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Welcome to MM150!
Unit 1 Seminar
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MM150 Unit 1 Seminar Agenda
• Welcome and Syllabus Review
• A Review of sets of Numbers
• Sections 1.1 - 1.6
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Syllabus
• Under Course Home: Syllabus and in Doc Sharing
• Attendance requirements• Due dates• Late policies• Plagiarism
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Seminar• Show up on time• Participate often• Participate in a respectful manner• Stay on topic• Stay until the end• Archived, so you can go back and review• Have 2 choices, you only have to attend once.
Wednesday, 10:00 PM ET, or Friday, 11:00 AM ET
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Discussion
• Respond to all discussion questions• Respond to at least 2 classmates for each
discussion question.• Say more than “Nice work.”
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Symbols
• Multiplication * (shift + 8) or () or []• Square root sqrt[16] = 4• Division /
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Sets of Numbers
• Natural Numbers: {1, 2, 3, 4, …}• Whole Numbers: {0, 1, 2,3, …}• Integers: {…-3, -2, -1, 0, 1, 2, 3, …}• Rational Numbers: ½, 0.5, -6, • Irrational Numbers: , √[2], √[3]• Real Numbers: all rational and irrational
numbers
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Example: Find all factors of 12
An easy way to approach this task is to think of pairs of factors you could use, then make the final list from them.
1*12 AND 2*6 AND 3*4 Make sure you have every factor pair listed!
Therefore, the factors of 12 (in numerical order) are 1, 2, 3, 4, 6, and 12.
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EVERYONE: Find all factors of 56.
1 * 56
2 * 28
4 * 14
7 * 8
Therefore, the factors of 56 (in numerical order) are 1, 2, 4, 7, 8, 14, 28, and 56.
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Example:Determine the GCF of 12 and 56.
We have already created these lists, so I will just put them under each other here:
12: 1, 2, 3, 4, 6, 12
56: 1, 2, 4, 7, 8, 14, 28, 56
Now, just plain old COMMON FACTORS of 12 and 56 include 1, 2, and 4.
The GCF is 4.
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Factor Tree72
2 36
2 18
2 9
3 3 72 = 2 * 2 * 2 * 3 * 3
72 = 23 * 32
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Steps to Finding GCF
There are two steps:• Write down only the COMMON PRIME
FACTORS (the big numbers; save the exponents for the next step).
• (For only the common prime factors) given the choice of powers, use the LOWEST POWER for each prime factor.
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GCF Example UsingPrime Factorization
Find GCF (72, 150).
72 = 23 * 32
150 = 2 * 3 * 52
GCF(72, 150) = 2? * 3?
GCF(72, 150) = 2 * 3 = 6
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Steps to Finding the Least Common Multiple
There are two steps:• Write down the PRIME FACTORS with
the greatest exponent.• Determine the product of the prime
factors.
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LCM example usingPrime Factorization
• LCM(72, 150)
• 72 = 23 * 32
• 150 = 2 * 3 * 52
• LCM(72, 150) = 2? * 3? * 5? • LCM(72, 150) = 23 * 32 * 52 = 1800
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Addition of Integers
Same sign
• 4 + 6 = 10
• 12 + 3 = 15
• -3 + (-8) = -11
• -2 + (-5) = -7
Opposite sign
• 3 + (-4) = -1 Think: 4 – 3 = 1. Then take sign of larger, -1
• -7 + 9 = 2 Think: 9 – 7 = 2. Then take sign of larger, 2.
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Subtraction of Integers
Example 1: 4 – 9 (positive four minus positive nine)= 4 + (-9) (positive four plus negative nine)= -5 (by the different signs rule of addition)
Example 2:-3 – 7 (negative three minus positive seven)= -3 + (-7) (negative three plus negative seven)= -10 (by the same sign rule of addition)
Subtraction of Integers
Example 3:-12 – (-14) (negative twelve minus negative
fourteen)
= -12 + 14 (negative twelve plus positive fourteen)
= 2 (by the different signs rule of addition)
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Multiplication and Division of Integers
• Two positives = positive
• Two negatives = positive
• One of each sign = negative
Examples:
(3)(-2) = -6
-9/ (-3) = 3
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Simplifying Fractions
• 15/45
• Divide both the numerator and denominator by 15.
• 15/45 = (15 / 15) / (45 / 15) = 1/3
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Mixed Numbers
• 2 7/8
• Write 2 7/8 as an improper fraction.• 2 7/8 = 2 + 7/8• 2/1 + 7/8• 16/8 + 7/8• 23/8
• 2 7/8 = (2*8 + 7) / 8 = 23/8 22
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Converting an Improper Fraction to a Mixed Number
• 58/4
• 14 2/4
• 14 1/223
144 / 58 4 18 16 2
Multiplying Fractions and Mixed Numbers
3 ½ * 1 ¼
= 7 * 5
2 4
= 35
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½ * 3/4
= 1 * 3
2 4
= 3
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Dividing Fractions
1 ÷ 2 3 7
1 * 73 2
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Dividing Mixed Numbers
9 ½ ÷ 4 3/5
= 19 ÷ 23 2 5
= 19 * 5 2 23
= 95 46
Radical ExpressionsThe radical symbol looks like this: √x and
the x that is located within or under the radical is called the radicand.
An expression that contains a radical is called a radical expression. The following is the square root of a: 2√a and this is also a radical expression. The small 2 in front of the radical is known as the index and it indicates that this is a square root. When no index is present, then the radical is understood to be a square root with an index of 2.
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Here are the perfect squares:(the right side of the equal
sign) 02 = 012 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81102 = 100
112 = 121
122 = 144
132 = 169
142 = 196
152 = 225
162 = 256
172 = 289
182 = 324
192 = 361
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√32
= √(16 * 2)
= √16 * √2
= 4 √2
√50= √(25 * 2)
= √25 * √2
= 5 √2
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Adding and Subtracting Radical Terms
• Radicals are “things”… Example:
2√5 + 4√5 = 6√52 apples + 4 apples = 6 apples
Example: 2√3 + 4√5 = 2√3 + 4√5 (can’t
combine)2 oranges + 4 apples = 2 oranges + 4
apples30
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Simplify: 8 sqrt[11] + 2 sqrt[11]
• (8 + 2) sqrt[11]• 10 sqrt[11]
Simplify: 13 sqrt[2] + 8 sqrt[2]• (13 + 8) sqrt[2]• 21 sqrt[2]
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Multiplying Radicals
• sqrt[25] * sqrt[4]• sqrt[100]• 10
• NOTE: sqrt[25] = 5 and sqrt[4] = 2• sqrt[25] * sqrt[4]• 5 * 2• 10• Either way you get the same answer 32
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Dividing Radicals• sqrt[36/9]• sqrt[36]/sqrt[9]• 6/3• 2
• NOTE: 36/9 = 4• sqrt[36/9]• sqrt[4]• 2 Either way you end up with same
answer
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Exponents
• 672
• 34
• -73
• 59
• So, if you’re presented with 2*2*2*2*2*2*2, you can rewrite this as 2^7 or 27.
• Beware of this situation:
• -24 vs. (-2)4
-24 = -(2)(2)(2)(2) = -16(-2)4 = (-2)(-2)(-2)(-2) = 16
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PRODUCT RULE OF EXPONENTS.
(ax) * (ay) = a(x + y) (KEEP THE BASE and ADD THE EXPONENTS.)
23 * 22 = 2 (3 + 2) = 25
57 * 58 = 5 (7+8) = 515
QUOTIENT RULE OF EXPONENTS.
(ax) / (ay) = a(x - y) (KEEP THE BASE & SBTRCT THE EXPONENTS)
57 = 5 (7-5) = 52 = 25 93 = 1 = 1
55 914 9 (14-3) 911
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POWER RULE OF EXPONENTS.
(ax)y = axy (Keep The Base and MULTIPLY THE EXPONENTS.)
(22)3 = 2 (2*3) = 26 = 64(811)4 = 8 (11*4) = 844
Anything to the zero power is 1. a0=1, a ≠ 0
40 = 1; (-10)0 = 1; 230 = 1; 1000 = 1
Anything to the first power is itself. a1=a
81 = 8; (-1/2)1 = -1/2; 251 = 2536
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A negative exponent moves the term to the other side of the fraction bar.
a-1 = 1/a and 1/a-1 = a
6(-3) = 1/6^3 19(-4) = 1/19^4
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Scientific Notation
3.1 x 104
9.2346 x 10-5
1.89 x 100
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Converting from Scientific Notation to Decimal Notation
9.2346 x 10-5 = 0.000092346
Another way to look at it:
9.2346 x 10-5
= 9.2346 x 1/100,000 = 9.2346/100,000
= 0.000092346
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Converting from Scientific Notation to Decimal Notation
• 1.89 * 103
• 1,890
• Another way to look at it• 1.89 * 103
• 1.89 * 1,000• 1,890
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Converting from Decimal Notation to Scientific Notation
Convert 45,678 to scientific notation
4.5678 x 104
Convert 0.0000082 to scientific notation
8.2 x 10-6
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