Transcript Alg 1.1 ant. set and Instruction

Number Sets
The numbers we can use vary
according to the context of the
problem we are doing as well as any
restraints put on us by the problem.
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
How old are you?
The age of a person is always
positive. We usually speak of being
a discrete number of years old.
In fact we are continually aging and
so can be a fractional age, say 14.375
years.
In either case, our age is always
positive.
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
What is today’s date?
Dates are always positive. They are
whole numbers. There is no 0 day
of the month. But there is a limit to
the numbers. The date will never be
more than 31.
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
What time is it?
Time is always given as a positive
number. It is continuous. But is is
limited from 1 to 12 in hours and 1 to
60 in minutes and seconds
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
What is the temperature?
Temperature is on a continuous scale.
It can be positive or negative. The cut
off point is zero degrees in Celsisus and
32 degrees in Farenheit. Theoretically,
there is no limit to temperature - either
cold or hot, but what is the hottest and
coldest it has ever been in the universe?
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
What is our elevation?
Elevation is given in height
above sea level (positive) or
below sea level (negative). It
is a continuous measure.
Does it have a limit or limits?
If so what could these be?
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
How long is the hypotenuse of
this right triangle?
We use the Pythagorean Theorem
to determine the length of the
hypotenuse:
2
2
2   4   h2
2 inches
4  16  h 2
4 inches
Clearly, we need positive
rational numbers for this!
20  h 2
20  h
4•5  h
2 5 inches  h
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
How do I quantify electicity?
Nickola Tesla invented
alternating current.
In order to write
equations to model its
use, Charles
Steinmetz introduced
a system using
imaginary numbers.
So whenever we turn
on a light, we make
use of complex
numbers.
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
When do we learn the Number Sets?
•
•
•
•
•
•
•
Natural or Counting Numbers - from childhood
Whole Numbers - in first grade
Integers - in fourth grade (or sooner)
Rational Numbers - fourth grade
Irrational Numbers - seventh grade
Real numbers - Algebra I
Complex Numbers - Algebra II
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
What are we going to learn
today?
• The number sets we work with in
Algebra
• Properties of the number sets
• How to use properties of numbers to
demonstrate whether assertions are true
or false.
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
The Number Family Tree
Meet my Family!
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
We are also called Counting Numbers.
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Natural Numbers
|
|
|
|
|
|
|
1
2
3
4
5
6
7
You can use me to solve equations like x - 2 = 1
But I can’t be used to solve x + 2 = 2
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
The Necessity of Nothing
This led to …..
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
The Birth of the Whole Numbers
|
|
|
|
|
|
|
0
1
2
3
4
5
6
You can use me to solve equations like x + 2 = 2
But I can’t be used to solve x + 2 = 1
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Perform the Following Operations
LXIII  CI  DXVI
MMCCLXIV  MCMXIII
CXVII • XIV
MCCXIV  VIII
XLIX
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Historical Note
•
•
Leonardo Pisano Fibonacci,
1170 - 1250
Liber Abaci introduced the
Hindu-Arabic place-valued
decimal system and the use of
Arabic numerals into Europe.
 5,http://www-history.mcs.st-andrews.ac.uk/BiogIndex.html
 3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Time Passes…….
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
I can also be negative!
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Integers
|
|
|
|
|
|
|
-3
-2
-1
0
1
2
3
You can use me to solve equations like x + 2 = 1
But I can’t be used to solve 2x = 1
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Historical Interlude
In 1759 the British mathematician Francis
Maseres wrote that negative numbers
"darken the very whole doctrines of the
equations and make dark of the things which
are in their nature excessively obvious and
simple". Because of their dark and
mysterious nature, Maseres concluded that
negative numbers did not exist. However,
other mathematicians were braver. They took
a leap into the unknown and decided that
negative numbers could be used during
calculations, as long as they had
disappeared upon reaching the solution.
 5,http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Maseres.html
 3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Then There were Fractions…
Part to whole or part to part
Whole to part or ratio
One the basis, one the heart
We are fractions, watch us grow!
Sanity is won or lost
Rational, irrational
What’s the number, whose the boss?
Big and little, some and all
Numbers move beyond the pall!
Are we real or nothing at all?
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Show me a picture of 2/3 of 1/2
1
61
13
6
1
261
113
62
1
1
16
13
6
2/3 of 1/2 = 2/6 or 1/3
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
A Little Bit of History
It is likely that the concept of fractional
numbers dates to prehistoric times.
Even the Ancient Egyptians wrote math
texts describing how to convert general
fractions into their special notation.
Classical Greek and Indian
mathematicians made studies of the
theory of rational numbers, as part of
the general study of number theory. The
best known of these is Euclid's
Elements, dating to roughly 300 BC.
http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Euclid.html
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Rational Numbers
• I can’t be written as a set!
• I can’t be pictured clearly on a number line!
• I do have a symbol: Q
• I can be used to solve the equation 2x = 1
• I can’t be used to solve the equation x2 = 2.
For this I need the …
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Irrational Irrationals Death by Number
The first existence proofs of irrational numbers is usually
attributed to Pythagoras, more specifically to the
Pythagorean Hippasus of Metapontum. The story goes
that Hippasus discovered irrational numbers when trying
to represent the square root of 2 as a fraction. However
Pythagoras believed in the absoluteness of numbers,
and could not accept the existence of irrational numbers.
He could not disprove their existence through logic, but
his beliefs would not accept the existence of irrational
numbers and so he sentenced Hippasus to death by
drowning.
http://www.anselm.edu/homepage/
dbanach/pyth3.htm
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
3
Irrational Numbers
2
• I can’t be written as a set!
• I can’t be pictured clearly on a number line!
• I don’t have a symbol.
• I can be used to solve the equation x2 = 2
• I can’t be used to solve the equation x2 = -2.
For this I need the …
• Wait a minute!
 1.21131114111151111161111117...
5
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Density
Find a rational and an irrational
number between √2 and √3
How many such numbers are there?
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Wait a Minute!
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Real Numbers
• I can’t be written as a set!
• I can be pictured clearly on a number line as I
am the whole line!
|
|
|
|
|
|
|
-3
-2
-1
0
1
2
3
• I have a symbol
R.
• I can be used to solve the equation x2 = 2
• I can’t be used to solve the equation x2 = -2.
For this I need the …
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Real Numbers at Last!
• In the latter part of the 19th
century attention turned to
irrational numbers.
• Real numbers were defined
by Dedekind as certain sets of
rationals.
• The theory of rational and
natural numbers were then
clarified in turn, ultimately
reducing all of these systems
to set theory and logic.
http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Dedekind.html
http://www.rbjones.com/rbjpub/maths/math008.htm
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Here at last, let me introduce
you to my ultimate incarnation
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Descartes said, “Cogito ergo sum.”
•How do we bring ourselves to believe in
the existence of nonexistent numbers?
•Why does language blur the issue?
•What are imaginary numbers?
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
When did we begin imagining the
imaginary?
Closed formulas for the roots of
cubic and quartic polynomials
were discovered by Italian
mathematician Gerolamo
Cardano. It was soon realized
that these formulas, even if one
was only interested in real
solutions, sometimes required the
manipulation of square roots of
negative numbers.
http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Cardan.
http://en.wikipedia.org/wiki/Complex_number#History
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Imaginary Numbers
• I can’t be written as a set!
• I can be pictured clearly on a number line!
• I have no symbol.
• I can be used to solve the equation x2 = -2
• When you marry me to the Real numbers I
become the ultimate set of numbers…
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
The Complex Numbers
You can now solve any of our equations.
Now, for the piece de resistance, view
my complete family tree!
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Complex
Numbers
Real
Numbers
Irrational
Numbers
Natural
Numbers
Whole
Numbers
Integers
Rational
Numbers
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,
Name
Symbol
Set Notation
Natural
Numbers
or Counting
Numbers
Whole
Numbers
N
{1, 2, 3, 4, …}
Number Line
W
3 4 5
{0, 1, 2, 3, 4, …}
| | | | | | | | | | |
I, Z, o r J
{…-4, -3, -2, -1, 0, 1, 2,
3, 4}
1 2
1 2
Irrational
Numbers
R
Complex
Numbers
C
| | | | | | | | | | |
-5 -4 -3 -2 -1 0
1 2
The Natural nu mbers and their
opposites and zero.
3 4 5
Q
Real
Numbers
The Counting or N atural numbers
and zero.
3 4 5
| | | | | | | | | | |
-5 -4 -3 -2 -1 0
Rational
Numbers
1 2
3 4 5
Why we Need it
To solve equations
similar to
x-2 = 5
| | | | | | | | | | |
-5 -4 -3 -2 -1 0
-5 -4 -3 -2 -1 0
Integers
Verbal Description
 Numbers that can b e writt en
as the quo tient of 2 in tegers.
 Numbers that can b e writt en
as terminating o r repeating
decimals.
 Numbers that can no t be
written as the quotient o f 2
integers.
 Numbers that can no t be
written as terminating or
repeating de cimals.
The set of a ll rational and
irrational numb ers
The set of a ll real and imaginary
numbers.
To solve equations
similar to
5 +x = 5
To solve equations
similar to
7 +x = 4
To solve equations
similar to
5x = 4
To solve equations
similar to
x2 = 10
To solve equations
similar to
5x = 4 o r
x2 = 10
To solve equations
similar to
x2 = -2
 5,  3, 0, 1  2, 5...  7i, i, i  2, i  3, i2 , i3 , i4
9, 10, …-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . .
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … -> 0,1, 2, 3, 4, 5, 6, 7, 8,
9, 10, …-> -10, -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,