Transcript natural numbers

Welcome to MM150!
Unit 1 Seminar
Louis Kaskowitz
[email protected]
AIM: xEqualsPi
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MM150 Unit 1 Seminar Agenda
• Welcome and Syllabus Review
• Number Theory
• The Real Number System
• Operations with Real Numbers
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Seminar Info
• Arrive on time and stay until the end.
• Seminars are brisk: Be prepared by previewing the
topics. Have book, pencil, and paper.
• You will seldom master a unit’s worth of work in only
one seminar. It takes practice, practice, practice, …
and patience.
• All seminars are archived, and the Power Points* are
posted in DocSharing after the seminar time.
• There are several “flex” seminar options for you.
• Be courteous and encouraging to others. Be positive!
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Flex Seminars
Flex seminar cohort for 1101B:
• Thursday 10 PM ET (this class), Louis Kaskowitz
• Wednesday 11 AM ET, Kimberly White
• Sunday 8 PM ET, Mark Johnston
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Syllabus
• Syllabus can be downloaded from Doc Sharing
or found under Course Home: Syllabus.
• Please read it! (I know it is long)
• Key things include grading rubrics, attendance
requirements, due dates and late policies, info
about your Project assignment, and doing
your own work.
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Kaplan University Math Center (KUMC)
•
•
•
•
Math Modules
Q&A: [email protected]
Math Workshops
Live Tutoring
Log in to the Kaplan homepage, click
“My Studies,” then “Academic Support
Center.”
Sunday, Wednesday,
Monday:
Thursday:
11 a.m. – 5 p.m. ET
8 p.m. – 12 a.m. ET
Tuesday: 11 a.m. – 12 a.m. ET 8 p.m. – 12 a.m. ET
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Chapter 1
Number Theory and the Real
Number System
• An introduction to number theory
• Prime numbers
• Integers, rational numbers, irrational numbers, and real
numbers
• Properties of real numbers
• Rules of exponents and scientific notation
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Number Theory
• The study of numbers and their properties.
• The numbers we use to count are called
natural numbers or counting numbers.
• The Natural Numbers = {1, 2, 3, 4, 5,…}
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Integers
• The set of integers consists of 0, the natural
numbers, and the negative natural numbers.
• Integers = {…–4, –3, –2, –1, 0, 1, 2, 3 4,…}
• On a number line, the positive numbers
extend to the right from zero; the negative
numbers extend to the left from zero.
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Factors
• The natural numbers that are multiplied
together to equal another natural number are
called factors of the product.
• Example:
The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
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Prime and Composite Numbers
• A prime number is a natural number greater
than 1 that has exactly two factors, itself and
1.
• A composite number is a natural number that
is divisible by a number other than itself and
1.
• The number 1 is neither prime nor composite,
it is called a unit.
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The Fundamental Theorem of
Arithmetic
• A prime number is a natural number greater than
1 that has exactly two factors, itself and 1.
• A composite number is a natural number that is
divisible by a number other than itself and 1.
• Every composite number can be expressed as a
unique product of prime numbers.
• This unique product is referred to as the prime
factorization of the number.
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Finding Prime Factorizations
•
Branching Method:
– Select any two numbers whose product is the
number to be factored.
– If the factors are not prime numbers, continue
factoring each number until all numbers are
prime.
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Example of branching method
140
14
7
10
2 5
2
Therefore, the prime factorization of
140 = 2 • 2 • 5 • 7.
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Commutative Property
• Addition
a+b=b+a
• Multiplication
a * b = b *a
• 8 + 12 = 12 + 8 is a true statement.
• 5 * 9 = 9 * 5 is a true statement.
• Note: The commutative property does not hold true
for subtraction or division.
• Is putting on your socks commutative? How about
putting on shoes and socks?
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Distributive Property
• Distributive property of multiplication over
addition
a * (b + c) = a * b + a * c
for any real numbers a, b, and c.
• Example: 6 * (r + 12) = 6 * r + 6 * 12
= 6r + 72
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Exponents
• When a number is written with an exponent,
there are two parts to the expression:
base
exponent
• The exponent tells how many times the base
should be multiplied together.
45  4  4  4  4  4
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Scientific Notation
• Many scientific problems deal with very large
or very small numbers.
• 93,000,000,000,000 is a very large number.
• 0.000000000482 is a very small number.
• Scientific notation is a shorthand method used
to write these numbers.
• 9.3 x 1013 and 4.82 x 10–10 are two examples
of numbers in scientific notation.
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To Write a Number in Scientific Notation
1. Move the decimal point in the original number to
the right or left until you obtain a number greater
than or equal to 1 and less than 10.
2. Count the number of places you have moved the
decimal point to obtain the number in step 1.
If the decimal point was moved to the left, the
count is to be considered positive. If the decimal
point was moved to the right, the count is to be
considered negative.
3. Multiply the number obtained in step 1 by 10
raised to the count found in step 2. (The count
found in step 2 is the exponent on the base 10.)
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Example
• Write each number in scientific notation.
a) 1,265,000,000.
1.265 x 109
b) 0.000000000432
4.32 x 1010
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Principal Square Root
• The principal (or positive) square root of a
number n, written n is the positive number
that when multiplied by itself, gives n.
• For example,
16 = 4 since 4  4 = 16
49 = 7 since 7  7 = 49
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Radicals
•
2, 17,
53 are all irrational numbers.
The symbol
is called the radical sign.
The number or expression inside the radical
sign is called the radicand.
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Product Rule for Radicals
a  b  a  b,
a  0, b  0
• Simplify:
a)
40
40  4 10  4  10  2  10  2 10
b)
125
125  25  5  25  5  5  5  5 5
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Example: Adding or Subtracting
Irrational Numbers
• Simplify: 4 7  3 7
• Simplify: 8 5  125
4 7 3 7
8 5  125
 (4  3) 7
 8 5  25  5
7 7
8 5 5 5
 (8  5) 5
3 5
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Fractions
• Fractions are numbers such as:
1
,
3
2
9
, and
.
9
53
• The numerator is the number above the
fraction line.
• The denominator is the number below the
fraction line.
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Reducing Fractions
• In order to reduce a fraction to its lowest
terms, we divide both the numerator and
denominator by the greatest common divisor.
72
• Example: Reduce
to its lowest terms.
81
72 72  9 8


• Solution:
81 81  9 9
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Mixed Numbers & Improper Fractions
• Rational numbers greater than 1 (or less than –1)
that are not integers may be written as mixed
numbers, or as improper fractions.
• A mixed number consists of an integer and a fraction.
For example, 3 ½ is a mixed number.
• 3 ½ means “3 + ½”.
• An improper fraction is a fraction whose numerator
is greater than its denominator. An example of an
improper fraction is 7/2.
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Converting a Positive Mixed Number
to an Improper Fraction
•
•
•
•
Multiply the denominator of the fraction by
the whole part.
Add the product obtained in step 1 to the
numerator of the fraction. This will be the
numerator of the improper fraction.
The denominator of the improper fraction is
the same as the denominator of the fraction
in the mixed number.
3 ½ = 7/2
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Example
7
• Convert 5 to an improper fraction.
10
7 (10  5  7) 50  7 57
5



10
10
10
10
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Terminating or Repeating Decimal
Numbers
• Every rational number when expressed as a
decimal number will be either a
terminating or a repeating decimal number.
• Examples of terminating decimal numbers
are 0.7, 2.85, 0.000045
• Examples of repeating decimal numbers
0.44444… which may be written 0.4,
and 0.2323232323... which may be written 0.23.
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