Download Set Theory: The study of sets

Midterm Study Guide
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Number Systems:
Counting Numbers or Natural Numbers: Whole numbers greater than zero.
Ex. 1, 2, 3, 4, 5…
Whole Numbers: The set of counting numbers plus zero. Ex. 0, 1, 2, 3, 4 …
Integers: The set of whole numbers and their opposites. or Positive and negative whole numbers.
Ex.
…-3, -2, -1, 0, 1, 2, 3…
Rational Numbers: Any number that can be written as the ratio of two integers in the form a , where b ≠ 0.
b
Irrational Numbers: Non-repeating, non-terminating decimals.
Factors: Any of 2 or more numbers multiplied together to form a product.
Multiples: The product of a given number and a whole number.
Prime number: A whole number greater than 1 that has only two factors, 1 and itself.
Composite Number: A whole number greater than 1, that has more than 2 factors.
**Because the number 1 has only 1 factor it is neither prime nor composite.
Fractions: The quotient of one number or expression to another
Numerator: The dividend, the top of the fraction.
Denominator: The divisor, the bottom of the fraction.
Proper fraction: A fraction whose numerator is less than its denominator.
Improper fraction: A fraction whose numerator is greater than its denominator.
Mixed number: An integer with a proper fraction.
Reciprocal: Change the position of the numerator and the denominator. Invert the fraction.
Adding fractions:
1. Create common denominators
2. Add the numerators
3. COPY the denominator - REDUCE
Subtracting fractions:
1. Create common denominators
2. Subtract the numerators
3.
COPY the denominator - REDUCE
Multiplying fractions*:
1. Multiply the numerator
2. Multiply the denominator
3. Reduce if possible. Reduce - divide the numerator and the denominator by the GCF.
Short-cut: take common factors from the numerator of one fraction the denominator
of another. Can be done in multiplication only!!!
Dividing fractions*:
1. Copy the first fraction
2. Change the division symbol to a multiplication symbol
3. Use the reciprocal of the second fraction, then multiply following multiplication rules,
4. Reduce
*Mixed numbers: For multiplying and dividing you must make improper fractions first!
Decimals:
Adding decimals: YOU MUST LINE-UP THE DECIMALS
Subtracting decimals: YOU MUST LINE-UP THE DECIMALS
Multiplying decimals: Multiply normally, place the decimal in the product so the number of
decimal places in the product is equal to the sum of the decimal places in the factors.
Dividing decimals: Move the decimal point so the divisor (the outside number) is a whole
number. Move the decimal in the dividend (the inner number) the same number of places. Move
decimal straight up. Divide.
Changing a decimal to a fraction: Terminating decimal: Say the decimal using the correct mathematical
place value. Reduce. OR Place the decimal (without the point) over the correct power of ten. Reduce.
Changing a fraction to a decimal: Divide the numerator by the denominator.
Changing a decimal to a percent: Multiply the decimal by 100. (move the decimal 2 places to the right.)
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Reference
Integers: Whole numbers and their opposites. Positive and negative whole numbers.
Absolute Value: The distance a number is from zero on the number line.
Rules-.
Adding with the same signs: Add and copy the sign. Ex. -4 + -6 = -10
Adding with different signs: Find the difference and use the sign of the larger number.
Ex. -8 + 4 = -4
Subtracting: ♫ Change subtraction to addition and the sign of the second. ♫
Ex. 3 - -5 =
↓ ↓
3 + +5 = 8
Multiplying: If there are an even number of negative signs, the product is positive.
Ex. -3 x -5 x 3 x -2 x -1 = +90 (Happy)
If there are an odd number of negative signs, the product is negative.
Ex. -5 x 7 x 2 = -70 (Sad)
Dividing: If there are an even number of negative signs, the product is positive.
Ex.
-56 ÷ -8 = +7
If there are an odd number of negative signs, the product is negative.
Ex.
-24 ÷ 3 = -8
**** Remember, these integer rules may be applied to all types of rational numbers!!!!!!!
Order of Operations: The standard order in which mathematical operations are performed.
1. Parentheses or brackets.
2. Exponents and roots, in order from left to right.
3. Multiplication and division, in order from left to right.
4. Addition and subtraction, in order from left to right.
Exponents and Square Roots:
Exponent: The number that shows how many times the base number is used as a factor.
* Remember, negative exponents, use the reciprocal.
Base: The repeated factor. The number that is multiplied together.
Square Root: One of two equal factors of a number.
Properties:
Associative Properties of Addition and Multiplication- The grouping does not change the sum or product.
Ex. (4 + 3) + 5= 4 + (3+ 5) or (4 x 3) x 5= 4 x (3 x 5)
Commutative Properties of Addition and Multiplication - The order does not change the sum or product.
Ex. 4 + 7 = 7 + 4
or
4 x 7= 7 x 4
Distributive Property – The outer number is distributed across the inner numbers. Ex. 2(4 + 5) = (2 x 4) + (2 x 5)
Additive Identity- IS ZERO-The sum of the number and its additive identity is the original number. Ex. 4 + 0= 4
Multiplicative Identity-IS ONE - The product of a number and its multiplicative identity is the original number. Ex. 5 x l = 5
Additive Inverse - OPPOSITE The sum of a number and its additive inverse is zero.
Multiplicative Inverse - RECIPROCAL The product of a number and its multiplicative inverse is one.
Algebra:
Variable: A letter used to represent a number in a mathematical expression. An unknown.
Coefficient: The numerical part of a term. Usually before the variable. Ex. 4n = 12; 4 is the coefficient.
Term: One of the addends of an algebraic expression.
Monomial: A polynomial with one term; it is a number, a variable or
the product of a number (coefficient) and one or more variables.
Ex.
-5, x, 8n, abc
Binomial: A polynomial with two terms.
Ex. 4a + 9
Trinomial: A polynomial with three terms. Ex. 9x2 + 7x - 4
Polynomial: A monomial or the sum of two or more monomials whose
exponents are positive.
Ex.
4x8, 7x + 5y + 11, 3x2 + 5x -12
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Equation: A mathematical sentence stating that two expressions are equal.
Expression: A mathematical representation containing numbers, variables and operation symbols. It
does not have an equality or inequality symbol.
Solving equations: Use the inverse operation in the reverse order of operations.
1-Step
Ex.
x+4=-6
- 4 -4
x = -10
5x = -45
5
5
x = -9
x-5=8
+5 +5
x = 13
(2) x = 5 (2)
2
x= 10
2-Step
5x+ 1 = 16
-1 -1
5x = 15
5
5
x=3
x- 6 = 3
2
+ 6 +6
(2)x = 9 (2)
2
x = 18
Combining like terms: You can only combine terms when the variables are exactly the same.
Ex. 3m + 5v + 4m = 7m + 5v
Once you have combine the like terms; solve the problem following the traditional steps.
Distributive property w/in the equation: Distribute the number that is outside the parentheses
to the all of the numbers inside the parentheses. Solve following algebra rules.
Ex.
2(3x + 8) = 64
6x + 16 = 64
-16 -16
6x = 48
6
6
x = 8
Adding and subtracting polynomials: Adding: Add the like terms. Follow all integer rules!!
When subtracting, be sure to subtract in the correct order. Read the question carefully for specific key words
regarding order. Ex. If -8x + 11 is subtracted from 5x – 15, the 5x – 15 is the first expression. Be sure to
distribute the negative sign to all terms within the expression!!!!
Ex.)
(5x – 15) – (-8x + 11)
5x – 15 + 8x – 11
13x - 26
Multi-step equations: Always simplify the equation first. Distribute if necessary, combine like terms of
possible.
Variables on both sides of the equation: Move the variable with the smaller coefficient to the opposite side
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Reference
by using the inverse operation. Solve the remaining 2-step equation .
Ex.
4x - 3 = 2x + 5
-2x
-2x
2x - 3 = 5
+3
+3
2x
= 8
2
2
x = 4
** BE SURE YOU HAVE SIMPLIFIED EACH SIDE BEFORE MOVING THE VARIABLE FROM
SIDE TO SIDE.
Solving Inequalities:
Inequality: The comparison of two expressions that uses one of the following symbols <, >, ≤, or ≥.
< - less than: the number before the symbol is smaller than the number after the symbol
> - greater than: the number before the symbol is larger than the number after the symbol
≤ - less than or equal to: the number before the symbol is smaller than or equal to the number after the symbol
≥ - greater than or equal to: the number before the symbol is larger than or equal to the number after the symbol
Graphing inequalities:
x=3
| | | | | | | | | | |
Single point on value
-5 -4 -3 -2 -1 0 1 2 3 4 5
x<2
| | | | | | | | | | |
Open point with arrow
-5 -4 -3 -2 -1 0 1 2 3 4 5
x ≥ -4
| | | | | | | | | | |
Closed point with arrow
-5 -4 -3 -2 -1 0 1 2 3 4 5
*You may need to solve the inequality before you graph it. Inequalities are solved following the same steps as an
equation. Remember- you must reverse the inequality symbol if you multiply or divide by a negative number!
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Reference