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Types of Numbers
Prepared by:
MS. RUPAL PATEL
Assistant Professor
CMPICA, CHARUSAT
Number Systems
• The Evolution of Numbers
• Q: What is the simplest idea of a number?
• A: Something to count with!
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
1. The Counting Numbers
• We can use numbers to count: 1, 2, 3, etc.
• Humans have been using numbers to
count with for thousands of years.
• It is a very natural thing to do.
• E.g. You can have "3 friends",
• A field can have "6 cows”, and so on.
Counting Numbers: {1, 2, 3, ..}
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
2. The Whole Numbers
• The idea of zero, though natural to us now,
was not natural to early humans.
• If there is nothing to count, how can you
count it?
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• E.g. : you can count dogs, but you can't
count an empty space:
• "I had 3 oranges, then I ate the 3 oranges,
• now I have zero oranges...!"
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• So, let us add zero to the counting
numbers to make a new set of numbers.
Whole Numbers: {0, 1, 2, 3, ..}
• Sometimes, It is also termed as
Natural Numbers (But sometimes they
does not include 0 in natural numbers).
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
3. The Integers
• "if you can go one way, can you go
the opposite way?”
• We can count forwards: 1, 2, 3, 4, ...
• ... but what if we count backwards?
• you get negative numbers:
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• E.g. If you had just sold two bulls, but can
only find one to hand over to the new
owner... you actually have minus one
bull ... you are in debt one bull!
• If we include the negative numbers with the
whole numbers, we have a new set of
numbers that are called integers.
Integers: {..., -3, -2, -1, 0, 1, 2, 3, ...}
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
4. The Rational Numbers
• E.g. If you have one orange and want to
share it with someone, you need to cut it in
half.
• You have just invented a new type of
number!
• You took a number (1) and divided by
another number (2) to come up with half
(1/2).
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• The same thing would have happened if
you had four biscuits (4) and needed to
share them among three people (3) ... they
would get (4/3) biscuits each.
• Any number that can be written as a
fraction is called a Rational Number.
• if "p" and "q" are integers (remember we
talked about integers), then p/q is a rational
number.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• E.g. : If p is 3 and q is 2, then:
• p/q = 3/2 = 1.5 is a rational number.
• The only time this doesn't work is when q is
zero, because dividing by zero is undefined
(i.e. Indeterminate).
• Only 2, is also a rational number, because
you could write it as 2/1.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• So, Rational Numbers include:
• all the integers
• and all fractions.
Rational Numbers:
{p/q : p and q are integers, q is not zero}
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
5. The Irrational Numbers
• E.g. If you draw a square (of size "1"), what
is the distance across the diagonal?
• The answer is the square root of 2, which
is 1.4142135623730950...(etc).
• But it is not a ratio of two integers.
• Square root of 2 ≠ p/q
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• So it is not a rational number.
• What is "Not Rational" ...? Irrational !
• So, the square root of 2 (√2) is
an irrational number.
• There
are
many
more
irrational
numbers. Pi (π) is a famous one.
Irrational Numbers: {Not Rational, ≠ p/q}
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
6. The Real Numbers
• A Real Number can be thought of as any
point anywhere on the number line.
• Real Numbers include:
– the rational numbers, and
– the irrational numbers
Real Numbers:
{x : x is a rational or an irrational number}
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
7. Complex Numbers
• “Is there a square root of minus one?”
• √-1 = ?
• If you multiply any number by itself you can't
get a negative result.
• E.g. 1x1 = 1,
• and also (-1)x(-1) = 1.
• It is not possible.
• Just imagine that the square root of minus
one exists.
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• Imagine that, √-1 = i
• E.g.: What is the square root of -9 ?
• √(-9) = √(9 x -1)
= √(9) x √(-1)
= 3 x √(-1)
= 3i
• i has this interesting property that if you
square it (i x i) you get -1.
• E.g. i = √-1
i2 = (-1)
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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• Imaginary Number: A number whose square
is a negative Real Number.
• A Complex Number is a combination of a
Real Number and an Imaginary Number.
• Examples: 1 + i
and
39 + 3i
• But either part can be 0.
Complex
Number
Real Part
Imaginary
Part
3 + 2i
3
2
5
5
0
-6i
0
-6
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
Summary
Type of Number
Counting Numbers
Quick Description
{1, 2, 3, ...}
Whole Numbers Or
{0, 1, 2, 3, ...}
Natural Numbers
Integers
{..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers
p/q : p and q are integers, q is not zero
Irrational Numbers
Not Rational
Real Numbers
Rationals and Irrationals
Imaginary Numbers Squaring them gives a negative Real Number
Complex Numbers
Sym.
Combinations of Real and Imaginary Numbers
Ms. Rupal Patel, Assistant Professor, CMPICA, CHARUSAT, Changa
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