Numerical Systems

Multiplicative Number Systems

Incorrect Answer!Go back and reviewTry Again!

Heres a hint!

Think of base 5 and base 10 systems.

Correct Answer!Fantastic! Youre doing great!

ContinueYou try!What is

in our modern number system?

9129, 1209, 1219.12Numerical SystemsNumerical systems are based on grouping systems that rely on certain bases. Represent a useful set of numbersGive every number represented a unique representationReflect the algebraic and arithmetic structure of the numbersOur numerical system is a base 10 system.Can you think of different ways that we use a base 10 system everyday?

NextA 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.

ContinueThere are many other bases that have been used in ancient times and in present day.Base 5- South American TribesBase 20- MayanBase 60- Babylonian Cuneiform

Can you think of things that are base 12 that we see everyday?

ContinueFrom what did some early numeral systems originate?CalendarsFingersThe SunSuccessive Kings

Correct Answer!Awesome work!

Keep it up!

Learn about Positional Number Systems

Correct Answer!Great job! Keep going!Next

Base 10 SystemsAlong 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?Lets Find Out!Hindu-Arabic Number SystemAs 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. Continuehttp://www.acs.ucalgary.ca/~vandersp/Courses/maps/fullmap2.jpgThe early engravings do not have a zero or the positional notation.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.Lets see what youve 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.

Simple Number SystemsMany 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.

Learn about Egyptian!Egyptian HieroglyphicsThis 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 neededLearn more!

Egyptian Hieroglyphics were used in Egypt throughout Persian rule in the 6th and 5th centuries B.C. and even after Alexander the Greats conquest during the Macedonian and Roman periods.Learn more!http://www.iziko.org.za/sh/resources/egypt/images/map_e1_l.gifEgyptian Hieroglyphics SymbolsHere are the symbols that the Egyptians used for numbers.

Now some examples!

Examples of numbers written in Egyptian Hieroglyphics13

457

More

More examples!6, 123

10, 268

You Try!

A few things to rememberWhenever there are more than five of the same symbol, stack the symbols to save room

Examples:

You Try!Write 342 in Egyptian Hieroglyphics.ABCD

Attic GreekAttic 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.

Learn more!

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.gifNextSpecial use of 5 ( )Another special feature is that when there are more than 5 of the same symbol, Greeks used with the symbol and wrote the remaining symbols.Examples8 is written as

700 is written as

ContinueAttic Green Symbols and Examples34 ||||

617 H|| 2341 XXHHH |

10,135 MH

http://www.jesus8880.com/chapters/gematria/images/Attic-Numerals.gifYou Try!What is HHH|| in our modern numeral system?50,32753,257503, 2125, 327

Roman NumeralsThe 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.

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

NextRoman Numerals

Learn the subtraction rule!Subtraction RuleIn modern times, the subtractive principle has become very common when writing Roman Numerals.I can precede only V or XExamples4 is written as IV9 is written as IXX can precede only L or CExamples40 is written as XL90 is written as XC C can precede only D or MExamples400 is written as CD900 is written as CM

Examples!Examples of Roman Numerals33XXXIII

54LIV

147CXLVII

999CMXCIX

MoreMore Examples1042MXLII

2741MMDCCXLI3001MMMI

5618MMMMMDCXVIII

Lets see what you know!

You Try!What is 798 in Roman Numerals?

DCCXCVIIICCCCCCCXCVIIIDCCHCIIIIIIIICCMXCVIII

Incorrect Answer!Dont give up! Try again!

Heres a hint!

Dont forget to use the subtraction rules!Go back and ReviewTry Again

Ancient Chinese-Japanese Number SystemThe 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.

Learn more!

http://www.earthquest.co.uk/china/chinamap.jp http://blog.asiahotels.com/wp-content/uploads/2008/05/japan_map.jpggIn 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.

ContinueAncient Chinese-Japanese Numerals159261037100481000

NextWriting Numbers in Ancient Chinese-JapaneseFirst, 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

610008100103

ExamplesAncient Chinese-Japanese Examples8,612=8 x 1000 + 6 x 100 + 1 x 10 + 2

354=3 x 100 + 5 x 10 + 4

You Try!

Positional Number SystemsBefore 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 60549 to base 20549 to base 609711799999More conversionsMore converting to base 20 and 6071855, 18, 7273939, 27Check your answer!Check your answers!Check your answer by multiplying the number out like this:179 in base 201 x 202 + 7 x 20 + 9 = 54999 in base 609 x 60 + 9 = 5495, 18, 7 in base 205 x 202 + 18 x 20 + 7 = 236739, 27 in base 6039 x 60 + 27 = 2367

You Try!Convert 791 to base 201, 19, 11119111, 20, 039.55

What is 791 in base 60?1813, 1111, 1313.18

Babylonian CuneiformThe 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.jpgNextPay 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 1s.10-1

9

Lets see some examples3

839

54Here are some examples of numbers written in Babylonian Cuneiform.

You Try!http://www.bibliotecapleyades.net/sociopolitica/codex_magica/images/acodex_60.jpgWhat is in our modern number system?19211.932Mayan Vegesimal systemThe 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.

Learn more!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.gifNextMayan NumeralsNotice that the numbers only range from 0 to 19 because the base is 20.

Examples450First, convert to base 20= 2, 10

Examples3,821First, convert to base 20= 9, 11, 1

MoreMore Examples6

18

600First, convert to base 20= 1, 10, 0

You Try!

What is 687 in Mayan numerals?ABCD

Youre Finished!Good job! You should now be an expert on ancient and modern number systems! Make sure to turn in your worksheet with all of your answers to the questions on it.

FINISH