Mathematics 106
Real Number Lab
NAME
1)
The basic set of numbers used by society is the set of numbers for counting: {1, 2, 3, 4,. . . }. The three dots
after the 4 mean that you have seen enough to know the pattern continues. The { } symbols (called braces ) are used
to indicate the opening and closing of a set. This set of counting numbers is called the natural numbers or the
finger numbers since we start out counting on our fingers.
What is 5 - 5 = ? Seems like a silly question but, the ancient Egyptians didn’t have a number for the value
zero. The inclusion of this number to the natural numbers created the whole numbers and satisfied society’s needs
until the invention of the check-book (see: overdrawn account). The natural numbers are a subset of the whole
numbers; that means the naturals are completely contained in the whole numbers.
Once people invented check-books they discovered the need for a new type of number; a number to
represent debt. Hence, the negative numbers were invented and annexed to our old set (the whole numbers) to create
the integers. The set of integers can be represented:
{. . . , -4, -3, -2, -1, 0, 1, 2, 3, 4, . . .}.
The integers are used quite extensively in the real world to measure things, such as temperature, distance below sea
level, a college students checking account balance etc... The integers introduce the concept of “which way and how
much” on the number line.
Origin
(where we start)
Negative direction
-8 -7 -6 -5 -4 -3 -2 -1
Positive direction
0
1
2
3
4
5
6
7
8
One unit
The positive or negative part of an integer tells us which way to move from the origin while the whole number part
tells us how much to move. We define the numbers on the number line to increase in size from the left to the right
and decrease in size from the right to the left.
a)
Which number is smaller: -8 or -5?
b)
Which number is larger: -237 or -236?
2)
In the case of the integers we move in whole unit increments. But, we all know that whole number
increments don’t always satisfy all of our needs. For example we might need a half of a cup of sugar or three
sixteenths of an inch of wire. This leads us to our next set of numbers, the rational numbers. The rational numbers
contain all of the fractions positive and negative. The rational numbers are the ratio (or quotient) of two integers. In
general we have a hard time writing all of the fractions down in a set as we did with the previous sets of numbers.
Hence, here we will just write the definition of these numbers in English.
Definition of the rational numbers:
The set of rational numbers is the set of numbers of the form
where a and b are integers and b does not equal zero.
a)
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Make-up some examples of rational numbers:
a
b
Mathematics 106
Real Number Lab
Notice the integers are completely contained by the rational numbers. Any integer can be written as a rational by
putting a one under it. If we let
a be any integer, then
a
!3
is a rational number. We can write -3 as
or we can
1
1
0
and the value remains the same. This leads us to another observation about the rational numbers, that
1
1
3
two different looking numbers can have the same value. Which would you rather have; of a pizza or
of a
2
6
write 0 as
pizza? Doesn’t make much difference, you still get the same amount of pizza.
b)
On the below number line, mark and label the following numbers, using their given letters:
(a)
-8
c)
-7
-6
2
,
11
-5
(b)
-4
-3
11
,
2
(c)
-2
-1
!3
,
8
0
(d)
1
12
,
3
2
(e)
3
!16
,
3
4
5
(f)
6
237
236
7
8
Write three equivalent fractions for each of the following fractions
1/2 = ____ = ____ = ____
3/8 = ____ = ____ = ____
-3/4 = ____ = ____ = ____
-1/2 = ____ = ____ = ____
3)
The rational numbers served society pretty darn well for thousands of years. But, back then they didn’t
need Ferrari sports cars or space shuttles. The advent of technology lead us to the need for numbers that are not
rational; the irrational numbers. The irrational numbers are made up of numbers that have non-repeating, nonterminating decimal representations. Since irrational numbers don’t have nice decimal representations, we use
special symbols like the Greek letter ! to represent them. The irrational numbers taken together with the rational
numbers form the set of numbers we will work with in this class; the set of real numbers. It is important to note that
the set of rational numbers and irrational numbers are disjoint sets which means you’re one or the other but not both.
Easy sources of irrational numbers are the square roots of prime numbers (a number only divisible by itself
and 1). For example, the square root of 7 is irrational since it has a non-repeating, non-terminating decimal
representation. This means try as you may, you can never completely write
7 down as a decimal. Using my
7 of 2.645751311. What are the next four decimals? It turns out
you have a 1 in 10,000 chance of correctly guessing since there is no pattern to values. To 14 places 7 is given by
calculator, I get the decimal approximation for
2.6457513110646.
a)
Below are some examples of irrational numbers. Get out your calculator and write the decimal
approximation for each number next to it to 6 places. Note: the ≈ is read “approximately equal to.”
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! " ____________________,
2 " ___________________,
2 3 " __________________,
! " ____________________.
Mathematics 106
b)
Real Number Lab
Using my calculator I found 2 to be 1.41421356237. Give a brief
argument why this is not the same as 2.
4)
The below diagram illustrates the set of real numbers. The loops in the figure enclose all of a particular set
of numbers. Since the irrational numbers are not contained in the rational loop, that excludes them from being
rational.
Irrationals
Naturals
Wholes
Integers
Rationals
Reals
List all sets of numbers that the following numbers belong to including their basic set (the diagram will help).
Example:
-2 is an integer (its basic set), a rational, and a real.
a)
2 is a
b)
! is a
c)
!
d)
0 is a
11
is a
32
6) Extension:
The formula Q = b ! 4ac turns out to quite useful for determining all sorts of things about a
quadratic equation. We'll be studying quadratic equations later this semester. What we want to know is: given a, b,
& c what basic set of numbers does Q belong to?
2
Example :
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If a = 1, b = 3 and c = -4 then
Mathematics 106
Real Number Lab
Q = b 2 ! 4ac
=
( 3)2 ! 4 (1) ( !4 )
= 9 + 16
So, Q is a natural number.
= 25
= 5.
For the following values of a, b, & c find Q and then determine which basic set of numbers Q belongs to. Show
Your Work Please!
a) a = 1, b = 5, c = 6
b) a = 1, b = 3, c = -1
c) a = 1, b = 5, c =
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