49750,
|
|
|
|
|
50000
|
|
Some
other important tests
Divisibility test for 7
The test holds good only for numbers with more
than three digits and is
applied as follows
1.
Group
the numbers in sets of three from the unit's digit.
2.
Add the odd
groups and the even groups separately.
3.
The difference of the odd and the even
groups should be either 0 or divisible by 7.
E16.Is 85437954 divisible by 7?
Sol. 85 437 954.
Adding up the first and the third sets, we
get 85 + 954 = 1039.
Now their
difference is 1039 - 437 = 602.
Since 602 + 7 =
86, hence the number is divisible by 7
Divisibility test for 13
The
test holds good only for numbers with more than three digits. The test
to be applied is as follows
1.
Group the numbers in sets of three from
the unit's digit.
2.
Add the odd
groups and the even groups separately.
3.
The difference of the odd and the even
groups should be either 0 or divisible by 13.
E17.Is 136999005 divisible by 13?
Sol. 136 999 005.
Adding up the first and the third sets, we get 136 + 5 = 141.
Now their
difference is 999 - 141 = 858.
Since 858÷13 = 66, so the number is divisible by 13.
E18. Find X & Y when
(1) 15X8351Y is divisible by 72.
(2)
2856354XY is
divisible by 99.
Sol.
(1) Since 72 = 8 x 9, so the number must be divisible both
by
8 and 9.
The last three
digits of the number should be divisible by 8. Hence 51Y/8 must be an integer (last 3 digits), i.e. Y = 2. Now, the given number should also be
divisible by 9.
►1+5+X+8+3+5+1+2=25+Xshould be divisible by 9. Thus X = 2.
Hence the number is 15283512.
(2) 99 = 9 x 11. Hence the number should
be divisible by 9 and 11 both.
33 + X + Y (sum of
the digits) should be divisible by 9. = (2+5+3+4+Y)-(8+6+5+X) =0or±11or±22...
= 14+Y-19+X=0 or ±11
or ... Solving the equations
Y-X-5=-11 and 33+X+Y= 45, we get
X
= 9 and Y = 3. Hence the number is 285635493.
Imp.
Ø If two numbers, say x
and y, are divisible by a third number, say z, then (x - y) and (x + y)
are also divisible by z.
For example, 20 and 64 are
divisible by 4. Also, (64 - 20) & (64 + 20) are divisible by 4.
Ø When any number with
even number of digits is added to its reverse, the sum is always
divisible by 11.
e.g.
2341 + 1432 = 3773 which is divisible by 11.
Ø When any number with odd number of digits is subtracted from
its reverse, the absolute difference is always divisible by 11&9.
e.g.
23411 - 11432 = 11979 which is divisible by 11 & 9.
Ø If X is a prime
number, then for any whole number "a", (ax - a) is
divisible by X.
For
example, Let X = 3 and a = 5.
Then according to our rule 53
- 5 should be divisible by 3. Now (53 - 5) = 120 which is divisible
by 3.
Fractions
A fraction denotes a part or parts of a unit. The
different types of fractions are as follows
Common
fractions: Fractions whose
denominator is not 10 or a multiple of 10. e.g. 2/3, 17/18 etc.
Decimal
fractions: Fractions whose
denominator is 10 or a multiple of 10.
Proper
fractions: Fractions whose
numerator < denominator e.g. 2/10, 6/7, 8/9 etc. Hence its value <
1.
Improper
fractions: Fractions whose
numerator > denominator e.g. 10/2 , 7/6, 8/7 etc. Hence its value
> 1.
Mixed Fractions: In these types of fractions, there are
two parts, an integral part and a fractional
part. e.g 1⅜, 5⅛ etc. are all mixed fractions.
Compounded Fraction: A fraction of a fraction is known as
a compounded fraction. e.g. 5 of 6 etc. are compounded fractions.
Complex
Fraction: If the numerator or the
denominator or both of a fraction are fractions, then the fraction is
called a complex fraction. e.g ⅛/7 is
complex fractions.
Operations
with Fractions
Operations
with fractions may be performed according to the following rules
The value of a fraction remains the same if its numerator and
denominator are both multiplied or divided by the same number provided the
number is not zero.
For example, 3/4= 3x2/4x2 =6/8 , 15/18 =
15÷3/18÷3 = 5/6.
Changing the sign of
either the numerator or the denominator of a fraction changes the sign
of the fraction.
For example -3/5 = -3/5 = 3/-5
Adding two fractions
with a common denominator yields a fraction whose numerator is the sum
of the numerators of the given fractions and whose denominator is the common
denominator.
For example, 3/5 + 4/5 = 3+4/5= 7/5
The sum or difference
of two fractions having different denominators may be found
by converting the fractions to a common denominator.
For example, 1/4+2/3 = 3/12 + 8/12
= 11/12
. 12
The product of two
fractions is a fraction whose numerator is the product of the
numerators of the given fractions and whose denominator
is the product of the denominators of the fractions.
For example
2/3 x 4/5 x = 2x4/3x5
= 8/15
The
reciprocal of a fraction is a fraction whose numerator is the denominator of the given fraction and whose
denominator is the numerator of the
given fraction. Thus the reciprocal of 3 i.e.
3/1 is
1/3. Similarly
the reciprocals of 5/8
and -4/3 are 8/5 and 3/-4 or -3/4, respectively.
To divide a
given fraction by another fraction, we multiply the first by the reciprocal of
the second.
For example,
a / b ÷ c / d= a/b x d/c
Decimal Fractions
Fractions in which the
denominators are the powers o0_are called decimal fractions.
In general, the decimal fractions are of the following types
Recurring Decimals: If in a decimal
fraction, a figure or a set of figures is repeated
continually, then such a number is called a recurring decimal.
If a single figure is
repeated, it is shown by putting a dot on it. But
if a set of figures is repeated, we express it either by putting one dot
at the starting digit and one dot at the last digit of the repeating digits or by placing a bar or a vinculum
on the repeating digit(s).
(i) 2/3 =
0.6666 .... = 0.6 = 0.6
(ii) 22/7 =
3.142857142857 = 3.142857 = 3.142857
(iii) 95/6 =
15.83333 .... = 15.83 = 15.83
Pure Recurring Decimals: A decimal in which all the figures after
the decimal point repeat is called a pure recurring decimal.
Ex. 0.6 , 3.142857 etc.
Mixed Recurring Decimals: A decimal in which some
figures do not repeat and some of them repeat is called a mixed
recurring decimal.
Ex.
15.83 etc.
Conversion of a
Pure Recurring Decimal into fraction
Rule:
Write the recurring figures only once in the numerator
and take as many nines in the denominator as the number of
repeating figures.
(1) 0.666…= 6/9 = 2/3.
(2) 16.6 = 16 + 0.6 = 16 + 6/9 = 16 +
2/3 = 50/3.
To convert a Mixed
Recurring Decimal into fraction
Rule: In the numerator,
write the difference between the number formed
by all the digits after decimal point (taking repeated digits only once) and that formed by the digits which are not
repeated.
In the
denominator, write the number formed by as many nines as there are repeating
digits followed by as many zeroes as in the number of non
repeating digits.
(1) 0.17
= (17 - 1)/90 = 16/90 = 8/45.
(2) 0.1254
= (1254 - 12)/9900 = 69/550.
(3) 2.536 = 2
+ (536 - 53)/900 = 2161/300
A quick summary
(1) 0.12345 =
12345/99999
(2) 0.12345 =
(12345 - 1)/99990
(3) 0.12345 =
(12345 - 12)/99900
(4) 0.12345 =
(12345 - 123)/99000
(5) 0.12345 =
(12345 - 1234)/90000
(6) 0.12345 =
12345/100000
E19. Let D be a decimal of the form, D = 0.a1
a2 al a2 al a2..., where digits al & a2
lie between 0 and 9. Then which of the following numbers necessarily produces an
integer, when multiplied by D?
(1) 18 (2)
108
(3) 198 (4) 208
Sol. It
is recurring decimal and can be written as D = 0.a,a2 . To convert this to fraction, we can write it
as ala2/99. Thus when
the number is multiplied by 99 or a multiple of it, we shall necessarily get an
integer. Of the given options, only (3) is a multiple of 99, hence Ans.(3)
Exponents
The product 10 x 10
x 10 can be written as 103 and is read as 10 raised to the third power. In general, a x a x a ....a (n
times) is written as a". The base a is
raised to the nth power and n is calleu the exponent or the
index.
Examples:
32 = 3 x 3 ................ read
as "3 squared"
23 = 2 x 2 x 2........... read as
"2 cubed"
54 = 5 x 5 x 5 x 5...... read as "5 to the fourth power". If the exponent is 1, it is usually
understood and not written; thus, al
= a.
Laws
1, am
x an = am+n
For example, 23 x 24 = 23+4 = 27
2.
am / a n = a
m - n = 1 / a n - m
( I f a
≠ 0 )
3.
( a m)
n = a m
n
4.
a-m
= 1/am
For example, 1/32 = 3-2
5.
a°=1 (Any
number with zero exponent is equal to 1)
6.
(axb)m=am
x bm For example, (4 x 5)2 = 42
x 52
7.
(a÷b)m =am÷bm(ifb#0)
8.
m√a = a1/m
9.
ap/q = q√ap
Squares & Square Roots
By the square of a number, we mean
the product of number by the number itself.
If a2 = b, we say that square root of b
is a and we write here square of b = a2.
From the above discussion, it is
clear that
Ø A square of a natural number cannot end with
2, 3, 7, 8 and an odd number of zeroes.
Ø The square of an odd number is odd and that
of an even number is even.
Ø Every square number is a multiple of 3 or
exceeds a multiple
of 3 by
unity.
Ø Every square number is a multiple of 4 or
exceeds a multiple
of 4 by
unity.
Ø If a square number ends in 9, the
preceding digit. is even.
Methods
for finding Square Roots
Factorization
When a given number is a perfect square, we
resolve it into prime factors and
take the product of prime factors choosing one out
of every pair of the same primes. For example
4624=2x2x2x2x17x17
4624 = 22
x 22 x 172
So, the square
root of 4624 = 4624 = 2 x 2 x 17 =68.
Long
division method
The nature of the MBA Entrance test precludes the possibility
of the student ever gainfully exploiting the long division method. In simple terms, what we mean is, DON? use the long
division method. The following example will illustrate the use of this method.
By approximation
In
order to find the square root in the actual test you should use the method of
approximation. In order to use the method of approximation effectively you must
know the
a.
Squares of
numbers upto 30 or more.
b. Rapid multiplication techniques etc.
Suppose you want to find √75, you know that √64 = 8 and √81 = 9. Hence ,.√75 will lie between 8 &
9. Now 75 is closer to 81 than 64, hence √75 will be closer to 9 than
8. Hence it will lie between 8.5 and 9 and
now you can approximate that its value is somewhere around 8.7.
Cube and Cube Roots
The
cube root of a number X is the number whose cube is the number X. We denote the
cube root of X by 3√X or X1/3.
We
resolve the given number into prime factors and take the product of prime
numbers choosing one out of three of each prime number.
= √9261=3√(3x3 x3 x7x7x7=3√(33 x73)=3X7=2129
= 3 √(9 X 9 X 9) =(93)1/3 = 93x1/3 = 9
Fourth root of Unity
If we find the square root of 1, we find its value as ±1
but further
we can't find the square root of the root -1 which is the fourth root of unity.
The fourth root of
unity is an imaginary number √-1 and is
designated by the letter i.
Thus we can say that √-4 = √ (4) (-1) = 2√-1 = 2ί
Also,
√-18 = √(18)√(-1) = 3√2i.
Also, since i = √-1 , we
have i2 = -1;
i3 = i2
x i = (-1) i = —i; i4 =( i 2)2 =
(-1)2 = 1;
i5 = i4
x i = 1 x i = i and similarly for any integral power.
Thus we can write i597 = (i596) x i = (i4)149 x i = 1149 x i = i. But we must be careful in applying a
few rules. For example,
√(-4)√(-4) = √16 = 4, which is incorrect.
To avoid such difficulties, always express √-m where m is a
positive number, as √m i; and use i2 = -1 whenever it
arises.
Thus: √-4√-4= (2i)(2i)
= 4i2 = - 4, which is correct.
Complex Numbers
As we discussed
earlier, a complex number is an expression of the
form a + bi, where a and b are real numbers and i = 1:1. In
the complex number a + bi, a is called the real part and bi is the imaginary
part. When a = 0, the complex number is called a pure imaginary. If b = 0,
the complex number reduces to the real number
a. Thus complex numbers include all real numbers and all pure imaginary
numbers. Two complex numbers a + bi and c + di are equal if and only if a = c and b = d. Thus a + bi = 0 if and only if
a = 0, b = 0.
If c + di = 3,
then c = 3, d = 0.
The conjugate of a complex number a + bi is a -
bi and conversely. Thus 5 - 3i and 5 + 3i are conjugates.
Algebraic
Operations with Complex Numbers
Addition
To add two complex numbers, add the real
parts and the imaginary parts separately.
(a + bi) +
(c + di) = (a + c) + (b + d)i
For example,
(5+4i)+(3+2i)=(5+3)+(4+2)i=8+6i
(-6+2i)+(4-5i)=(-6+4)+(2-5)i=-2-3i.
Subtraction
To subtract
two complex numbers, subtract the real parts and the imaginary parts
separately.
(a + bi) - (c + di) = (a - c) + (b - d)i
For
example,
(3+2i)-(5-3i)
= (3-5)+(2+ 3)i = -2 + 5i (-1+i)-(-3+2i)=(-1+3)+(1-2)i=2-i.
Multiplication
To multiply two complex numbers, treat the numbers as ordinary binomials
and replace i2 by -1.
(a + bi)(c + di) = ac + adi + bci + bdi2
= (ac - bd) + (ad + bc)i
For example, (5 + 3i)(2 - 2i) = 10 - 10i + 6i - 612
= 10-41- 6(-1) = 16- 4i.
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