Numerical Systems Miss Chrishele Hruska Pre-Calculus, Grade 11
April 19, 2009 EDLT 302-Electronic Literacy Dr. David D. Carbonara
Spring 2009 Start
Slide 3
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
Slide 4
Directions Dear Student, Complete this Interactive PowerPoint
to learn about different numerical systems. You will have three
days to complete all of the slides. Dont 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!
Slide 5
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
Slide 6
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
Slide 7
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
Slide 8
From what did some early numeral systems originate? Calendars
Fingers The Sun Successive Kings
Slide 9
Incorrect Answer! Go back and review Try Again! Heres a hint!
Think of base 5 and base 10 systems.
Slide 10
Correct Answer! Great job! Keep going! Next
Slide 11
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? Lets Find
Out!
Slide 12
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
Slide 13
The Arabs invaded Spain in 711 A.D., bringing their number
system along with them. Continue
http://www.acs.ucalgary.ca/~vandersp/Courses/maps/fullmap2.jpg
Slide 14
The 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 5 th
century developed the place value notation and Brahmagupta later
introduced the symbol zero in the 6 th century. Lets see what youve
learned!
Slide 15
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.
Slide 16
Incorrect Answer! Sorry, you didnt get the question right.
Heres a hint: Remember who brought the number system to Spain
during the conquest of 711 A.D. Go back and review Try Again!
Slide 17
Correct! Great Job! Youre ready to learn more about different
number systems! Next
Slide 18
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 b 2, b 3, b 4, 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!
Slide 19
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, 10 2,10 3,10
4, 10 5, and 10 6. Multiples of these values were expressed by
repeating the symbol as many times as needed Learn more!
Slide 20
Egyptian Hieroglyphics were used in Egypt throughout Persian
rule in the 6 th and 5 th 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.gif
Slide 21
Egyptian Hieroglyphics Symbols Here are the symbols that the
Egyptians used for numbers. Now some examples!
Slide 22
Examples of numbers written in Egyptian Hieroglyphics 13 457
More
Slide 23
More examples! 6, 123 10, 268 You Try!
Slide 24
A few things to remember Whenever there are more than five of
the same symbol, stack the symbols to save room Examples: You
Try!
Slide 25
Write 342 in Egyptian Hieroglyphics. A B C D
Slide 26
Incorrect Answer. Heres a hint! Remember to write in descending
order! Go back and review Try Again!
Slide 27
Correct Answer! Great Job! Youre ready to move on to Attic
Greek! Move on
Slide 28
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. Learn
more!
Slide 29
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/thucydi
des/greece_ancient_sm.gif Next
Slide 30
Special 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. Examples 8 is written as 700 is
written as Continue
Slide 31
Attic Green Symbols and Examples 34 |||| 617 H || 2341 XXHHH |
10,135 MH
http://www.jesus8880.com/chapters/gematria/images/Attic-Numerals.gif
You Try!
Slide 32
What is HHH || in our modern numeral system? 50,327 53,257 503,
212 5, 327
Slide 33
Incorrect Answer! Heres a hint! Remember that when there are
more than 5 of a symbol, the is used to hang one, and then the rest
of the symbols are written. Dont forget that a is also used for the
number 5. Go Back and Review Try Again!
Slide 34
Correct Answer! Fantastic work! Youre doing great! Move on to
Roman Numerals
Slide 35
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. Learn
more!
Slide 36
Although the Roman numerals are now written with letters of the
Roman alphabet, they were originally independent symbols. Next
Slide 37
Roman Numerals Learn the subtraction rule!
Slide 38
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!
Slide 39
Examples of Roman Numerals 33 XXXIII 54 LIV 147 CXLVII 999
CMXCIX More
Slide 40
More Examples 1042 MXLII 2741 MMDCCXLI 3001 MMMI 5618
MMMMMDCXVIII Lets see what you know!
Slide 41
You Try! What is 798 in Roman Numerals? DCCXCVIII CCCCCCCXCVIII
DCCHCIIIIIIII CCMXCVIII
Slide 42
Incorrect Answer! Dont give up! Try again! Heres a hint! Dont
forget to use the subtraction rules! Go back and Review Try
Again
Slide 43
Correct Answer! You are ready to move on to Multiplicative
Number Systems! Keep going!
Slide 44
Multiplicative Number Systems In a multiplicative system, there
are only symbols for 1-9, 10, 10 2, 10 3, etc. We need to first
write the number in expanded form. Examples 54= 5 x 10 + 4 613= 6 x
10 2 + 1 x 10 + 3 So, we dont 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.
http://bbs.chinadaily.com.cn/attachments/month_0901/chinese-paper-cutting-40120141324284_qOeMAVE7WlRD.jpg
Start
Slide 45
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. Learn more!
Slide 46
http://www.earthquest.co.uk/china/chinamap.jphttp://www.earthquest.co.uk/china/chinamap.jp
http://blog.asiahotels.com/wp-content/uploads/2008/05/japan_map.jpgg
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 14 th century BC. Continue
Slide 47
Ancient Chinese-Japanese Numerals 159 2610 37100 481000
Next
Slide 48
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
6 1000 8 100 10 3 Examples
Slide 49
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!
Slide 50
You try! What is in our modern number system? 912 9, 120 9, 121
9.12
Slide 51
Incorrect Answer! Heres a hint! In the number 405, the tenths
digit is a zero, instead of writing the symbol for zero, it is
omitted! Try Again Go back and review
Slide 52
Correct Answer! Awesome work! Keep it up! Learn about
Positional Number Systems
Slide 53
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
Slide 54
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
Slide 55
Converting to base 20 and 60 549 to base 20 549 to base 60 20
549 20 27 20 1 60 549 60 9 9 9 7 7 1 1 179 9 9 9 9 99 More
conversions
Slide 56
More converting to base 20 and 60 20 2,367 20 118 20 5 60 2,
367 60 39 7 7 18 5 5 5, 18, 7 27 39 39, 27 Check your answer!
Slide 57
Check your answers! Check your answer by multiplying the number
out like this: 179 in base 20 1 x 20 2 + 7 x 20 + 9 = 549 99 in
base 60 9 x 60 + 9 = 549 5, 18, 7 in base 20 5 x 20 2 + 18 x 20 + 7
= 2367 39, 27 in base 60 39 x 60 + 27 = 2367 You Try!
Slide 58
Convert 791 to base 20 1, 19, 11 11911 1, 20, 0 39.55
Slide 59
Incorrect Answer! Heres a hint! Make sure to only use digits
from 0 to 19! Go back and review! Try again!
Slide 60
Correct Answer! Wonderful! Keep up the great work! Convert to
Base 60!
Slide 61
What is 791 in base 60? 18 13, 11 11, 13 13.18
Slide 62
Incorrect Answer! Heres a hint! Make sure to write the
remainders backwards! Go Back and Review Try Again!
Slide 63
Correct Answer! Good work! Youre ready to learn more about
Babylonian Cuneiform! Learn now!
Slide 64
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
Slide 65
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
Slide 66
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!
Slide 67
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 examples
Slide 68
3 8 39 54 Here are some examples of numbers written in
Babylonian Cuneiform. You Try!
http://www.bibliotecapleyades.net/sociopolitica/codex_magica/images/acodex_60.jpg
Slide 69
What is in our modern number system? 19 21 1.9 32
Slide 70
Incorrect Answer!
http://www.timewarptrio.com/show/who/images/nebuchadnezzar.gif
Remember that a triangle facing down beside a triangle facing right
is a subtraction sign! Go back and review Try Again!
Slide 71
Correct Answer! Fabulous work! Keep going!
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IcQVCBx9w=&h=705&w=600&sz=162&hl=en&start=5&sig2=U-
mN4wGIn0_bMz_AZO_EVw&tbnid=8AUd1bIFrKPydM:&tbnh=140&tbnw=119&prev=/images%3Fq%3Dnebuchadnezzar%26gbv%3D2%26hl%3Den%26sa%3DX&ei=VfrkSditFarq
lQe_js2lDg Move on to Mayan
Slide 72
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. Learn more!
Slide 73
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/13183/10640 4/159685_mayamap_L.gif
Next
Slide 74
Mayan Numerals Notice that the numbers only range from 0 to 19
because the base is 20. Examples
Slide 75
450 First, convert to base 20 = 2, 10 Examples 3,821 First,
convert to base 20 = 9, 11, 1 More
Slide 76
More Examples 6 18 600 First, convert to base 20 = 1, 10, 0 You
Try!
Slide 77
What is 687 in Mayan numerals?
Slide 78
Incorrect Answer! Dont give up! Try again! Heres 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!
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