pure math part 3

8/13/2019 Pure Math Part 3

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Indices

The Laws of Indices

Indices - Multiplication

remembering that:

Examples

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Indices - Division

remembering that:

Examples:



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Examples:



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The Laws of Logarithms

The Laws of Logarithms



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Changing the base

Remember that the change of base occurs in the term where the base is 'x' or some other

variable.

Example



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Simultaneous equations

'Substitution' simultaneous equations are common problems. First find what x is in terms

of y. Then substitute for x in the other equation. Solve for y.

Example



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Variable in the index

Take logs on both sides. Move the indices infront of the logs. Expand the equation. Collect

x-terms to the left. Sum the numbers to the right. These problems can be tricky with the

amount of arithmetic involved. So make sure you write everything down to make checking

your working easier.

Example



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Surds

Rules

Surds are mathematical expressions containing square roots. However, it must be

emphasized that the square roots are 'irrational' i.e. they do not result in a whole number,

a terminating decimal or a recurring decimal.

The rules governing surds are taken from the Laws of Indices.

rule #1

examples

rule #2



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expression #2 - (the difference of two squares)

Rationalising Surds - This is a way of modifying surd expressions so that the square root is

in the numerator of a fraction and not in the denominator.

The method is to multiply the top and bottom of the fraction by the square root.



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Rationalising expressions using 'difference of two squares'

Remembering that : .... .....from 'useful expressions' above.

Example #1 - simplify

multiplying top and bottom by



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Example #2 rationalise

multiply top and bottom by

Reduction of Surds - This is a way of making the square root smaller by examining its

squared factors and removing them.



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Rational and Irrational Numbers - In the test for rational and irrational numbers, if a surd

has a square root in the numerator, while the denominator is '1' or some other number,

then the number represented by the expression is 'irrational'.

examples of irrational surds:



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Inequalities

Symbols

The rules of inequalities (sometimes called 'inequations')

These are the same as for equations i.e that whatever you do to one side of theequation(add/subtract, multiply/divide by quantities) you must do to the other.

However, their are twoexceptions to these rules.

When youmultiply each side by a negative quantity

'', or '>' becomes '


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Examples

Inequalities with one variable

Example #1 - Find all the integral values of x where,

The values of x lie equal to and less than 6 but greater than -5, but not equal to it.

The integral(whole numbers + or - or zero) values of x are therefore:

6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4

Example #2 - What is the range of values of x where,

Since the square root of 144 is +12 or -12(remember two negatives multiplied make a

positive), x can have values between 12 and -12.

In other words the value of x is less than or equal to 12 and more than or equal to -12.

This is written:



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Inequalities with two variables - Solution is by arranging the equation into the form

Ax + By = C

Then, above the line of the equation, Ax + By < C

and below the line, Ax + By > C

Consider the graph of -2x + y = -2

note - the first termAmust be made positive by multiplying the whole equation by -1

The equation -2x + y = -2becomes 2x - y =2

look at the points(red) and the value of 2x - yfor each. The table below summarises the

result.

point(x,y) 2x - y valuemore than 2

?above/below

(1,1) 2(1)1(1) 1 no - less above

(1,4) 2(1)-(4) -2 no - less above

(2,3) 2(2)-(3) 1 no - less above

(3,3) 2(3)-(3) 3 yes - more below

(2,1) 2(2)-(1) 3 yes - more below

(4,2) 2(4)-(2) 6 yes - more below



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Quadratic Equations

Introduction

The general form of a quadratic equation is:

ax2+bx+ c

where a, b& care constants

The expression b

2

- 4acis called the discriminant and given the letter

(delta).

All quadratic equations have two roots/solutions. These roots are either REAL, EQUALor

COMPLEX*.

*complex - involving the square root of 1



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Solution by factorising - This is best understood with an example.

solve:

You must first ask yourself which two factors when multiplied will give 12 ?

The factor pairs of 12are : 1 x 12, 2 x 6 and 3 x 4

You must decide which of these factor pairs added or subtracted will give 7?

1 : 12 ...gives 13, 11

2 : 6 .....gives 8, 4

3 : 4 .....gives 7, 1

Which combination when multiplied makes +12 and which when added gives -7?

these are the choices:

(+3)(+4),

(-3)(+4),

(+3)(-4)

(-3)(-4)

Clearly, (-3)(-4) are the two factors we want.

therefore

Now to solve the equation .

factorising, as above

either



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or

for the equation to be true.

So the roots of the equation are:

Completing the square

This can be fraught with difficulty, especially if you only half understand what you are

doing.

The method is to move the last term of the quadratic over to the right hand side of the

equation and to add a number to both sides so that the left hand side can be factorised as

the square of two terms.

e.g.

However, there is a much neater way of solving this type of problem, and that is by

remembering to put the equation in the following form:



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using the previous example,

Using the Formula - the two solutions of quadratic equations in the form

are given by the formula:



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Proof

The proof of the formula is by using 'completing the square'.



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Example find the two values of x that satisfy the following quadratic equation:



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Partial Fractions

some definitions:

Proper FractionWhen the degree(index) of the function is higher in the denominator

than the numerator.

Improper Fraction When the degree(index) of the function is higher in the numerator

than the denominator.

Partial Fractions Factorising the denominator of a proper fraction means that the fraction

can be expressed as the sum(or difference) of other proper fractions.

Simple addition/subtraction of algebraic fractions

As with simple fraction arithmetic, a common denominator is found from the denominators

of either fraction and the numerators altered to be fractions of the new denominator.

Equations & Identities

Equations are satisfied by discretevalues of the variable involved.

Example:

Identities are satisfied by anyvalue of the variable used. Note the equals sign '=' is

modified to reflect this.



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Example:

When we make partial fractions(below) we are creating an identity from the original

expression.

Denominator with only 'linear factors'

By 'linear' we mean that x has a power no higher than '1' . In other words, this method

does not work with x2, x3, x4etc.

For each linear factor of the type:

there is a partial fraction:

Example:

where x is a variable and A,B,a,b,c,d are constants, where 'a' is not equal to 'b'.



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Example

Denominator with 'repeated' linear factors

For each 'repeated' linear factor of the type:


Example:



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Example

Denominator with a quadratic factor

For each quadratic factor of the type:




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Example:

Example



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Polynomials

Introduction

A polynomial is an expressionwhich:

consists of a sum of a finitenumber of terms

has terms of the form kxn

(xa variable, ka constant, na positive integer)

Every polynomial in one variable (eg 'x') is equivalent to a polynomial with the form:

Polynomials are often described by their degree of order. This is the highest index of thevariable in the expression.

eg: containing x5order 5, containing x7order 7 etc.

These are NOTpolynomials:

3x2+x1/2+x

second term has an index which is not an integer(whole number)

5x-2+2x-3+x-5

indices of the variable contain integers which are not positive

examples of polynomials:

x +5x +2x5 2 +3

(x +4x7 2)(3x-2)

x+2x2-5x +x3 4-2x5+7x6



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Algebraic long division

If

f(x)the numerator and d(x) the denominator are polynomials

and

the degree of d(x)


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The Remainder Theorem

If a polynomial f(x) is divided by (x-a), the remainder is f(a).

Example

Find the remainder when (2x3+3x+x) is divided by (x+4).

The reader may wish to verify this answer by using algebraic division.



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The Factor Theorem

( a special case of the Remainder Theorem)

(xa)is a factor of the polynomial f(x) if f(a) = 0

Example



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The Binomial Theorem

Introduction

This section of work is to do with the expansion of (a+b)nand (1+x)n.

Pascal's Triangle and the Binomial Theorem gives us a way of expressing the expansion as

a sum of ordered terms.

Pascal's Triangle

This is a method of predicting the coefficients of the binomial series.

Coefficients are the constants(1,2,3,4,5,6 etc.) that multiply each variable, or group of

variables.

Consider (a+b)nvariables a, b .

The first line represents the coefficients for n=0.

(a+b)0= 1

The second line represents the coefficients for n=1.

(a+b)1= a + b

The third line represents the coefficients for n=2.

(a+b)2= a2 + 2ab + b2

The sixth line represents the coefficients for n=5.

(a+b)5 = a5+ 5a4b + 10a3b2+ 10a2b3+ 5ab4 + b5

The Binomial Theorem builds on Pascal's Triangle in practical terms, since writing out

triangles of numbers has its limits.



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The General Binomial Expansion ( n1 )

This is a way of finding all the terms of the series, the coefficients and the powers of the

variables.

The coefficients, represented by nCr, are calculated using probability theory. For a deeper

understanding you may wish to look at where nCrcomes from; but for now you must

accept that:

where 'n' is the power/index of the original expression

and 'r' is the number order of the term minus one

If n is a positive integer, then:

Example #1



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Example #2

It is suggested that the reader try making similar questions, working through the

calculations and checking the answer here(max. value of n=8)

http://www.kusashi.com/binomial.phphttp://www.kusashi.com/binomial.php


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The Particular Binomial Expansion

This is for (1+x)n, where n can take any value positive or negative, and x is a fraction( -

1


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Iteration

Introduction

Repeatedly solving an equation to obtain a result using the result from the previous

calculation, is called 'iteration'. The procedure is used in mathematics to give a more

accurate answer when the original data is only approximate.

Problems usually involve finding the root of an equation when only an approximate value is

given for where the curve crosses an axis.

Direct/Fixed Point Iteration

method:

1. rearrange the given equation to make the highest power ofxthe subject

2. find the power root of each side, leavingxon its own on the left

3. the LHSxbecomes xn+1

4. the RHSxbecomes xn

The equation is now in its iterative form.

We start by working outx2from the given valuex1.

x3is worked out using the valuex2in the equation.

x4is worked out using the valuex3and so on.



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Example

Find correct to 3 d.p. a root of the equation

f(x)= x3- 2x+ 3

given that there is a solution near x= -2



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Iteration by Bisection

method:

1. reduce the interval where the root lies into two equal parts

2. decide in which part the solution resides

3. repeat the process until a consistent answer is achieved for the degree of accuracy

required

Example


f(x) = 2x2- 2x+ 7

given that there is a solution near x= -2



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Example


f(x) = 2x2+x- 6

given that there is a solution near x= 1.4



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Set Theory

Introduction

A setis a collection of objects, numbers or characters.

{abcdef....wxyz} {1,2,3,4,...45, 46, 47} etc.

Note how the set is enclosed in brackets {.....}

A definite setis one in which all its members are known.

Setsare given uppercase letters: A, B, C, etc.

The elementsof sets are given lowercase letters: a, b, c,..etc.

An elementxbelonging to the set Ais written:

A constraint bar{...|...} is for setting a property that all members satisfy.

A{x l x has the colour blue} - all elements of A are blue

Common Sets



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Venn Diagrams

Venn diagrams are used to visualise sets and their relations to one another.

Above is a diagramatic representation of set A. The set can be represented mathematically

as: A{1,3,5,7,9} .

Note that set A(the circle) is a subset of the Universal set(the rectangle).

A' (A-dash)is called the complement of A. It contains all elements which are not members

of A.

A and A' together make up the Universal set.



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The unionof sets A and B contains all of the elements from both sets.

Theintersection of sets A and B contains a particular group of elements that exist in set

A and in set B.



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Subsets

If B is a subsetof A. Then all of the elements of B are also in A.



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Functions

Introduction

To thoroughly understand the terms and symbols used in this section it is advised that you

visit 'sets of numbers' first.

Mapping(or function)

This a 'notation' for expressing a relation between two variables(say xand y).

Individual values of these variables are called elements

eg x1x2x3... y1y2y3...

The first set of elements ( x) is called the domain.

The second set of elements ( y) is called the range .

A simple relation like y= x2can be more accurately expressed using the following format:

The last part relates to the fact that xand yare elements of the set of real numbers R(anypositive or negative number, whole or otherwise, including zero)

One-One mapping

Here one element of the domain is associated with one and only one element of the range.

A property of one-one functions is that a on a graph a horizontal line will only cut the

graph once.



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Example

R+the set of positive real numbers

Many-One mapping

Here more than one element of the domain can be associated with one particular element

of the range.

Example

Z is the set of integers(positive & negative whole numbers not including zero)



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Completefunction notation is a variation on what has been used so far. It will be used

from now on.

Inverse Function f -1

The inverse functionis obtained by interchangingxand yin the function equation and

then rearranging to make ythe subject.

If f -1exists then,

ff-1(x) = f-1f(x) = x

It is also a condition that the two functions be 'one to one'. That is that the domain of fis

identical to the range of its inverse function f -1.

When graphed, the function and its inverse are reflections either side of the line y=x.

Example

Find the inverse of the function(below) and graph the function and its inverse on the same

axes.



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Composite Functions

A composite functionis formed when two functionsf,gare combined.

However it must be emphasized that the order in which the composite function is

determined is important.

The method for finding composite functions is:

find g(x)

findf[g(x)]



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Example

For the two functions,

find the composite functions (i fg (ii g f



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Exponential & Logarithmic Functions

Exponential functions have the general form:

where 'a' is a positive constant

However there is a specific value of 'a' at (0.1) when the gradient is 1 . This value,

2.718...or 'e' is called the exponential function.

The function(above) has one-one mapping. It therefore possesses an inverse. This inverse

is the logarithmic function.



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pure math part 3

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