Indices & Algebraic Methods
When working with indices, there are eight laws that must be followed:
These can be used to factorise and expand expressions.
Surds
Surds are examples of irrational numbers, meaning they do not follow a repeating pattern but go on forever, uniquely. Pi is the most common example of an irrational number, but surds are slightly different - they are the square roots of non-square numbers.
√4 = 2 4 is a square number, so gives a rational square root
√2 = 1.4142... 2 is not a square number, so its square root is irrational
Like with indices, there are rules that apply to surds:
These can be used to rationalise denominators:
For fractions in the form 1 / √a, multiply both numerator and denominator by √a
For fractions in the form 1 / (a + √b), multiply both numerator and denominator by (a - √b)
For fractions in the from 1 / (a - √b), multiply both numerator and denominator by (a + √b)
This is known as the conjugate pair (switching the sign of the denominator)
Algebraic Fractions
To simplify algebraic fractions, factorise whatever can be factorised so that parts of the numerator and denominator can cancel:
Multiplication
To multiply fractions, any common factors can be cancelled before multiplying the numerators and denominators.
Division
To divide fractions, multiply the first fraction by the reciprocal of the second fraction (flip the second fraction).
Addition & Subtraction
To add or subtract one fraction from another, a common denominator must be found.
Long Division of Polynomials
Polynomials are expressions that contain only rational numbers, positive indices and/or variables in the numerators.
3x + 5 and 3x² + 5x + 7 are examples of polynomials
3/x, √x and 5x-² are not polynomials
Polynomials can be divided by (x ± p), where p is a constant, using long division:
The Factor Theorem
The example above divides perfectly - it does not have a remainder. This means that (2x+1) must be a factor of the initial expression we divided it into. There is a quicker way to check this: the factor theorem.
The factor theorem states that if f(x) is a polynomial, then:
if f(p) = 0, then (x-p) is a factor of f(x)
if (x-p) is a factor of f(x), then f(p) = 0
The Remainder Theorem
This can be used to find the remainder of a long division, without actually doing the division.
If (x-a) is not a factor of f(x), then the remainder is given as f(a)
Partial Fractions
If a fraction has two or more distinct factors in its denominator, it can be separated into partial fractions.
Two Linear Factors
Three Linear Factors
This method cannot be used if two of the factors are the same (repeated)
Repeated Factors
Improper Partial Fractions
An improper algebraic fraction is a fraction where the numerator has an equal or higher power to the denominator. These must first be converted into proper fractions before they can be expressed as partial fractions. There are two methods of doing this:
Use algebraic long division and add the remainder divided by the divisor to the quotient
Multiply by the divisor and add the remainder
You can find the remainder using the remainder theorem.
Mathematical Proof
A mathematical proof is a logical argument to show that a conjecture (a mathematical statement) is always true. Typically, a proof begins with a theorem (a pre-established fact).
There are a number of requirements for a valid mathematical proof:
All information and assumptions being used must be stated
Every step must be shown explicitly
Every step must lead on logically from the previous step
All cases must be covered
A proof must always end in a statement of proof.
To prove an identity, like (a+b)(a-b) ≡ a² - b², you start with the expression on one side, and manipulate it algebraically until it is exactly the same as the other side. Again, the requirements above apply.
Proof by Deduction
Proof by deduction means starting from a fact or definition and using logical steps to prove the conjecture.
Conjecture:
The product of two odd numbers is also odd
Proof:
If a and b are integers, then 2a+1 and 2b+1 are definitely odd integers
(2a+1)(2b+1) = 4ab + 2a + 2b +1 = 2(2ab+a+b) + 1
This is in the same form as 2m+1, the standard form for an odd number
Therefore, the product of two odd numbers is always odd
Proof by Exhaustion
This involves breaking a proof into smaller proofs and dealing with these all individually. Since it requires every possible instance to be calculated, it is impossible for an infinite range (such as the example above) but can only be used on smaller scales between set limits.
Conjecture:
The sum of perfect cubes between zero and 100 is a multiple of 10.
Proof:
The only perfect cubes between zero and 100 are 1, 8, 27 and 64
1 + 8 + 27 + 64 = 100
100 = 10(10)
Therefore, the sum of cubes between zero and 100 is a multiple of 10.
Proof by Counter-Example
Perhaps the simplest form of mathematical proof (though often the most frustrating), proof by counter-example works by finding a single occurrence when the conjecture is false. If the conjecture is false once, it is always false.
Conjecture:
All even multiples of five are also multiples of 4
Counter-Example:
5x2 = 10
10 is even
10 is not divisible by 4
Therefore, not all even multiples of five are also multiples of four
Proof by Contradiction
Proof by contradiction works by first assuming the conjecture is untrue. Then, you show through logical steps that this is impossible, and so you conclude that the initial conjecture is in fact true.
The contradiction can either be with the initial assumption, or something else that is known to be true.
Conjecture:
√2 is irrational
Assumption:
√2 is not irrational
Proof by contradiction:
Rational numbers can be expressed in the form a/b, where a and b have no common factors
So, √2 can be written as a/b: √2 = a/b
Squaring both sides gives 2 = a²/b²
This can be rearranged to give a² = 2b²
This means that a² must be even, and so a is also even, and can be expressed as 2n (where n is an integer)
Therefore, (2n)² = 2b²
This equals 4n² = 2b², which cancels to 2n² = b²
This shows that b² is also even, and so is b
If a and b are both even, they have a common factor, 2
This contradicts the statement that a and b have no common factors, so √2 must be irrational