Friday, 7 August 2015
2. REAL AND COMPLEX NUMBER SYSTEM
One of he most common uses of numbers is in counting and measuring distances. The natural numbers were known as counting numbers. People were well acquainted with the use of natural numbers and fractions for measuring distances. We are surprised to learn that a Greek intellectual, Pythagoras in the middle of the sixth century B. C., Discovered that there are fractions. For example square root 2, square root 3, etc. He felt that corresponding to every distance there ought to be a number. Symbols 0, 1, 2, 3, 9 used to denote numbers are called numerals.
3. IRRATIONAL NUMBERS
Irrational number cannot be written in p/q form Irrational number is any real numbers that cannot be expressed as a ratio of integers. Irrational numbers cannot be represented as terminating or repeating decimals.for example: π , √2, √3, √5, 3.1416…….., etc
4. RATIONAL NUMBERS
Every rational number can be representing
in the form of p/q and q is not equal to zero.Both are real number, rational numbers are frequently
expressed as decimals.
For example, ¼= 0.25, 5/8=
0.625,
1/3=
0.3333…., 6/11=0.5454…,
etc.
The decimal said to be “terminating” if the remainder is
zero, otherwise it is called “non-terminating”.
Non-terminating is of two types. If the expansion contains
repeated blocks of digits, as in the case of 1/3 and 6/11, this is a rational
number. If the expansion does not contain repeated blocks then the number is
irrational...
The term
rational in reference to the set Q refers
to the fact that a rational number represents a ratio of two integers.Note: In retinal numbers the common fraction is changed in decimal
fractions and decimal is changed in common fractions.
Natural, Integers (positive and negative) prime, whole, odd, even and
compound numbers etc are rational numbers.
5. COMPLEX NUMBERS
A complex number is a NUMBER that can be expressed
in the form a + bi, where
a and b are Real
numbers and i is
the Imaginary unit that satisfies
the equation x2 = −1, that
is, i2 = −1.[1] In this expression, a is the real part and b is
the imaginary part of the
complex number. The complex number a + bi can
be identified with the point (a, b) in
the complex plane
A Complex Number is a combination
of a Real Number and an Imaginary Number
Imaginary number:
The ordered
pair (0, 1) is denoted by the letter i , read as “IOTA”. Then , i2 = i.i
= (0, 1) . (0, 1)
= (-1, 0) ∈ Complex
= -1 ∈ R
so .i= √-1.
“i” is called an imaginary number because
there is no real number x satisfying the property x2 =
-1.
The
number of the form ib is called an imaginary number,∀b ∈ R.Thursday, 6 August 2015
6. REAL NUMBERS
A number x is a real number if x∈Q or x∈I. Thus the set of R numbers, R=Q∪I such that N∪W∪Z∪Q∪R.
The type of
number we normally use such as 1, 15.83, -1.2, 4/5, etc. Positive or negative,
large or small, whole numbers or decimal numbers are all Real Numbers. They are
called “Real Numbers” because they are
not Imaginary Numbers.For example 1, 2,
-4, π, 3, 5, 0.0003, 1/2 , etc.Real numbers have
no starting or ending. There are infinite real numbers between 1 and 2 as well
as 2 and 3 and so on.
Real Numbers |
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