Class Notes

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MTH 208 Notes for Week 1

SubjectExplanationExamples (if available)
Numbers
1. Natural numbers: all numbers from numerical one forward
2. Whole Numbers: all natural numbers and including zero (a.k.a. "origin)
3. Integers: all positive and negative numbers, including zero
4. Rational numbers: Fractions
1. Natural Numbers: 1, 2, 3, 4, 5, ...
2. Whole Numbers: 0, 1, 2, 3, 4, ...
3. Integers: ..., -2, -1, 0, 1, 2, ...
4. Rational Numbers: 1/2, 2/4, 3/6, 4/8, ...
Absolute ValuesAbsolute values are the values of numbers based on their position from origin (zero)|-3| = 3, because -3 is three places away from zero on the number line. Also, |3| = 3, because it is also three places away from zero.
Fractions
Multiplying FractionsMultiply straight across on both top and bottom to get your answer.
Formula: a/b * c/d = ac/bd
Problem:
3/6 * 5/6
=(3)(5)/(6)(6)
= 15/32
Dividing FractionsTo divide two fractions, invert (flip!) the second fraction and multiply across. (Please note- I have no division sign, so a capital D will be standing in. :] )
Formula: a/b D c/d = a/b * d/c = ad/bc
Problem:
1/2 D 2/5
= 1/2 * 5/2
= (1)(5)/(2)(2)
= 5/4 OR 1 1/4
Adding Fractions (Must have a common denominator)This one's simple! Just cross multiply and add the products for the top half, then multiply straight across for the bottom half.
Formula: a/b + c/d = ad + bc/bd
Problem:
2/3 + 4/5
= (2)(5) + (3)(4)/(3)(5)
= 10 + 12/15
= 22/15
= 1 7/15
Subtracting Fractions (Must have a common denominator)Same as with addition, only after we cross multiply for the top, we subtract the products!
Formula: a/b - c/d = ad - bc/bd
Problem:
3/5 - 4/3
= (3)(3) - (5)(4)/(5)(3)
= 9 - 20/15
= -11/15
Like/Unlike Signs
1. Adding a negative and a positive is like subtracting the negative number from the positive one.
2. Adding two negatives is the same as adding two positives, just with a little dash in front.
Subtraction:
1. Subtracting a positive from a negative is like adding the two numbers and putting a dash in front.
2. Subtracting a negative from anything is like adding the negative to what's in front of it.
1. 3 + (-1) = 2
2. -1 + (-5) = -6
Subtraction:
1. -8 - 3 = -11
2. 3 - (-3) = 6. Also, -4 - (-5) = 1
MultiplicationWhen multiplying signed numbers:
1. neg * pos = neg
2. pos * neg = neg
3. pos * pos = pos
4. neg * neg = pos
Examples:
1. -3 * 3 = -9
2. 4 * -5 = -20
3. 9 * 3 = 27
4. -2 * -4 = 8
DivisionWhen dividing signed numbers:
1. pos D neg = neg
2. neg D pos= neg
3. neg D neg = pos
4. pos D pos = pos
Examples:
1. 5 D -5 = -1
2. -4 D 2 = -2
3. -6 D -3 = 2
4. 20 D 4 = 5
Division by Origin (Zero)When dividing with the origin (or zero):
1. 0 D x = 0
2. x D 0 = infinity/undefined
Examples:
1. 0 D 3 = 0
2. 3 D 0 = undefined
Exponential ExpressionsPlease note- I can't superscript the text. x^y therefore represents x to the y power.
Exponents are simple! If you have a^4 it is the same as multiplying a four times (i.e. a*a*a*a). a^3 = a*a*a, and so on.
Problems:
3^2 = 3*3 = 9
2^5 = 2*2*2*2*2 = 32
Algebraic EquationsWhen doing algebraic equations, make sure you remember to "Please Excuse My Dear Aunt Sally" -- Parenthesis Exponents Multiplication/Division Addition/Subtraction. That's the order you need to go in to get the right answer!Example: 1+2*3(4^3-6 D 3)
= 1+2*3(16-2)
= 1+2*3(14)
= 1+2*42
= 1+84
= 85

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