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Following are some items relating to arithmetic discussed in the history of mathematics.

Contents of this Page
Egyptian Arithmetic
The Venn Diagram
Arithmetic Around the World: Multiplication
Rules for Divisibility
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Egyptian Arithmetic

 

The Egyptians were one of the first civilizations to use mathematics in an extensive setting. Their system was derived from base ten and this was probably so because of the number of fingers and toes. In later years the Greeks would use the abstract qualities of math, however, it appears the Egyptians were only concerned with the practical aspects of numbers. For example while the Greeks might actually use see and think the number six, the Egyptians would need concrete items such as such as six sphinxes. Egyptian numbers were represented by symbols in the following way: a rod for the number one, a heal bone for ten, a snare for 100, a lotus flower for 1,000, a bent finger for 10,000, a burbot fish for 100,000, and a kneeling figure for 1,000,000.

Decimal
Number 		Egyptian
Symbol 		 
1 =
a rod
10 =
a heel bone
100 =
a snare
1000 =
a lotus flower
10,000 =
a pointing finger
100,000 =
a burbot fish
1,000,000 =
a kneeling figure

Their number system worked very well when doing addition or subtraction. The numbers were grouped together in no particular order and the operation was performed. In one example, from the Rhind Papyrus, addition and subtraction signs were represented through figures which resemble the legs of a person advancing for addition, and departing for subtraction.

Example.

Our Way Egyptian Way
255 =   = 
+ 827 =  
1082 

When it came time to multiply is when the Egyptians faced problems using their system. They over came this by devising a very ingenious solution. Instead of multiplying, the Egyptians would simply double one of the numbers and add the numbers which were being doubled to equal the other portion of the problem. An example of the way the Egyptians would perform a multiplication problem is exemplified in the following: Suppose an Egyptian scribe wanted to multiply 52 by 14.

1   52 
2   104 
4   208 
8   416We quit here because 8 doubled = 16 would be larger than 14
Because 8 + 4 + 2 = 14, we add the corresponding integers 416 + 208 +104 = 728

Division could be done in a the reverse fashion.
For example when dividing 132 by 11, the ancient Egyptians would pose the question in what must be multiplied by 11 to equal 132. Again the process was repeated.

1   11 
2   22 
4   44 
8   88We quit here because 88 doubled = 176 would be larger than 132

The numbers which equal 132 would be added up which are 44 + 88 and then by adding the corresponding numbers in the left column the quotient, 12, could be found.

The Egyptians also worked with fractions. The symbol for fractions was an oval written over the denominator. Nothing was written in the numerator's position because all of the fractions in the Egyptian system used one. For example say an Egyptian wanted to write 7/8 he would have to write 1/2+1/4+1/8.

Contributed by Chris Pinaire
References

  1. http://cgi.math.tamu.edu/~don.allen/history/egypt/egypt.html (1999) .
  2. http://riceinfo.rice.edu/armadillo/Rice/Resources/math.html (1999) .
  3. Burton, D. (1997). The history of mathematics, an introduction. McGraw- Hill Co. New York.
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The Venn Diagram

The Venn Diagram was popularized and named after the logician John Venn. John lived from 1834 to 1923 in England. During his life he was a priest who taught at Gonville and Caius College of Cambridge. He wrote several books including two on logic.

The Venn Diagram helps to describe and compare any number of elements and characteristics (usually just 2 or 3) of items, such as; Events, People, Situations, Wants, Needs, Ideas, and Concepts.

To begin creating a Venn Diagram, you may want to start by asking yourself...

  • What do I know about this particular situation?
  • What are the most important elements of this situation?
  • What do the characteristics have in common?
  • What do the characteristics not have in common?
The interesting part is the overlaping areas.

The overlapping areas are the most interesting part of the circles. Some say life occurs in the overlapping areas. One uses the Venn Diagram to identify the commonality or differences of the situation.

To begin a Venn Diagram, you must start with the univeral set, which is denoted by U, is represented by rectangle. Any set within the universe is represented by a closed circle lying within the rectangle. The area inside the circle is associated with the elements in the set.

An example of how a Venn Diagram might work according to a sport activities and the students participating in these sports are as follows.

Of the seventh graders at Darmuth middle school, 7 played basketball, 9 played volleyball, 10 played soccer, and 3 played basketball and volleyball, 3 played basketball and soccer, 4 played volleyball and soccer, and 2 played volleyball, basketball, and soccer. There are a total of 25 seventh graders that attend Darmuth middle school.

After charting them out, you should find out that 3 students played only basketball, 4 played only volleyball, and 5 played only soccer. 2 students played all the sports. And 7 students at Darmuth middle school played no sports.

Contributed by Jennifer Goodwin
References:

  1. Long, Calvin T. & DeTemple, Duane W. Mathematical Reasoning for Elementary Teachers. pp. 82. Addison Wesley Longman, Inc.
  2. http://www.venndiagram.com/intro.html
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Arithmetic Around the World: Multiplication

I remember in school when I just learned multiplication. I was consistently drilled on my facts and that was the only was to multiply. As I got older, I realized that there were other ways of multiplying. Even a few simple ways which could have helped me a great deal.

No one is quite sure, but the Egyptians may have used a duplation process of multiplication. Duplation may have been used for centuries which Stifel displayed a successive process of it.

	42 x 31
			    1 * 42 =  42
			    2 * 42 =  84
			    4 * 42 = 168
			    8 * 42 = 336
			   16 * 42 = 672
			   31 * 42 = 1302
In Russia, they used duplation and mediation in the early 1900’s.
		49 x 28
		49	24	12	6	3	1
 		28	56	112	224	448	896
They would double the smaller number and halves the larger number, then add the lower row numbers that were under the odd numbers.
28 + 448 + 896 = 1372

Our common form of multiplication was called by Pacioli, "Multiplication bericocoli vel scachierij". The Venetians called this method "per scachier" because it looked like a chess board. Florentines called it "per bericuocolo" because it looked like cakes. In Verona, it was called "per organetto" because it looked like the lines of pipe organs, then sometimes "little stairs".

Leonardo de Pisa would turn his work upside down, even though he would say that his work was correct.
About 1478 in Treviso, Italy, Treviso Arithmetic showed 4 ways of multiplying. These were called the Gelosia Method.
Most of these methods were adapted in many different cultures. As one can tell, different societies had many to choose from. There are so many ways of multiplying, one hardly knows which method to use. Problem solving is finding ways to find a solution. If one method does is not suited for you, never hurts to try another.

Contributed by Tina Gonzales
References:
  1. Glenn, W. H. & Johnson, D. A. (1960). Understanding numeration systems. St. Louis: Webster.
  2. O’Conner, J. & Robertson, E. F. (September 1996). Babylonian and egyptian mathematics. [Online] Available:http://www-groups.dcs.st-andrews.ac.uk/`history/HistTopics/Babylonian_and_Egyptian.html
  3. Smith, D. E. (1953). History of mathematics. Vol II. New York: Dover.
  4. Smith, D. E. (1995). Number stories of long ago. Washington DC: National Council of Teachers of Mathematics.
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Rules of Divisibility

The following is a list of divisibility rules for the numbers 1-13, 17, and 19. For some numbers, other ways have been included to check for divisibility. (Things that make you go hmmm?!?)

Divisibility by 1:
Any number divided by one is equal to itself: 3/1 = 3, 5/1 = 5

Divisibility by 2:
All numbers ending in 0,2,4,6,8 are divisible by 2.

54278 is even since it ends in an even number, therefore is divisible by 2.

Divisibility by 3:
If the sum of the digits is divisible by 3 then so is the original number.

Is 5136 divisible by 3? Let’s check: 5 + 1 + 3 + 6 = 15. Since 15 is divisible by three then 5136 is divisible by 3.

Divisibility by 4:
If the last two digits are divisible by 4 then so is the original number.

Is 87532 divisible by 4? Let’s check: 32/4 = 8. Since 32 is divisible by 4 then 87532 is divisible by 4.

Divisibility by 5:
If the number ends in 0 or 5 then it is divisible by 5.

Is 345,740 divisible by 5? Let’s check: Since 345,740 ends in 0 then it is divisible by 5.

Is 560,985 divisible by 5? Let’s check: Since 560,985 ends in 5 then it is divisible by 5.

Divisibility by 6:
If a number is divisible by 2 and 3, then it is divisible by 6.

Is 5964 divisible by 6? Let’s check: Since 5964 is an even number and 5 + 9 + 6 + 4 = 24, which is divisible by 3, then 5964 is divisible by 6.

Is 670 divisible by 6? Let’s check: 670 is an even number, but 6 + 7 + 0 = 13 which is not divisible by 3, therefore 670 is not divisible by 6.

Divisibility by 7:
If a number is divisible by 7, take the last digit, double it, and subtract it from the rest of the number. If the result is divisible by 7 then the original number is divisible by 7.

Is 217 divisible by 7? Let’s check: 7*2 = 14. 21-14 = 7. Since 7 is divisible by 7 then 217 is also.

Is 896 divisible by 7? Let’s check: 6*2 = 12. 89-12 = 77. Since 77 is divisible by 7 then 896 is also.

Divisibility by 8:
If the last three digits are divisible by 8, then the original number is divisible by 8. How do you know if the last three digits are divisible by 8? If the last two digits are divisible by eight and the first digit is even, the number is divisible by 8. When the first digit is odd, subtract 4 from the last two digits and if the result is divisible by 8 then so is the original number.

Let’s try two examples:

  • 34,856: last three digits 856. Since 56 is divisible by 8 and the first digit, 8, is even, then 34,856 is divisible by 8.

  • 980,744: last three digits 744. Since the first digit is odd we subtract 4 from 44. 44 - 4 = 40. Because 40 is divisible by eight, then 980,744 is also.

Divisibility by 9:
If the sum of the digits is divisible by 9 then so is the original number.

Is 675 divisible by 9? Let’s check: 6 + 7 + 5 = 18. Since 18 is divisible by 9, then so is 675.

Divisibility by 10:
If a number ends in 0, then the number is divisible by 10.

Is 5430 divisible by 10? Let’s check: Since 5430 ends in 10, it is divisible by 10.

Divisibility by 11:
Subtract the last digit from the rest of the number. Continue to do so until you come to a number you know is divisible by 11.

Example: For 352, 35 - 2 = 33. Since 33 is divisible by 11, 352 is also.

 For 61,897,
 			6189-7 = 6182
			618-2 = 616
			61 - 6 = 55
Since we know 55 is divisible by 11, then 61897 is also.

Another check for the divisibility of 11 is: when the sum of the even place digits is subtracted from the sum of the odd place digits and the difference is divisible by 11, then so is the original number. 

291,874  	Odd place digits: 9 + 8 + 4 = 21
	Even place digits: 2 + 1 + 7 = 10
			21 - 10 = 11
Since 11 is divisible by 11 then 291,874 is also.

And yet another way to check for the divisibility of 11.
Take 10 times the number of hundreds and subtract the remaining two digit number. If the difference is divisible by 11, then the original number is also.
You may have to do this process until you come to a number that is divisible by 11 as in the following example. 

61,897    10*618 = 6180-97 = 6083
	 6,083	   10*60 = 600-83 = 517
	    517     10*5 = 50-17 = 33 
Since 33 is divisible by 11 then 61,897 is also.

Divisibility by 12:
If the number is divisible by 3 and 4, then the number is divisible by 12.

Is 6756 divisible by 12? Let’s check: 6+7+5+6 = 24, which is divisible by 3 and 56/4 = 14, which means it is divisible by 4, so 6756 is divisible by 12.

Divisibility by 13:
Delete the last digit from the number and multiply it by 9. Subtract this number from what was left of the original number. If the result is divisible by 13, then so is the original number.

Is 8294 divisible by 13?  Let’s check: 4*9 = 36.  829-36 = 793
		3*9 = 27   79-27 = 52
Fifty-two is evenly divisible by 13, therefore 8294 is divisible by 13.

Another check for 13 is as follows:
Take 4 times the number of hundreds and subtract the remaining two digit number. If the difference is divisible by 13 then so is the original number. 

Is 8294 divisible by 13?  Let’s check: 4*82 = 328 - 94 = 234
		4* 2 = 8 - 34 = -26
Since -26 is divisible by 13, then so is 8294.

Divisibility by 17
Multiply the number of hundreds by 2 and subtract the last two digits from it. If the difference is divisible by 17 then so is the original number. These steps may need to be repeated until you know your result is divisible by 17.

Divisibility by 19
Multiply the number of hundreds by 14 and subtract the last two digits from it. If the difference is divisible by 19 then so is the original number. These steps may need to be repeated until you know your result is divisible by 19.

Contributed by Linette Liby
References:
  1. http://dmoz.org/Science/Math/Number_Theory/Divisibility
  2. http://mathforum.com/
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