Download Mastery Test 1 Study Guide

Mastery Test 1 Study Guide
Number Systems & Number Theory
Subsets of Real Numbers:
 Counting #’s: 1, 2, 3,….
 Whole #’s: 0, 1, 2, 3,….
 Integers: …., -3, -2, -1, 0, 1, 2, 3,….
 Positive #’s: all numbers greater than 0
 Negative #’s: all numbers less than 0
 Rational #’s: all Numbers that are NOT IRRATIONAL
 Irrational #’s: CRAZY, most well know example is Pi (3.1416……..)
#’s that go on forever without repeating or terminating
Perfect Squares and Square Roots:
 A perfect square has a square root that works out with no remainder.
 List of square roots from 1-15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121,
144, 169, 196, 225.
 Examples of perfect squares: √16 = 4 because 4 x 4 = 16;
√225 = 15 because 15 x 15 = 225
 Square roots for non-perfect squares: using a “Square Root Sandwich”:
1. Find the √5: Think about the perfect squares that are one
above and one below √5
2. They are the √9 and √4
3. Find the square root of the one above & below: √9 = 3; √4 = 2
4. Estimate the square root of the non-perfect square: √5 ≈ 2
because 5 is closest to four.
Factors and GCF (greatest common factor):
 A FACTOR is a # that divides evenly into a counting #
Examples: Factors of 6: 1, 2, 3, 6 (1 x 6 = 6; 2 x 3 = 6)
Factors of 9: 1, 3, 9 (1 x 9 = 9; 3 x 3 = 9)
Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
 Prime #’s: numbers that have ONLY 2 FACTORS: 1 and itself
Ex: 2, 3, 5, 7, 11, 13, 17, 19,….
 Composite #’s: numbers that have 3 OR MORE FACTORS
Ex: 4, 6, 8, 9, 10, 12, 14, 15, ….
 The number 1 is neither prime nor composite (it only has 1 factor: 1)
 The GCF of a numbers is the largest factor that 2 or more #’s have in
common
Example: Find the GCF of 18 & 27
1. List the factors of each #
18: 1, 2, 3, 6, 9, 18
27: 1, 3, 9, 27
2. Find the greatest number they have in common: 9
Prime Factorization (factor trees):
 The goal of Prime Factorization is to find all of the prime factors of a given
number.
Example:
36
(think: what 2 #’s = 36 when I multiply them)
^
94
(ask yourself: are these prime? NO)
^^
3 3 2 2 (ask yourself: are these prime? YES)
Next you write your answer using exponents: 36 = 22 x 32
Finding the GCF using factor trees:
 Make a factor tree for each number; write each answer using exponents
Example: Find the GCF of 36 and 54:
36
^
6 6
^ ^
2323
54
^
9 6
^ ^
3323
36: 22 x 32
54: 2 x 33
Ask yourself: What do they have in common?
GCF: 2 x 32
Now multiply to get the GCF
GCF: 2 x 3 x 3 = 18
GCF: 18
Multiples and LCM (Least Common Multiple):
 Multiples are the numbers that you count when you say the multiplication
tables of a given number.
Example: Multiples of 3: 3, 6, 9, 12, ….
Multiples of 5: 5, 10, 15, 20, ….
 Common Multiples: the multiples that 2 given numbers have in common.
Example: List the common multiples of 3 & 5: 15, 30, 45, …
Once you find the first number you just continue to add it each time
Finding LCM using prime factorization:
 When given a set of numbers you can use factor trees to find their LCM
Example: Find the LCM of 33, 15 & 50:
33
^
3 11
15
^
35
50
^
5 10
^
25
List the factors of each using exponents:
33: 3 x 11
15: 3 x 5
50: 2 x 52
Next make a problem with the MOST of each number
LCM of 33, 15, & 50 = 2 x 3 x 52 x 11
Multiply to get the answer: 2 x 3 x 5 x 5 x 11 = 240
Changing Fractions to Decimals:
 Terminating decimals: decimal numbers that end
Example: .23
 Repeating decimals: decimal numbers that never end (they repeat)
Example: .2323232323 (you would put a _ over the first .23 to
write this answer.
 To change a fraction to a decimal, use long division:
Example: change _4_ to a decimal
9
.444 .4 (you would put a _over the 4)
9 |4.000
-36
40
-36
40
-36
Placing numbers on a number line:
 For whole numbers, put a dot on the number, label with the letter.
 For fractions, divide to make a decimal, estimate where the decimal would
be on the number line, label with the letter.
 For decimals, estimate where it would be on the number line, label with
the letter.
 Pi is always a little past the 3
 For square roots, find the square root, plot the point on the number line,
label with the letter