Download M19500 Precalculus Chapter 1.1: Real numbers

Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Welcome to Math 19500 Video Lessons
Prof. Stanley Ocken
Department of Mathematics
The City College of New York
Fall 2013
An important feature of the following Beamer slide presentations is that you, the
reader, move step-by-step and at your own pace through these notes. To do so, use
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use the index dots at the top of this slide (or the index at the left, accessible from the
Adobe Acrobat Toolbar) to access the different sections of this document.
To prepare for the Chapter 1.1 Quiz (September 11th at the start of class), please
Read all the following material carefully, especially the included Examples.
Memorize and understand all included Definitions and Procedures.
Work out the Exercises section, which explains how to check your answers.
Do the Quiz Review and check your answers by referring back to the Examples.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Classifying real numbers
Numbers are used for counting and measuring.
Counting numbers are 1,2,3,... They tell how many objects are in a collection: 12
people, 7 marbles, 8 sentences. Any such object is destroyed if it is split into pieces.
Counting numbers, usually called natural numbers, can be used to measure things: A
stick might be 7 feet long and weigh 17 pounds. But most quantities require
in-between measurements. We need measuring numbers to describe physical quantities
such as length, mass, and duration ( = length of time).
Things that are measured can be split into pieces. When you split a length 1 stick into
7 parts, you get 7 small sticks, each with length 1/7. When you glue together 17 such
small sticks, you get a stick with length 17/7. In the symbol 17/7, the denominator 7
tells you the number of parts into which the unit was split, while the numerator 17
tells you how many of these parts were glued together to form a longer stick. All such
numbers (stick lengths) are called rational numbers.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Not every length is a rational number. In other words, some (in fact most) numbers
are not rational: they are called irrational numbers. Two examples:
√
• If a square’s side length is 1 meter, then the length of its diagonal is D = 2 meters.
Although D can’t be written as a fraction, it satisfies a simple equation: D2 = 2.
• If a circle’s diameter is 1 meter, its circumference is π meters. This number doesn’t
satisfy any polynomial equation whatsoever.
Real numbers consist of measuring numbers and their negatives. They are best
pictured as an infinite straight line on which we place two reference numbers: 0 and 1.
Positive real numbers are the lengths of sticks placed on the number line with their left
end at 0. Negative real numbers are the reflections of positive real numbers through
the number 0. You know the picture:
-5
-4
-3
-2
-1
0
Stanley Ocken
1
2
3
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M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
We often use decimal numbers, which are simply rational numbers whose denominator
is a power of 10. For example, 38476
= 38476
10000 = 3.8476. Numbers of this sort are, more
104
properly, called terminating decimals: they can be written with a finite number of
digits to to the right of the decimal point. Every terminating decimal number is a
rational number.
All real numbers
are approximately equal to (written ≈) decimal numbers. For
√
example, 2 ≈ 1.41421356237 while π ≈ 3.14159265359.
However, lots of rational numbers are not terminating decimal numbers. For example,
1/3 can be written only as a non-terminating decimal .33333333333..... This sort of
symbol is obviously suspicious: always beware of “....” To justify such shenanigans we
need the theory of infinite series, which is usually encountered in third semester
calculus and beyond.
Example 1: Rewrite 37.123 as a rational number.
1000
Answer: 37.123 = 37.123 · 1000
= 37123
1000 .
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Factoring whole numbers
Before you work with fractions, you need to understand the meaning of factoring and
how factoring is connected with fractions. Factoring is used to rewrite an algebra
expression as a simplified product. Lets start with factoring whole numbers.
When we write 12 = 3 · 4 , we break 12 down as a product of whole numbers. Each
such breakdown is called a factorization of 12.
The word factor is both a noun and a verb.
Noun: A factor of 12 is a number that appears in some factorization of 12.
Since 12 = 1 · 12 = 3 · 4 = 2 · 6, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Notice that 3 is a factor of 12 precisely because 12/3 = 4 is a whole number.
Verb: We say that 12 factors as 12 = 3 · 4, or as 12 = 2 · 6, or as 2 · 3 · 2.
Factorizations often use powers to abbreviate multiplication:
For example, 108 = 22 · 33 is a factorization of 108.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Some factorizations are more obvious than other. For example, the two obvious
factorizations of 12 are 12 = 12 · 1 and 12 = 1 · 12. Some of the less obvious
factorizations are 12 = 3 · 4, 12 = 2 · 3 · 2, 12 = 2 · 6, etc.
Definition
A prime number is a positive whole number that has exactly two different factors: 1
and itself.
For example, 7 is prime because its only factors are 1 and 7.
Notice that 1 is not prime, since it has only one factor.
The list of primes starts off as 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...
Assume n and p are whole numbers.
Definition
n is divisible by p means: n/p is a whole number.
Therefore p is a factor of n if and only if n is divisible by p.
A basic fact about primes: n is prime if and only it has a factor p with p2 < n.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Procedure
To decide if n is prime, divide n by primes p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ..., with
p2 < n. If n/p is a whole number for any such p, then n is not prime. Otherwise, n is
prime.
Example 2: Is n = 437 prime?
Solution: You need (at most) to divide 437 by primes p = 2, 3, 5, 7, 11, 13, 17, 19, since the
next prime is 23 and 232 = 529 is greater than 432. Since 437/19 = 23, conclude that 437 is
not prime.
Example 3: Is n = 191 prime?
Solution: You need (at most) to divide 191 by 2, 3, 5, 7, 11, 13, since the next prime is 17 and
172 = 289 is greater than 432. None of these is a factor of n, and so n = 191 is prime.
Definitions
A number is factored completely (or a factorization is complete) if it is written
as a product of primes.
The prime power factorization of a number is the factorization that expresses
the number as a product of powers of (distinct) prime numbers that increase from
left to right.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
There is only one prime power factorization of a number.
This would not be true if we allowed 1 to be prime.
Example 4: The (unique) prime power factorization of 240 can be obtained as follows:
240 = 2 · 120 = 2 · 2 · 60 = 2 · 2 · 2 · 30 = 2 · 2 · 2 · 2 · 15 = 2 · 2 · 2 · 2 · 3 · 5 = 24 · 3 · 5.
The basic idea was to list primes 2, 3, 5, ... in order and then pull out the maximum
power of each prime that is a factor of 240.
Factor out 2 four times to get 240 = 24 · 15 .
Factor out 3 one time to get 240 = 24 · 3 · 5 .
We are finished since the remaining factor 5 is prime.
This could have been done in other, less systematic, ways. For example:
240 = 24 · 10 = 6 · 4 · 2 · 5 = 2 · 3 · 2 · 2 · 3 · 5 = 2 · 2 · 2 · 2 · 3 · 5 = 24 · 3 · 5.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Basic fraction rules
Procedure
To reduce a fraction
Factor the top completely.
Factor the bottom completely.
Cancel common factors.
A
AC
=B
. We say that the numerator AC and the
The critical identity used here is BC
A
C
A
denominator BC have common factor C and we write B
.
=B
C
When we cancel powers, there is a trick for speeding up cancellation:
ax5
axxxxx
axx
x
x
x
axx
ax2
=
=
=
=
bx3
bxxx
b
x
x
x
b
b
Explanation:
The numerator started out with 5 factors of x.
The denominator started out with 3 factors of x.
The 3 factors of x on the top cancelled all 3 factors of x on the bottom.
Therefore we were left with 5 − 3 = 2 factors of x on the top.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Quiz review
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Summary: when the numerator has the higher power of x:
Similarly, when the denominator has the higher power of x:
Absolute value
Exercises
Quiz review
ax5
ax2
ax5−3
=
.
=
bx3
b
b
ax3
a
a
= 5−3 = 2 .
bx5
bx
bx
Here are a few examples:
27 38
y4
38−4
34
x7 y 8
y 8−4
= 9−7 = 2
• 9 4 = 9−7 = 2
9
4
2 3
2
2
x y
x x
x
120
by first finding the prime power factorizations
Example 5: Reduce the fraction
432
of the numerator and denominator.
•
y
y
23 y
= 7−3 = 4
27 x
2 x
2 x
•
Solution:
120
8 · 15
23 · 3 · 5
23 · 31 · 5
23 · 3 · 5
5
5
5
=
= 2 2 3 =
=
= 4−3 3−1 = 1 2 =
4
3
4
3
432
4 · 108
2 ·2 ·3
2 ·3
2 ·3
2
·3
2 ·3
18
120
by using any factorizations you like.
432
120
4 · 30
2 · 15
3 · 5
5
Sample solution:
=
=
=
=
.
432
4 · 108
2 · 54
3 · 18
18
Example 6: Reduce the fraction
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
In practice, you are unlikely to encounter fractions with such big numerators and
denominators. That’s because you should reduce fractions that arise in your work as
soon as they appear.
Procedure
To add fractions with the same denominator, add numerators:
A
C
+
B
C
=
A+B
C
Before you add fractions with different denominators, you need to build them up so
that they have the same denominator. The most general way to do this gives a
formula you know. But that method isn’t always easy to work with:
Procedure (basic, will be improved....)
To add fractions with different denominators
A
B
A D
B C
AD
BC
AD+BC
C + D = C · D + D · C = CD + CD =
BD
Example 7:
3
5
+
2
3
=
3·3+5·2
3·5
=
29
15 .
This method is less useful than it looks, since, as we will soon see, the denominator
BD is often too large to work with comfortably. A better approach: first rewrite the
fractions so that they have a common ( the same) common denominator.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Adding Fractions by first finding their LCD
A
C
As stated above, the basic fraction addition rule is B
+D
= AD+BC
BD .
33
5
However, this formula doesn’t work well for examples like 1000 + 2000
(try it).
33
A better method is to rewrite the first fraction 1000 as follows: multiply numerator and
33
33·2
66
33
= 1000
· 22 = 1000·2
= 2000
.
denominator by 2 to get 1000
Then the fractions can be added easily because their denominators are the same.
33
5
66
5
71
1000 + 2000 = 2000 + 2000 = 2000
The best way to add fractions is to rewrite them with a common denominator that is
as small as possible. This will be the denominators’ least common multiple (LCM).
Procedure
To find the least common multiple (LCM) of two or more natural numbers:
Write down the prime power factorization of each number.
For each prime that appears, determine the highest power of that prime in those
factorizations.
The LCM is the product of those highest powers.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Example 8: Find the LCM of 36 and 120
Solution: 36 = 4 · 9 = 22 · 32 and 120 = 8 · 15 = 23 · 31 · 51 .
The primes that appear are 2, 3, and 5.
The highest power of 2 is 23 , which appears in the factorization of 120.
The highest power of 3 is 32 , which appears in the factorization of 36.
The highest power of 5 is 51 , which appears in the factorization of 120.
The product of those highest powers is 23 · 32 · 51 .
Answer: The LCM of 36 and 120 is 23 · 32 · 5 = 8 · 9 · 5 = 360.
Finding the LCD of fractions
The LCD of fractions is the LCM of their denominators.
Procedure
To add fractions by using their LCD:
Build (rewrite) each fraction with that LCD as its denominator.
Add the rewritten fractions by adding their numerators.
Reduce the resulting fraction.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Exercises
Quiz review
Welcome
Classifying real numbers
Example 9: Find
Factoring
5
36
+
Fraction rules
7
120
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
by using their LCD.
Solution:
Find the LCD quickly, just as above: 36 = 4 · 9 = 22 · 32 and
120 = 8 · 15 = 23 · 31 · 51 . and so the LCD is 23 · 32 · 5 = 360
5
5·10
50
Since 360 = 36 · 10, we write 36
= 36·10
= 360
7·3
21
7
120 = 120·3 = 360
50
21
71
360 + 360 = 360 . This
Since 360 = 120 · 3, we write
Add the rewritten fractions:
can’t reduce: 71 is prime.
You may have seen the following procedure. It always works, but it can take a long
time. Furthermore, it doesn’t work when the fractions involve variables.
Procedure
To find the LCD of numerical fractions, list multiples of the smallest denominator
until you get a number that is a multiple of the other denominators.
Example 10: Find the LCD of 36 and 120.
Solution: List the multiples of 36 until you get a multiple of 120, as follows:
36, 72, 108, 144, 180, 216, 252, 288, 324, 360, which equals 3 · 120. Thus the LCD is 360.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Intervals on the real number line
As we have discussed earlier, the real number line includes:
natural numbers 1, 2, 3, ... ;
integers −3, −2, −1, 0, 1, 2, 3, ... ;
decimals such as 3.14567;
rational numbers (fractions of integers), such as 327/244;
√
irrational numbers such as as 2 or π .
The real number line is drawn as a horizontal line on which the numbers increase from
left to right.
-5
-4
-3
-2
-1
0
1
2
3
4
5
The number line mixes algebra and geometry: algebra is used to measure distance.
Definition
The distance between real numbers a and b on the number line equals the right
number minus the left number.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Basic facts about the number line
Numbers to the left of zero are negative.
Numbers to the right of zero are positive:
-5
-4
-3
-2
-1
0
1
2
3
4
5
The number −5 (read as ’negative five’ or ’minus five’ ) is five units to the left of
zero. The number 5 (occasionally written +5 and read as ’positive five’ or ’five’)
is five units to the right of zero.
A number is called non-negative if it is zero or positive.
The negative (or opposite) of a given number is its reflection through 0. A
number and its negative are the same distance from zero, but are on opposite
sides of 0. For example, the negative of 5 is −5, and the negative of −5 is 5 .
The distance between two numbers on the number line is the larger number (on
the right) minus the smaller number (on the left). Below, the distance between
−3 and 5 is 5 − (−3) = 5 + 3 = 8
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
The distance between points is a positive number. There is no such thing as
negative distance.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Comparing numbers. Suppose you want to compare the positions of the numbers −3
and 5 on the number line. The following are different ways of saying the same thing:
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
−3 is less than 5;
5 is greater than −3;
−3 < 5;
5 > −3;
−3 is to the left of 5;
5 is to the right of −3.
The symbols ≤ and ≥ are read ’less than or equal to’ and ’greater than or equal to.’
x ≤ 3 means: x = 3 or x < 3.
x ≥ 3 means: x = 3 or x > 3.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Definition
An interval is a connected piece of the number line.
Example 11: Describe in three different ways the interval consisting of all real
numbers to the right of −3 and also to the left of (or equal to) 5.
Solution:
inequality notation: −3 < x ≤ 5;
interval notation: (−3, 5];
a number line graph:
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
−3 is drawn as a hollow dot, to show that it is not included in (is missing from) the
interval.
5 is drawn as a solid dot, to show that it is included in the interval.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Quiz review
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
There are four types of intervals, depending on which endpoint(s) are excluded:
Example 12: There are four intervals with left endpoint −3 and right endpoint 5.
Write the four intervals using interval notation, inequality notation, and a graph.
Solution:
(−3, 5) is an open interval, written in inequality form as −3 < x < 5.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
5
6
[−3, 5] is a closed interval, written in inequality form as −3 ≤ x ≤ 5.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
(−3, 5] is a half open interval, written in inequality form as −3 < x ≤ 5.
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
[−3, 5) is a half open interval, written in inequality form as −3 ≤ x < 5.
-6
-5
-4
-3
-2
-1
Stanley Ocken
0
1
2
3
4
5
M19500 Precalculus Chapter 1.1: Real numbers
6
Quiz review
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Infinite intervals on the real line
The symbol ∞ or +∞, for (positive) infinity, is used when an interval contains all numbers to
the right of a certain point.
Example 13: Use three kinds of notation to describe all numbers to the right of −3.
inequality notation −3 < x;
interval notation: (−3, ∞);
a number line graph:
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
The symbol −∞, for negative infinity, is used when an interval contains all numbers to the left
of a certain point.
Example 14: Use three kinds of notation to describe all number to the left of and including 3.
inequality notation: x ≤ 3;
interval notation: (−∞, 3] ;
a number line graph:
-6
-5
-4
-3
-2
-1
Stanley Ocken
0
1
2
3
4
5
M19500 Precalculus Chapter 1.1: Real numbers
6
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Absolute value and distance
Definition:
The absolute value of a real number a, written |a|, is the positive number defined
by |a| = a if a ≥ 0 and |a| = −a
if a ≤ 0.
a if a ≥ 0
This is often written as |a| =
.
−a if a ≥ 0
The length of the line segment joining points a and b is the distance between
them.
The distance between points a and b is the absolute value of their difference, in
either order: |a − b| = |b − a|.
In math classes (unlike in physics), distance is a number expressed without a
specific unit of length.
Here are some examples:
|38| = 38 = | − 38|
| − 76.123| = 76.123
|0| = 0.
There’s nothing special about 38. The basic rule |x| = | − x| is true for all real x.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
In particular, let x = b − a. Then −x = −(b − a) = −b + a = a − b and so −x = a − b.
Since |x| = | − x|, it follows that |a − b| = |b − a|.
For example, |8 − 3| and |3 − 8| = | − 5| are both equal to 5.
Example 15: Use three methods to find the distance between −3 and −7.
Since −3 is to the right of −7, the distance is −3 − (−7) = −3 + 7 = 4.
The distance is the absolute value of the first number minus the second number:
| − 3 − (−7)| = | − 3 + 7| = |4| = 4.
The distance is the absolute value of the second number minus the first number:
| − 7 − (−3)| = | − 7 + 3| = | − 4| = 4.
We seem to have demonstrated the following:
Theorem
If a and b are real numbers, then |a − b| = |b − a| is the distance from a to b on the
number line.
To understand why this is true, recall that the distance between two numbers
represented as points on the number line is the right number minus the left number.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
There are two possibilities:
If b is to the right of (or equal to) a, then b − a ≥ 0 and so the distance between
a and b is b − a = |b − a|, or
If a is to the right of (or equal to) b, then a − b ≥ 0 and so the distance between
a and b is a − b = |a − b|.
In the above statement, take b = 0. Then |a − b| = |a − 0| = |a| is the distance
between a and 0 on the number line. That’s an important fact in itself:
Theorem
The absolute value of a real number is its distance from 0 on the number line.
Absolute value inequalities are straightforward (see page 9 in SRW), but watch out:
The statement |a + b| = |a| + |b| is FALSE.
The above formula is not always wrong. Indeed, it’s correct if a and b have the same
sign but is incorrect if a and b have opposite signs. So it’s wrong just half the time!
But the laws of algebra must be correct all the time! Please:
DON’T INVENT FORMULAS!
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Exercises for Chapter 1.1: Real numbers
Click on
Wolfram Calculator
Click on
Wolfram Algebra Examples
to find an answer checker.
to see how to check various types of algebra problems.
1. WebAssign homework.
2. Find the prime power factorizations of a) 3000, b) 4000, c) 1250, and d) 768
3. Find each of the following sums by finding and using the fractions’ LCD.
1
7
3
7
7
7
7
5
1
a) 3000
+ 5000
b) 40
− 50
+ 11
c) 200
+ 300
d) 12
+ 16
+ 15
60
4. Rewrite each of the following intervals using inequality notation
a) [−5, 17)
b) (−9, −4)
c) (−∞, 8]
d) (80, ∞)
5. Rewrite each of the following inequalities using interval notation
a) −5 ≤ x < 17
b) x ≥ 80
c) x < 8
d) −9 < x ≤ 42
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Quiz review
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Quiz review
Example 1: Rewrite 37.123 as a rational number.
Example 2: Is n = 437 prime?
Example 3: Is n = 191 prime?
Example 4: Find the prime power factorization of 240.
120
by first finding the prime power factorizations
Example 5: Reduce the fraction
432
of the numerator and denominator.
120
Example 6: Reduce the fraction
by using any factorizations you like.
432
Example 7: Find 53 + 23 .
Example 8: Find the LCM of 36 and 120 by using prime power factorizations.
Example 9: Find
5
36
+
7
120
by using their LCD.
Example 10: Find the LCD of 36 and 120 by listing multiples.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Welcome
Classifying real numbers
Factoring
Fraction rules
Adding fractions; LCD
Intervals
Infinite intervals
Absolute value
Exercises
Example 11: Describe in three different ways the interval consisting of all real
numbers to the right of −3 and also to the left of (or equal to) 5.
Example 12: There are four intervals with left endpoint −3 and right endpoint 5.
Write the four intervals using interval notation, inequality notation, and a graph.
Example 13: Use 3 kinds of notation to describe all numbers to the right of −3.
Example 14: Use 3 kinds of notation to describe all number to the left of and
including 3.
Example 15: Use three methods to find the distance between −3 and −7.
Special practice: See the interval plotter in the M195 movie file posted on
Blackboard.
Stanley Ocken
M19500 Precalculus Chapter 1.1: Real numbers
Quiz review