Maths ppt.....7766

Math’s power point
presentation

Made By : Blossom Shrivastava ,

Class : 9 ‘A’

Submit to : Mrs. Urmila Dixit

Made by : Purnima bohare
Class : 9 ‘a’

Natural numbers : the numbers which starts from 1 these numbers
are callled natural numbers.

Whole Numbers : The numbers which starts from 0 are called
whole numbers.

Rational Numbers : A numbers r is called a rational number, if it
can be written in the form of p/q , where p & r integers & q is not
equal to 0.

Irrational numbers : A number s is called a irrational number, if it
can not be written in the form p/q, where p and q are integers
and q is not equal to 0.

The decimal expansion of a rational number is either terminating
or non terminating recurring moreover a number whose decimal
expansion is terminating or non-terminating recurring is rational .

PYTHAGORAS

Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras ho
Sámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about 570 – d. about 495 BC[1][2]) was an Ionian Greek
philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information
about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He
was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking
knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect.
His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical
theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The
Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his
days in Metapontum.
Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often
revered as a great mathematician, mystic and scientist, but he is best known for the Pythagorean theorem which
bears his name. However, because legend and obfuscation cloud his work even more than with the other pre-Socratic
philosophers, one can give account of his teachings to a little extent, and some have questioned whether he
contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may
actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that
everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he
was the first man to call himself a philosopher, or lover of wisdom,[3] and Pythagorean ideas exercised a marked
influence on Plato, and through him, all of Western philosophy.

Chapter 2
Polinomial

 A polynomial have one term is called monomial.

 A polynomial have 2 terms is called binomial.

 A polynomial have 3 terms is called a trinomial.

 A polynomial of degree one is called linear polynomial.

 A polynomial of degree 2 is called a quadratic polynomial.

 A polynomial of degree 3 is called a cubic polynomial.

 A real number ‘a’ is a zero of polynomial p(x) if p(a) = 0. In this case , a is
also called a root of the equation p(x) = 0.

 Remainder theorem : If p(x) is any polynomial of degree greater than or
equal to 1 and p(x) is divided by the linear polynomial x – a, than the
remainder is p(a).

 Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if
x –a is a factorof p(x), then p(a) = 0.

R.Dedekind

While teaching calculus for the first time at the Polytechnic, Dedekind came up with
the notion now called a Dedekind cut (German: Schnitt), now a standard definition of
the real numbers. The idea behind a cut is that an irrational number divides the
rational numbers into two classes (sets), with all the members of one class (upper)
being strictly greater than all the members of the other (lower) class. For example, the
square root of 2 puts all the negative numbers and the numbers whose squares are
less than 2 into the lower class, and the positive numbers whose squares are greater
than 2 into the upper class. Every location on the number line continuum contains
either a rational or an irrational number. Thus there are no empty locations, gaps, or
discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind
cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational
numbers");[1] in modern terminology, Vollständigkeit, completeness.
In 1874, while on holiday in Interlaken, Dedekind met Cantor. Thus began an enduring
relationship of mutual respect, and Dedekind became one of the very first
mathematicians to admire Cantor's work on infinite sets, proving a valued ally in
Cantor's battles with Kronecker, who was philosophically opposed to Cantor's
transfinite numbers.
If there existed a one-to-one correspondence between two sets, Dedekind said that
the two sets were "similar." He invoked similarity to give the first precise definition of
an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern
terminology, is equinumerous to one of its proper subsets. (This[clarification needed] is
known as Dedekind's theorem.[citation needed]) Thus the set N of natural numbers can be
shown to be similar to the subset of N whose members are the squares of every
member of N, (N → N2):

Georg Ferdinand Ludwig Philipp Cantor ( /ˈ kæntɔr/ KAN-tor; German: [ɡeˈ ɔʁk ˈfɛʁdinant
ˈ luˈtv ˈ ɪp ˈ
ɪç fiˈl kantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German
mathematician, best known as the inventor of set theory, which has become a fundamental theory
in mathematics. Cantor established the importance of one-to-one correspondence between the
members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are
"more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem
implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and
their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. [2]
Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even
shocking—that it encountered resistance from mathematical contemporaries such as Leopold
Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig
Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-
Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the
nature of God - [4] on one occasion equating the theory of transfinite numbers with pantheism[5] - a
proposition which Cantor vigorously refuted. The objections to his work were occasionally fierce:
Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6]
and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific
charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs
that the algebraic numbers are countable, and that the transcendental numbers are uncountable,
results now included in a standard mathematics curriculum. Writing decades after Cantor's death,
Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms
of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong". [8] Cantor's
recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile
attitude of many of his contemporaries,[9] though some have explained these episodes as probable
manifestations of a bipolar disorder.[10]
The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded
Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.[11] It has been
suggested that Cantor believed his theory of transfinite numbers had been communicated to him by
God.[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us from
the Paradise that Cantor has created."[13]

Made by --- Blossom Shrivastava
Class --- 9 ‘ A ’
Roll no : 16

 Grid – A pattern of horizontal
and vertical lines, usually
forming squares.

Copyright © 2000 by Monica Yuskaitis

Definition
 x axis – a horizontal number
line on a coordinate grid.

0 1 2 3 4 5 6
x

Definition
 Coordinate grid – a grid used
to locate a point by its
distances from 2 intersecting
straight lines.
A
B What are the
C coordinates
D for the house?
E
1 2 3 4 5

Hint
• x is the shortest and likes to lie
down horizontally.

0 1 2 3 4 5 6
x

 y axis – a vertical number
line on a coordinate grid.
6
y 5
4
3
2
1
0

Hint
 y is the tallest and stands upright
or vertically.
6
5
y 4
3
2
1
0

 Coordinates – an ordered pair
of numbers that give the
location of a point on a grid. (3,
4)
6
5
4 (3,4)
3
2
1
00 1 2 3 4 5 6

How to Find Ordered Pairs
 Step 1 – Find how far over
horizontally the point is by counting to
the right (positive) or the left (5,
(negative).6 )
5
4
y 3
2
1
00 1 2 3 4 5 6 x

How to Find Ordered Pairs
 Step 2 – Now count how far
vertically the point is by counting
up (positive) or down (negative).
(5,4)
6
5
4
y 3
2
1
00 1 2 3 4 5 6 x

What is the ordered pair?

(0,5)
6
5
4
y 3
2
1
00 1 2 3 4 5 6 x

What is the ordered pair?

(1,1)
6
5
4
y 3
2
1
00 1 2 3 4 5 6 x

Four Quadrants of Coordinate Grid
 If the y is negative you move
down below the zero. y = -3
3
y
2
1
0 x
-1
-2
-3
-3 - -1 0 1 2 3

How to Plot in 4 Quadrants
 Step 1 - Plot the x number first
moving to the left when the
number is negative. (-3, -2)
3 y
2
1
0 x
-1
-2
-3
-3 -2 -1 0 1 2 3

• Step 2 - Plot the y number
Copyright © 2000 by
Monica Yuskaitis

moving from your new position
down 2 when the number is
negative.
3 y
2
(-3, -2) 1
0 x
-1
-2
-3
-3 - -1 0 1 2 3

 When x is negative and y is
positive, plot the ordered pair
in this manner. y
3
2
(-2, 2) 1
0 x
-1
-2
-3
-3 - -1 0 1 2 3

Plot This Ordered Pair
(-3, -3)

3
y
2
1
0 x
-1
-2
-3
-3 - -1 0 1 2 3

(-1, 2)
y
3
2
1
0 x
-1
-2
-3
-3 - -1 0 1 2 3

(2, -2)

3
y
2
1
0 x
-1
-2
-3
-3 - -1 0 1 2 3

(-3, -2)

3
y
2
1
0 x
-1
-2
-3
-3 -2 -1 0 1 2 3

Created by : Blossom Shrivastava
Class : 9 ‘ A ‘
Roll no. : 16


Definition of a Linear Equation

A linear equation in two variable x is
an equation that can be written in the
form ax + by + c = 0, where a ,b and c are
real numbers and a and b is not equal to
0.

An example of a linear equation in x
is .

Equations of the form ax + by = c are
called linear equations in two variables.

Equations of the form ax + by = c are (0,4)
called linear equations in two variables.

The point (0,4) is the y-intercept.

The point (6,0) is the x-intercept.
-2 2

Solution of an Equation in Two Variables

Example:
Given the equation 2x + 3y = 18, determine
if the ordered pair (3, 4) is a solution to the
equation.

We substitute 3 in for x and 4 in for y.
2(3) + 3 (4) ? 18
6 + 12 ? 18
18 = 18 True.
Therefore, the ordered pair (3, 4) is a
solution to the equation 2x + 3y = 18.

CARTESIAN PLANE

Quadrant II Quadrant I
( - ,+) (+,+)

Quadrant III Quadrant IV
(-,-) (+, - )

Finding Solutions of an Equation
Find five solutions to the equation y = 3x + 1.
Start by choosing some x values and then computing the
corresponding y values.
If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5)
If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2)
If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1)
If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4)
If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7)

Maths ppt.....7766

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