Math Club Week 1 Chapters 1 and 2 of AoPS Volume 1 By John Cao

Note: I did not put any memes on this power point so it won’t be “racist”. Just kidding

Oops At least it’s not “racist”.

First Things First Hi Everyone! Welcome to the 1 st meeting of the Manvel High School Math Club. I am very excited to kick off this club today, and hopefully you would walk out of this room next year knowing more about math than you ever did. Also, it would be great to have a notebook or something so that you can make sure you still have the information from the powerpoints.

Stuff to think about Officers T-shirts Payments (books) Trips Chess If you have any ideas me at: or

Officers We will elect officers next week. Sorry guys, President/Founder is taken already. The other spots: Vice-President (will help the president in guiding lessons) Secretary/Treasurer (may split up) (keeps track of meetings, funds, books) Historian: takes pictures Any other officer ideas? We will do nominations next week.

T-Shirts I am not a very artistic person (except music), so one of you guys can make a t-shirt design. A possible deadline for the t-shirt design is the beginning of next year (August 2015). Any ideas?

Payments I don’t know anything about the math department funds or whatever, so…………. The Secretary/Treasurer will take care of that once he/she is elected. Ordering books for all of us will be great, and that could put us ahead of many other schools. Also I have not decided on any payments that members will make to the club. ($10??)

Trips There are some trips that we are going to make next year to math competitions, such as the A&M math tournament, the U of H math tournament, etc.

Chess Club? The math department used to run chess club. Do you guys want to make Chess Club part of math club, or separate? Do you think it’s a good idea?

OK, now lets actually start learning (or reviewing)

Chapter 1 Exponents and Logs 1.1: Integer Exponents Problem 1: What is 2 5 x 2 6 You can multiply and divide exponents if they have the same base. (the answer is 2 11 ) Problem 2: ( )/( ) Answer: 1/3 If there is an equation (x y ) z, it would equal x yz What is (4 -3 ) -2 ? Answer: 4 6

Chapter 1 cont. 1.2 Fractional Exponents What is 25 1/2 ? We know that 5 squared is 25. We can do: (5 2 ) 1/2 = 25 1/2 Since (5 2 ) 1/2 = 5 2(1/2) = 5 1 = 5, then 25 1/2 =5. Something to the power of ½ is basically a square root. 81 1/2 is just √81 which is 8. When working with fractional powers in which the numerator is not 1: you can split up the fraction. For example: 8 2/3 = 8 (1/3)(2) = (8 1/3 ) 2 =2 2 =4 Try the following problem: (√8) 2/3 =? Solution: (8 1/2 ) 2/3 = 8 (1/2)(2/3) = 8 1/3 = 2

Chapter 1 cont. You may also realize that 5 is not the only number, when squared, equals 25. In fact, -5 squared is also 5. In AMC, just give the positive root. What is the difference between (-1) 1/2 and –1 1/2 ? The first one is the square root of negative 1, and the 2 nd one is the negative square root of 1.

Chapter 1 cont. 1.3 Simplifying Radical Expressions √8 is not “simplified”. We can reduce it even more, since 4x2 is 8. √8 = (8) 1/2 = (4x2) 1/2 = 4 1/2 x 2 1/2 = 2 x 2 1/2 = 2√2 2√2 would be the correct answer instead of negative 1. Problem: Simplify √1440. Answer: 12√10

Chapter 1 cont. 1.4: Rationalizing Denominators: The simplest way to put it: It is not allowed to have a radical on the bottom of a fraction. You have to multiply it to the top. For example: 1/√2 can be multiplied by √2/√2 to get √2/2 Try the example: 2/ 3 √24 Answer: 3 √9/√3 We all know that we can cancel out (x-y) by multiplying by its conjugate, or (x+y). This is exactly what you can do with fractions with radicals in them too! Try the problem: 6/(√15-√6) Answer: (2√15+2√6)/3

Chapter 1 cont. 1.5 Logs If x y = z, then log x z=y Write this as a log functions: 16 1/4 = 2 Answer: log 16 2=1/4 What is x if log √5 3 √5= x ? Answer: √5 x = 3 √5, 5 (1/2)(x) = 5 1/3, (1/2)(x) = 1/3, divide both sides by ½, x=2/3

Chapter 2: Complex Numbers 2.1 Square root of -1 It is represented as i. i 2 is -1, i 3 is –i, and i 4 is 1. Problem: What is i 1972 ? Answer: 1 because 1972 is divisible by Complex Number Operations It is represented by a+bi, where a and b are both numbers. Every number is actually a complex number, for example, 2 would just be 2+0i.

Chapter 2: Complex Numbers Try adding complex numbers: What is 3+4i-3+8i? Answer: 12i How about (3+4i)(-3+8i) Hint: use foil Answer: i Note: i 2 is -1, so 4i x 8i = 32i 2 = -32 When we divide 2 complex numbers, we clear all instances of 1 from the denominator. What we do is we multiply the bottom by its conjugate. If the bottom has a+bi, we multiply it by a-bi to create a 2 –(b 2 ) which equals a 2 +b 2.

Chapter 2: Complex Numbers Try dividing 3+4i by -3+8i Answer: (23/73)-(36i/73)

3 problems for Chapter 1 1.Find the value of log 5 (125)(625)/(25) Answer: 5 2. Express 2+√2 + 1/(2+√2) +1/(√2-2) Answer: 2 3. Find the sum of (1/2) -1/2 + (3/2) -3/2 + (5/2) -5/2 Answer: 2 1/2 + (2/3) 3/2 +(2/5) 5/2 =√2 + (√2/√3) 3 + (√2/√5) 5 = √2+ (2√2)/(3√3) +(4√2)/(25√5)

3 Problems for Chapter 2 Evaluate √-1 (√-1) 2 √(-1) 2 Answer: -i Simplify (√-6 √2)/√3 Answer: 2i If F(x)=3x 3 -2x 2 +x-3, find F(1+i) Answer: -8+3i

Other stuff If you are interested in UIL Calculator, Number Sense, or Math, you should try to start taking practice tests. If you want to do Number Sense, go to MathMagic.com to get some tricks. As for Calculator, get familiar with the formulas. Also, because you can use 2 calculators in calculator, you can also try to get familiar with the ti-86 since it has unit conversions. If you want to do Math, the best thing to do is to get familiar with all concepts of math. Some practice tests can be found here: