Pure Mathematics

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Pure Mathematics

• Algebra, Calculus and Coordinate Geometry

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??(Betrand Arthur William Russell)1872 -- 1970
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Suggested Teaching Sequence
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Suggested Teaching Sequence
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Suggested Teaching Sequence
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Unit 1

• Preliminary Chapter

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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(a) Definition of a term is a statement giving
the strict and precise meaning of the term. To
define a term is to give the definition of that
term.
e.g. A parallelogram is a quadrilateral with two
pairs of opposite sides parallel.
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(b) Axiom is a self-evident and generally
accepted principle.
e.g. Things equal to the same thing are equal to
each other. If a c and b c, then a b
e.g. If equals are added to equals, the results
are equal. If a b and c d, then a c b d
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(c) There is no sharp distinction between
postulate and axiom. Postulates appear almost in
geometry and axioms in all branches of
mathematics.
e.g. A straight line can be drawn passing through
two distinct points.
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(c) There is no sharp distinction between
postulate and axiom. Postulates appear almost in
geometry and axioms in all branches of
mathematics.
e.g. If two straight line lying on a plane are
cut by a third, making the sum of the interior
angles on one side less than two right angles,
then those straight lines will meet, if
sufficiently produced, on the side on which the
sum of the angles is less than two right angles.
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(d) Theorem is a general conclusion proved
logically upon the basis of certain given
assumptions.
e.g. Base angles of an isosceles triangle are
equal.
e.g. In a right-angled triangle, the square on
the hypotenuse equals the sum of squares on the
other two sides.
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(e) Corollary is a theorem that follows so
obviously from the proof of some other theorems
that no, or almost no, proof is necessary. It is
a by-product of a theorem.
e.g. The factor theorem can be considered as the
corollary of the remainder theorem.
i.e. Let f(x) be a polynomial. If f(a) equals
zero, then (x - a) is a factor of f(x).
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

(f) Lemma is less important theorem used in the
proof of another theorem.
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1.1 Terminologies and Number System

• (A) Some Useful Terminologies

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1.1 Terminologies and Number System

• (B) Number Systems

(a) Positive integers or natural numbers
e.g. 1, 2, 3, 4, ..
(b) Negative integers
e.g. 1, -2, -3, -4,..
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1.1 Terminologies and Number System

• (B) Number Systems

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1.1 Terminologies and Number System

• (B) Number Systems

(e) Real numbers (rational numbers together with
irrational numbers)
(f) Complex numbers (numbers of the form a
bi, where a, b are real numbers and i2 -1)
a real part bi
imaginary part
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1.1 Terminologies and Number System

• (B) Number Systems

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1.1 Terminologies and Number System

• (C) Basic Mathematical Operation

An operation is a way of combining two objects.
e.g. addition, subtraction, multiplication,
division
(a) An operation is said to be associative if the
result of the combination of three objects (order
being preserved) does not depend on the way in
which the objects are grouped.
e.g (5 x 6) x 7 5 x (6 x 7)
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1.1 Terminologies and Number System

• (C) Basic Mathematical Operation

(b) An operation is said to be commutative if the
result of the combination of two objects does not
depend on the order in which the objects are
given.
e.g 5 x 6 6 x 5
5 6 6 5
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1.1 Terminologies and Number System

• (C) Basic Mathematical Operation

(c) An operation is said to be distributive if
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1.2 Synthetic Division
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1.2 Synthetic Division
quotient
divisor
remainder
dividend
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1.2 Synthetic Division
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P.8 Ex1A
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1.3 Polynomials and the Method of Undetermined
coefficients
For any polynomials P(x), the degree of P(x)is
usually denoted by deg(P(x)).
Let P(x) and Q(x) be any two polynomials in x.
e.g. if P(x) x4, Q(x) x3
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1.3 Polynomials an the Method of Undetermined
coefficients
Let P(x) and Q(x) be two polynomials with
variable x. The equality P(x) Q(x) is called
(i) an identity if P(x) is always equal to Q(x)
for any value of x.
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1.3 Polynomials an the Method of Undetermined
coefficients
Let P(x) and Q(x) be two polynomials with
variable x. The equality P(x) Q(x) is called
(ii) an equation if P(x) is equal to Q(x) only
for some particular values of x but not all
values of x, or no such x exists such that P(x)
equals Q(x).
e.g. 2x 1 2x 3, x has no solution
Solution must be checked logarithmic equation,
rational equation, fractional equation, equation
with absolute value
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1.3 Polynomials an the Method of Undetermined
coefficients
A polynomials P(x) is called zero polynomial is
P(x) is identically equal to zero, i.e. P(x) 0
for any value x.
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1.3 Polynomials an the Method of Undetermined
coefficients
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1.3 Polynomials an the Method of Undetermined
coefficients
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1.3 Polynomials an the Method of Undetermined
coefficients
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1.3 Polynomials an the Method of Undetermined
coefficients
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1.3 Polynomials an the Method of Undetermined
coefficients
This is called the method of undetermined
coefficients (comparing coefficients) which
states that if two polynomials are identically
equal then their corresponding coefficients of
like powers of x are equal.
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P.14 Ex1B
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1.4 Remainder Theorem and Factor Theorem
Remainder Theorem
Factor Theorem
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1.4 Remainder Theorem and Factor Theorem
Applying factor theorem and synthetic division,
we have the following facts
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1.4 Remainder Theorem and Factor Theorem
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P.17 Ex1C
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1.5 Factorization of Cyclic Symmetric Expressions
A polynomials in several variables is called a
homogeneous polynomial if the degrees of all its
terms are the same.
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1.5 Factorization of Cyclic Symmetric Expressions
A polynomial f(x, y, z) in variables x, y, z is
called cyclic symmetric about x, y, z if
is unaltered if x is replaced by y, y is replaced
by z and z is replaced by x.
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1.5 Factorization of Cyclic Symmetric Expressions
are also cyclic symmetric polynomials about x, y,
z.
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1.5 Factorization of Cyclic Symmetric Expressions
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1.5 Factorization of Cyclic Symmetric Expressions
i.e. f(y, y, z) 0
i.e. f(-y, y, z) 0
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1.5 Factorization of Cyclic Symmetric Expressions
i.e. f(0, y, z) 0
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1.5 Factorization of Cyclic Symmetric Expressions
i.e. f(yz, y, z) 0
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1.5 Factorization of Cyclic Symmetric Expressions
i.e. f(y - z, y, z) 0
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1.5 Factorization of Cyclic Symmetric Expressions
i.e. f(y, y, z) 0
i.e. f(-y, y, z) 0
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1.5 Factorization of Cyclic Symmetric Expressions
i.e. f(yz, y, z) 0
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1.5 Factorization of Cyclic Symmetric Expressions
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P.21 Ex.1D
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1.6 Expression of a polynomial in Terms of Another
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1.6 Expression of a polynomial in Terms of Another
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1.6 Expression of a polynomial in Terms of Another
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1.6 Expression of a polynomial in Terms of Another
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1.6 Expression of a polynomial in Terms of Another
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P.24 Ex.1E
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1.7 Partial Fractions
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1.7 Partial Fractions
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1.7 Partial Fractions
Partial fraction is just the reverse process
which breaks a given fraction into an algebraic
sum of several simpler proper fractions.
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1.7 Partial Fractions
There are four categories for resolving proper
fraction into partial fractions.
(1) The denominator contains non-repeated linear
factors only.
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1.7 Partial Fractions
There are four categories for resolving proper
fraction into partial fractions.
(2) The denominator contains non-repeated and
quadratic prime factors.
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1.7 Partial Fractions
There are four categories for resolving proper
fraction into partial fractions.
(3) The denominator contains linear factors only
but may be repeated.
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1.7 Partial Fractions
There are four categories for resolving proper
fraction into partial fractions.
(4) The denominator contains linear and quadratic
prime factors, which may be repeated.
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P.32 Ex.1F
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1.8 ? and ? Notations
?? is a symbol denoting the the sum of and ?
is one denoting the product of.
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1.8 ? and ? Notations
?Some notes for ?

• The notation ? is distributive over addition and
  subtraction.

(2) Constant factor can be taken out from the ?
notation.
(3)
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1.8 ? and ? Notations
?
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1.8 ? and ? Notations
Let n be a positive integer, we define the
symbols
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1.8 ? and ? Notations
For any positive integer n,
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P.41 Ex.1G