what is number system
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Easier-A number system is a way of counting
things. It's a way of identifying the quantity ofsomething.
Harder A number system is the set of symbols
used to express quantities as the basis for
counting, determining order, comparing amounts,performing calculations, and representing value. It
is the set of characters and mathematical rules that
are used to represent a number. Examples include
the Arabic, Babylonian, Chinese, Egyptian, Greek,
Mayan, and Roman number systems. The ISBN
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Counting numbers are known as natural
numbers.
Thus, 1, 2, 3, 4, 5, 6, 7, ...,etc.,are all
natural numbers.
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All natural numbers together with 0 form the
collection of all whole numbers.
Thus, 1, 2, 3, 4, 5, 6, 7, ...,etc.,are all
whole numbers. Every natural number is a whole number.
0 is a whole number which is not a natural
number.
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All natural numbers, 0 and negatives of
natural numbers form the collection of all
integers.
Thus, ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4,5,...,etc., are all integers.
Every natural number is an integer.
Every whole number is an integer.
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A rational number is a number that can be
expressed as a fraction p/q where p and q
are integers and q0, are known as rational
number. 0 is a rational number, since we can write, 0
= 0/1
Every natural number is a rational number,
since we can write, 1=1/1, 2=2/1, 3=3/1,etc.
Every integer is a rational number.
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A rational number p/q is said to be in
simplest form, if p and q are integers having
no common factor other than 1 and q0.
Thus, the simplest form of each of 2/4, 3/6,4/8, 5/10, etc., is .
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Every rational number is expressible either as
a terminating decimal or as a repeating
decimal.
Every terminating decimal is a rationalnumber.
Every repeating decimal is a rational number.
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A number which can neither be expressed as
a termnating decimal nor as a repeating
decimal, is called an irrational numbers.
EXAMPLES OF IRRATIONAL NUMBERS
TYPE - I
Clearly, 0.01001000100001... is a
nonterminating and non repeating decimals
and therefore, it is irrational.
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TYPE II
If m is a positive interger which is not a
perfect square, then m is irrational.Thus 2, 3, 5, 6, 7, etc., are all
irrational numbers.
The Number : is a number whose exact
value is not 22/7.
In fact has a value which is
nonterminating and non repeating.
So, is irrational, while 22/7 is rational.
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Terminating Decimal
Every fraction p/q can be expressed as a
decimal.If the decimal expression of p/q
terminates, i.e., comes to an end, then thedecimal so obtained is called a terminating
decimal.
Examples
=0.25
5/8=0.625
2 &3/5=2.6
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A decimal in which a digit or a set of digits
repeats periodically, is called a repeating or
recurring decimal.
In a recurring decimal, we place a bar overthe first block of the repeating part and omit
the other repeating blocks.
Examples
2/3=0.666...=0.6
3/11=0.272727...=2.142857
11/6=1.8333...=1.83
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A number whose square is non-negative, is
called a real number.In fact, all rational and
all irrationalnumbers form the collection of
all real number. Every real number is either rational or
irrational.
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Closure property-The product of two real
numbers is always a real number.
Associative law- (a+b)+c=a+(b+c) for all real
numbers a, b, c.Commutative law- a+b=b+a for all real
numbers a & b.
Existence of Additive Identity-Clearly, 0 is a
real number such that 0+a=a+0=a for everyreal number a. 0 is called the additive
identity for real numbers.
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Existence of Additive Inverse-For each real
number a, there exists a real number(-a)
such that a+(-a)=(-a)+a=0.a and (-a) are called the additive inverse (or
negative) of each other.
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Closure Property-The product of two real
numbers is always a real number.
Associate Property- (ab)c=a(bc) for all real
numbers a, b, c.Commutative Law- ab=ba for all real
numbers a and b.
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Suppose we are given a number whose
denominator is irrational.Then, the process
of converting its denominator to a rational
number by multiplying its numerator anddenominator by a suitable number, is called
rationalisation.