Numerical Systems

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# Numerical Systems - PowerPoint PPT Presentation

Numerical Systems. Miss Chrishele Hruska Pre-Calculus, Grade 11 April 19, 2009 EDLT 302-Electronic Literacy Dr. David D. Carbonara Spring 2009. Start. Student Objectives.

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### Numerical Systems

Miss ChrisheleHruska

April 19, 2009

EDLT 302-Electronic Literacy

Dr. David D. Carbonara

Spring 2009

Start

Student Objectives
• The students will be able to write numbers in simple, multiplicative, and positional number systems after completing the Interactive PowerPoint to 100% correctness.
• The students will be ale to write numbers in base 20 and base 60 after completing the Interactive PowerPoint to 100% correctness.
• The students will be able to identify Egyptian Heiroglyphics, Mayan numerals, Babylonian Cuneiform, Ancient Chinese- Japanese numerals, and Attick Greek after completing the Interactive PowerPoint to 100% correctness.

Directions

Directions

Dear Student,

Complete this Interactive PowerPoint to learn about different numerical systems. You will have three days to complete all of the slides.

Don’t worry about getting a question wrong! You can always go back to review the previous slides and try again. Remember to record your answers to the questions on your worksheet that you will hand in.

If you have any questions while you are completing the PowerPoint, please contact me before you move on.

Good luck!

Get started!

Numerical Systems
• Numerical systems are based on grouping systems that rely on certain bases.
• Represent a useful set of numbers
• Give every number represented a unique representation
• Reflect the algebraic and arithmetic structure of the numbers
• Our numerical system is a base 10 system.
• Can you think of different ways that we use a base 10 system everyday?

Next

A base-5 system has been used in many cultures for counting. It originated from counting the number of fingers on a human hand.

• A base-8 system was devised by the Yuki tribe of Northern California, who used the spaces between the fingers to count, corresponding to the digits one through eight.

Continue

There are many other bases that have been used in ancient times and in present day.

• Base 5- South American Tribes
• Base 20- Mayan
• Base 60- Babylonian Cuneiform
• Can you think of things that are base 12 that we see everyday?

Continue

From what did some early numeral systems originate?

The Sun

Fingers

Calendars

Successive Kings

Here’s a hint!

Think of base 5 and base 10 systems.

Go back and review

Try Again!

• Great job! Keep going!

Next

Base 10 Systems
• Along with our modern number system that we use everyday, many other civilizations also use base 10 systems like the Egyptians, Ancient Chinese-Japanese, Attic Greek, and Romans.
• But how did our system come to be how it is today?

Let’s Find Out!

Hindu-Arabic Number System
• As early as 250 B.C., the Hindus of India invented a new number system.
• The oldest preserved examples are found on stone columns erected in India by King Asoka.
• It is likely that traders and travelers of the Mediterranean coast introduced the new number system to the Arabs.
• In 711 A.D., the Arabs invaded Spain and the number system emerged into Europe.

Next

The Arabs invaded Spain in 711 A.D., bringing their number system along with them.

http://www.acs.ucalgary.ca/~vandersp/Courses/maps/fullmap2.jpg

Continue

• The Hindu-Arabic number system became popular because it was easier to write out calculations.
• Later, when zero and the positional system were developed, the Hindu-Arabic number system became superior than any of the other number systems being used at the time.
• Aryabhatta of Kusumapura who lived during the 5th century developed the place value notation and Brahmagupta later introduced the symbol zero in the 6th century.

Let’s see what you’ve learned!

Why is our modern number system called the Hindu-Arabic number system?

The Hindus and the Arabs developed the number system at the same exact time.

The Hindus invented the number system, and then Arabs continued to spread it around Europe.

The Hindus and the Arabs were at war with each other over the number system.

The Arabs invented the number system and the Hindus stole it from them.

• Sorry, you didn’t get the question right.

Here’s a hint: Remember who brought the number system to Spain during the conquest of 711 A.D.

Go back and review

Try Again!

Correct!

Next

Simple Number Systems
• Many numeral systems are simple.
• Simple means that there is a symbol for the base number, b, and also a symbol for b2, b3, b4, etc.
• A number is expressed by using these symbols additively, repeating the symbol a certain number of times.
• Examples of simple number systems are: Egyptian Hieroglyphics, Attic Greek, and Roman Numerals.

Egyptian Hieroglyphics
• This system was used in Egypt until the first century B.C.
• Hieroglyphics are based on a scale of 10 and consecutive bases of 10.
• There are symbols for 1, 10, 102,103,104, 105, and 106.
• Multiples of these values were expressed by repeating the symbol as many times as needed

Egyptian Hieroglyphics were used in Egypt throughout Persian rule in the 6th and 5th centuries B.C. and even after Alexander the Great’s conquest during the Macedonian and Roman periods.

http://www.iziko.org.za/sh/resources/egypt/images/map_e1_l.gif

Egyptian Hieroglyphics Symbols
• Here are the symbols that the Egyptians used for numbers.

Now some examples!

More examples!
• 6, 123
• 10, 268

You Try!

A few things to remember…
• Whenever there are more than five of the same symbol, stack the symbols to save room
• Examples:

You Try!

### Write 342 in Egyptian Hieroglyphics.

A

B

C

D

Here’s a hint! Remember to write in descending order!

Go back and review

Try Again!

• Great Job! You’re ready to move on to Attic Greek!

Move on

Attic Greek
• Attic Greek was developed sometime before the third century B.C.
• Like Egyptian Hieroglyphics, there are symbols for 1, 10, 100, 1000, and 10,000. But, there is also a symbol for 5.

Attic Greek was used until the 4th century BC, when it was replaced by Koine Greek, known as “the Common Dialect”.

http://www.thucydides.netfirms.com/thucydides/greece_ancient_sm.gif

Next

Special use of 5 ( Γ )
• Another special feature is that when there are more than 5 of the same symbol, Greeks used Γ with the symbol and wrote the remaining symbols.
• Examples
• 8 is written as
• 700 is written as

Continue

Attic Green Symbols and Examples
• 34 ΔΔΔ||||
• 617 HΔΓ||
• 2341 XXHHHΔ Δ Δ Δ|
• 10,135 MHΔΔΔΓ

http://www.jesus8880.com/chapters/gematria/images/Attic-Numerals.gif

You Try!

Here’s a hint! Remember that when there are more than 5 of a symbol, the Γ is used to hang one, and then the rest of the symbols are written. Don’t forget that a Γis also used for the number 5.

Go Back and Review

Try Again!

• Fantastic work! You’re doing great!

Move on to Roman Numerals

Roman Numerals
• The last type of simple grouping system is Roman Numerals.
• Roman Numerals were the standard numbering system in Ancient Rome and Europe until around 900 AD, when the Hindu-Arabic system emerged.
• There is no symbol for 0 in Roman Numerals.

Although the Roman numerals are now written with letters of the Roman alphabet, they were originally independent symbols.

Next

Roman Numerals

Learn the subtraction rule!

Subtraction Rule
• In modern times, the subtractive principle has become very common when writing Roman Numerals.
• I can precede only V or X
• Examples
• 4 is written as IV
• 9 is written as IX
• X can precede only L or C
• Examples
• 40 is written as XL
• 90 is written as XC
• C can precede only D or M
• Examples
• 400 is written as CD
• 900 is written as CM

Examples!

Examples of Roman Numerals
• 33
• XXXIII
• 54
• LIV
• 147
• CXLVII
• 999
• CMXCIX

More

More Examples
• 1042
• MXLII
• 2741
• MMDCCXLI
• 3001
• MMMI
• 5618
• MMMMMDCXVIII

Let’s see what you know!

You Try!

What is 798 in Roman Numerals?

DCCXCVIII

CCCCCCCXCVIII

DCCHCIIIIIIII

CCMXCVIII

• Don’t give up! Try again!

Here’s a hint!

Don’t forget to use the subtraction rules!

Go back and Review

Try Again

• You are ready to move on to Multiplicative Number Systems!

Keep going!

Multiplicative Number Systems
• In a multiplicative system, there are only symbols for 1-9, 10, 102, 103, etc.
• We need to first write the number in expanded form.
• Examples
• 54= 5 x 10 + 4
• 613= 6 x 102 + 1 x 10 + 3
• So, we don’t need to have a number for 40. Instead, we can write it as 4 x 10.
• One example of a multiplicative system is that of the Ancient Chinese-Japanese.

Start

Ancient Chinese-Japanese Number System
• The traditional Chinese-Japanese number system has characters for the numerals 0 through 9, 10, 100, and 1000.
• Numbers are written in expanded form from top to bottom instead of left to right.
• Since this system has a symbol for 0, it is used as a place holder.

In 1899 a major discovery was made at the archaeological site at the village of Xiao dun .Thousands of bones and tortoise shells were discovered there which had been inscribed with ancient Chinese numerals. Archaeologists think that they date back to the Late Shang dynasty from the 14th century BC.

Continue

6

1000

8

100

10

3

Writing Numbers in Ancient Chinese-Japanese
• First, write the number in expanded form.
• Then, fill in the symbols and remember to write the number vertically.

6813 = 6 x 1000 + 8 x 100 + 1 x 10 + 3

Examples

Ancient Chinese-Japanese Examples
• 8,612

=8 x 1000 + 6 x 100 + 1 x 10 + 2

• 354

=3 x 100 + 5 x 10 + 4

You Try!

You try!

What is

in our modern number system?

912

9, 120

9, 121

9.12

Here’s a hint!

In the number 405, the tenths digit is a zero, instead of writing the symbol for zero, it is omitted!

Go back and

review

Try Again

Awesome work!

Keep it up!

Positional Number Systems
• Before writing a number in positional numeral system, it is necessary to convert the number to a different base.
• If the base is b, there are basic symbols for 0, 1, 2, …b-1. These are called the digits.

More

There are two civilizations that used positional number systems (other than our modern Hindu-Arabic system)

• Mayan (base 20)
• Babylonian Cuneiform (base 60)
• Sometimes, if the number in a position is bigger than 10, we use a comma to separate the digits.
• Example- in a base 20 system, digits are from 0-19. It would be very confusing if the digits were 18, 19, and 5. If someone wrote 18195, no one would know where the distinct digits are. So, use commas to separate digits to eliminate confusion.

Learn to compute in different bases

Converting to base 20 and 60
• 549 to base 20
• 549 to base 60

9

9

7

9

1

99

179

More conversions

More converting to base 20 and 60

7

27

18

39

5

39, 27

5, 18, 7

• 179 in base 20
• 1 x 202 + 7 x 20 + 9 = 549
• 99 in base 60
• 9 x 60 + 9 = 549
• 5, 18, 7 in base 20
• 5 x 202 + 18 x 20 + 7 = 2367
• 39, 27 in base 60
• 39 x 60 + 27 = 2367

You Try!

Convert 791 to base 20

1, 19, 11

11911

1, 20, 0

39.55

Here’s a hint!

Make sure to only use digits from 0 to 19!

Go back and review!

Try again!

• Wonderful! Keep up the great work!

Convert to Base 60!

What is 791 in base 60?

18

13, 11

11, 13

13.18

Here’s a hint!

Make sure to write the remainders backwards!

Go Back and Review

Try Again!

Learn now!

Babylonian Cuneiform
• The Babylonians used clay tablets to write with. They pressed into the wet clay with a stylus that was shaped like a triangle.
• Remember that the Babylonians used a base 60 system.
• First, convert the number to base 60.
• Only use digits from 0-59 when writing the numerals in cuneiform.

Next

Knowledge of cuneiform was lost until 1835 A.D., when Henry Rawlinson, an English army officer, found inscriptions on a cliff at Behistun in Persia.

http://kbagdanov.files.wordpress.com/2008/10/map-1.jpg

Next

Pay attention to the way that the stylus is facing, different ways mean different numbers!

• A triangle facing right is the symbol for 10.
• A triangle pointing down is the symbol for 1.
• A triangle pointing down beside a triangle facing the left is the symbol for subtraction.

Keep going!

Why the subtraction sign?!?!
• The Babylonians decided to use the subtraction sign to limit the number of symbols being used.
• For example, it is easier and uses less symbols to write 10-1 than it is to write 9 1’s.
• 10-1
• 9

Let’s see some examples

• 3
• 8
• 39
• 54

You Try!

Remember that a triangle facing down beside a triangle facing right is a subtraction sign!

Go back and review

Try Again!

• Fabulous work! Keep going!

Move on to Mayan

Mayan Vegesimal system
• The Mayans used a base 20 system.
• During Spanish expeditions in the Yucatan in the early sixteenth-century, this number system was discovered.
• The numbers are written very simply with dots and dashes which probably originated as pebbles and sticks.

The Mayans had a sophisticated number system, but a little complex because it is base 20. The Mayans probably chose five and twenty as the two bases of their system as there are five fingers on one hand, and twenty fingers and toes on one person.

http://archives.zinester.com/13183/106404/159685_mayamap_L.gif

Next

Mayan Numerals
• Notice that the numbers only range from 0 to 19 because the base is 20.

Examples

Examples
• 450

First, convert to base 20

= 2, 10

• 3,821

First, convert to base 20

= 9, 11, 1

More

More Examples
• 6
• 18
• 600

First, convert to base 20

= 1, 10, 0

You Try!

• Don’t give up! Try again!

Here’s a hint!

Remember to first convert the number to base 20 and then write the remainders from bottom to top!

Go Back and Review!

Try Again!