1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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?

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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?

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

How long is the hypotenuse of this right triangle?

2 inches

4 inches

We use the Pythagorean Theorem to determine the length of the hypotenuse:

2( )2 + 4( )2 =h2

4 + 16 = h2

20 = h2

20 = h

4 • 5 =h

2 5 inches=h

Clearly, we need positive rational numbers for this!

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

The Number Family Tree

Meet my Family!

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

We are also called Counting Numbers.

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

The Necessity of Nothing

This led to …..

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Perform the Following Operations

LXIII  + CI  + DXVI

CXVII  • XIV

MMCCLXIV  − MCMXIII

MCCXIV  ÷ VIII

XLIX

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

http://www-history.mcs.st-andrews.ac.uk/BiogIndex.html

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Time Passes…….

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

I can also be negative!

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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.

http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Maseres.html

http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20060309.shtml

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Then There were Fractions…Part to whole or part to part

Whole to part or ratioOne the basis, one the heart

We are fractions, watch us grow!

Sanity is won or lostRational, 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?

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Show me a picture of 2/3 of 1/2

11

2

1

2

1

3

1

3

1

3

1

61

6

1

61

6

1

61

6

2/3 of 1/2 = 2/6 or 1/3

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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 …

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Irrational Numbers

• 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!

23

5Π 1.21131114111151111161111117...

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Density

Find a rational and an irrational number between √2 and √3

How many such numbers are there?

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Wait a Minute!

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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!

• 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 …

| | | | | | |

-3 -2 -1 0 1 2 3

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Here at last, let me introduce you to my ultimate incarnation

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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?

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

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…

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

The Complex Numbers

You can now solve any of our equations.

Now, for the piece de resistance, view my complete family tree!

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Natural Numbers

Whole Numbers

Integers

Rational Numbers

IrrationalNumbers

Complex Numbers

Real Numbers

1, 2

, 3, 4

, 5, 6

, 7, 8

, 9, 1

0, …

->

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, 9, 10, …

-> - 3 1/4, -2/3. -1/2, 0, 2, 7/3, 2 1/2, . . . − 5, − 3, 0, 1 + 2,  5...− > −7i, −i, −i + 2, i + 3, i2 , i3 , i4

Name Symbol Set Notation Number Line Verbal Description Why we Need it

NaturalNumbers

or CountingNumbers

N {1, 2, 3, 4, …} To solve equationssimilar tox - 2 = 5

WholeNumbers

W {0, 1, 2, 3, 4, …} The Counting or Natural numbersand zero.

To solve equationssimilar to5 + x = 5

Integers I, Z, or J {…-4, -3, -2, -1, 0, 1, 2,3, 4}

The Natural numbers and theiropposites and zero.

To solve equationssimilar to7 + x = 4

RationalNumbers

Q • Numbers that can be writtenas the quotient of 2 integers.

• Numbers that can be writtenas terminating or repeatingdecimals.

To solve equationssimilar to

5x = 4

IrrationalNumbers

• Numbers that can not bewritten as the quotient of 2integers.

• Numbers that can not bewritten as terminating orrepeating decimals.

To solve equationssimilar tox2 = 10

RealNumbers

R The set of all rational andirrational numbers

To solve equationssimilar to5x = 4 orx2 = 10

ComplexNumbers

C The set of all real and imaginarynumbers.

To solve equationssimilar tox2 = -2

| | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5

| | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5

| | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5

| | | | | | | | | | | -5 -4 -3 -2 -1 0 1 2 3 4 5