High School Mathematics

1. High School Mathematics Revision Lecture 1: Sep 7 2010

2. Table of Contents General Mathematics • Power and Logarithms • Numbers and Polynomials • Simple Identity and Quadratic Equations • Number sequences, AP, GP, summation Additional Mathematics • Mathematical Induction • Limits and Derivatives • Integration Skip Linear Algebra, Probabilities, Geometry, Complex numbers.

3. Power Definition: given integers a and n, Identities: a picture of 2x (source: wikipedia – logarithm) What is a0?

4. Logarithm Logarithm is the “inverse” operation of power. Given y, find x such that ax = y, the answer is called logay Power: given x, compute y Logarithm: given y, compute x a picture of 2x

5. Logarithm logb(xp) = p logb(x).

6. Numbers Integers Z Rational number Q - a/b for a,b integers Real number R Irrational number - real number but not rational number Complex number – won’t discuss

7. Polynomials Polynomial is an expression of one variable and numbers, using only addition, subtraction, multiplication, and non-negative integer exponents. For example, x2 − 4x + 7 is a polynomial, but x2 − 4/x + 7x3/2 is not Polynomial addition: Polynomial division: Polynomial multiplication:

8. Polynomial Equations Given compute x. (wikipedia – quadratic equation) Exercise: solve

9. Number Sequences Arithmetic sequence: the difference between two consecutive terms is the same e.g. 2,5,8,11,14,17… 3,10,17,24,31,… 5,2,-1,-4,-7,-10,… Geometric sequence: the ratio between two consecutive terms is the same e.g. 1,2,4,8,16,32,64… 3,-9,27,-81,243,… 4,2,1,1/2,1/4,1/8,… e.g.

10. Sum of Number Sequences What is the sum of an arithmetic sequence: a+(a+d)+(a+2d)+…+(a+(n-1)d)? The answer is What is the sum of a geometric sequence: a+ar+ar2+…+arn-1? The answer is We will prove these in a later lecture. The proofs can also be found here: http://en.wikipedia.org/wiki/Arithmetic_progression http://en.wikipedia.org/wiki/Geometric_progression

11. Mathematical Induction Fact: If m is odd and n is odd, then nm is odd. Statement: for an odd number m, mk is odd for all non-negative integer k. Idea of induction. • m is odd by definition. • m2 is odd by the fact. • m3 is odd because m2 is odd and the fact. • mi+1 is odd because mi is odd and the fact. • So mi is odd for all i.

12. Mathematical Induction Prove Basis: Show that the statement holds for n = 0. Inductive step: Show that if statement is true for k, then it is also true for k+1. That is, assuming 0+1+…k=k(k+1)/2, want to prove 0+1+…(k+1)=(k+1)(k+2)/2. 0+1+2+ … +k+(k+1) = (0+1+2+ … +k)+k+1 = k(k+1)/2 + k+1 = (k+1)(k/2+1) = (k+1)(k+2)/2 assumption (induction hypothesis)

13. Mathematical Induction Objective: Prove a statement is true for any non-negative integer This is to prove The idea of induction is to first prove P(0) unconditionally, then use P(0) to prove P(1) then use P(1) to prove P(2) and repeat this to infinity… Don’t worry, we are going to study mathematical induction in details.

14. Limits and Derivatives Given a function, want to find the “slope” of a given point. (Recall that the slope is defined as (y2-y1)(x2-x1).) Informally, we’d like to determine how y changes as x changes. Say we’d like to compute the slope of x2 at point p1. Take p2 closer, the answer is better. So the answer is f(x)=x2

15. Limits and Derivatives We say dx2/dx = 2x That is, when we increase x by a small number c, then x2 increases by 2c. f(x)=x2 In principle, we can do differentiation in this way for every function f. In practice, we have a table and a set of rules for convenience of computation. See http://www.cse.cuhk.edu.hk/~chi/csc2110/notes/calculus.pdf

16. Integration • There are two definitions of integration • It is the “reverse” operation of differentiation (i.e. undo the differentiation). • It computes the area of a function For (1), there is not much to say, mostly look at the rules and do the calculations. See http://www.cse.cuhk.edu.hk/~chi/csc2110/notes/calculus.pdf For (2), we mean given a starting point and an ending point, compute the area “under” the curve. As you see in the figure, one can approximately compute the area using smaller and smaller rectangles. (source: wikipedia – integral)

17. Integration What is the connection between (1) and (2)? The answer is: where dF(x)/dx = f(x) To see why it makes sense, consider the following area that represents your income Call this function F(x). the total income after b years is simply F(b). f(x)=x F(x) To compute the area, one can plot another graph to do so.

18. Integration e.g. slope at x=5 is equal to 5 What is the relation between f(x) and F(x)? f(x)=x The slope of every point in F(x) is equal to f(x) F(x) To compute the area, one can plot another graph to do so. e.g. slope at x=4 is equal to 4 Therefore, dF(x)/dx = f(x) This is why we first compute the “reverse differentiation” F(x) and then compute the area from under f from a to b by F(b)-F(a).

19. Summary You should remember what you learnt in general mathematics, especially power and logarithm, polynomial and number sequences. Mathematical induction will be covered in details in this course. Differentiation and integration will not be essential in this course, but they are important in other courses (e.g. numerical analysis, probability, etc). Tutor will tell you more about how to do calculations. We just try to give you a very brief idea of what they are. They are not very difficult once you understood it. In additional mathematics it took many months to do calculations for more complicated functions, but these can be picked up gradually.