natural no
TRANSCRIPT
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8/7/2019 natural no
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Natural numbers on the number line
| | | | | |
1 2 3 4 5 6
1 . 1 is the smallest natural number.
2 . There exist infinite natural numbers.
3 . Here the natural numbers arerepresented by blue marks.
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8/7/2019 natural no
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Closure property
in N
+
1 2 3 4
1 0 -1 -2 -3
2 1 0 -1 -2
3 2 1 0 -1
4 3 2 1 0
z 1 2 3 4
1 1 1/2 1/3 1/
2 2 1 2/3 1/
3 3 3/2 1 3/
4 4 2 3/4 1
-
1 2 3 4
1 1 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
X 1 2 3 4
1 1 2 3 4
2 2 4 6 8
3 3 6 9 12
4 4 8 12 16
1 2 3 4
1 0 -
1 -2 -3
2 1 0 -1 -2
3 2 1 0 -1
4 3 2 1 0
z 1 2 3 4
1 1 1/2 1/3 1/4
2 2 1 2/3 1/2
3 3 3/2 1 3/4
4 4 2 3/4 1
N is not
closed
under-
&z since
all the
numbers
in thetable
doesnot
belong
to N.
N is
closed
under
+&x,since all
the
numbers
in the
tablebelongs
to N.
1 2 3 4
1 1 3 4 5
2 3 4 5 6
3 4 5 6 7
4 5 6 7 8
X 1 2 3 4
1 1 2 3 4
2 2 4 6 8
3 3 6 9 12
4 4 8 12 16
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8/7/2019 natural no
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| | | | | | | | | | | | |
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5
INTEGERS ON THE NUMBER LINE
All positive numbers , all negative
numbers together with zero are calledintegers. There are infinite Integers.Number line is vacant in between twointegers.At any point on the number line ,youfind the numbers of greater magnitudeas you move to the right and theconverse is also true.
>>>
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8/7/2019 natural no
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Q, the set of all rational numbers
In Q, in between any two rationalnumbers you can find infinitely many
rational numbers. Therefore rational
numbers are dense on the number line.
Q
Number line corresponding to Q has lotof gaps .Gaps corresponds to irrationalnumbers. Therefore rational numbersare not complete on the number line.
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8/7/2019 natural no
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An addition table for integers
-2 -1 0 1 2
-2 -4 -3 -2 -2 0-1 -3 -2 -1 0 1
0 -2 -1 0 1 2
1 -1 0 1 2 3
2 0 1 2 3 4
+ As all the numbers inthis addition table areintegers, Z is closed
under the operationaddition .
Same is the case withsubtraction {additionof negative integer}and multiplication{repeated addition}.
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8/7/2019 natural no
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Fundamental operation in Z & Q
-2 -1 0 1 2
-2 1 1/2 n.d -1/2 -1
-1 1/2 1 n.d -1 -1/2
0 0 0 n.d 0 0
1 -1/2 -1 n.d 1 2
2 -1 -2 n.d 2 1
z
Q - { 0 } is closed under the operation division.
z
n . d standsfor
not defined.+ does
not belongto Z.ThereforeZ is not
Closed under
the operationdivision.
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8/7/2019 natural no
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Belongs to
does not belong to
N
+
x N
N
Z+
-
x
Z Z
Q Q- {0} Q
+
-
x
z
-
z
z
Q is closed under all the four fundamentalmathematical operations, of course, Q - {0} is closed
under division because division by 0 in not defined in
mathematics.
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8/7/2019 natural no
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REAL NUMBERS ON THE NUMBER LINE
Q
I
Q
U =R
Numberline corresponding to R
has no gaps. Real numbers are
dense and complete on the number line.
To every point on this number line therecorresponds a real number and to everyreal number there corresponds a point
on the number line.
I
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8/7/2019 natural no
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Square roots in Q & R
1 2 3 4 5 6 7 8 9
1 n.d n.d 2 n.d n.d n.d n.d 3
number
Square
root in Q
2 3 5 6 7 8Square
root in R
1 2 3 4 5 6 7 8 91 n.d n.d 2 n.d n.d n.d n.d 3
number
2
Q I I Q I I I I Q
number
Squareroot in R
-3 -2 -1 0 1 2 3
n.d n.d n.d 0 1 2 3
Q U I = R
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8/7/2019 natural no
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R and mathematical operations
R R
+
-
Xz
R
Q Q
+
-
X
z
Q
Square
roots
NonPerfect
Square
Numbers
Square roots
Non-Perfect
Square Numbers
Square
rootsnegative
Numbers
Belongs to
does not belong to
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8/7/2019 natural no
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N
Z
Q I
R
C
N
Z
Q
I
R
C
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8/7/2019 natural no
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By adding irrational numbers to
Q we get a system called set of all real numbers [ R ]Q and I are disjoint sets.
Q
I
QUI = R
Q I =
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8/7/2019 natural no
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In R square
root of anegativeinteger is
not possible.
By adding new type [Square root -venumbers] of numbers to R , we
can get a bigger number systemcalled the system of complexnumbers in which every real
number has a square root .
C
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8/7/2019 natural no
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Tell me the smallest numbersystem to which the following
numbers belong ?
1. 4 + 4
2. 1 - 3
3. -7 x 2
4. 3 z 2
5. 4
6. 0. 47. ( - 4 )
8. 2. 5