Topics in Arithmetic
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|The Venn Diagram|
|Arithmetic Around the World: Multiplication|
|Rules for Divisibility|
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.
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.
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.
Division could be done in a the reverse fashion.
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|
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...
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|
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 = 1302In Russia, they used duplation and mediation in the early 1900ís.
49 x 28 49 24 12 6 3 1 28 56 112 224 448 896They would double the smaller number and halves the larger number, then add the lower row numbers that were under the odd numbers.
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".
|Contributed by Tina Gonzales|
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:
Divisibility by 2:
54278 is even since it ends in an even number, therefore is divisible by 2.
Divisibility by 3:
Is 5136 divisible by 3? Letís check: 5 + 1 + 3 + 6 = 15. Since 15 is divisible by three then 5136 is divisible by 3.
Divisibility by 4:
Is 87532 divisible by 4? Letís check: 32/4 = 8. Since 32 is divisible by 4 then 87532 is divisible by 4.
Divisibility by 5:
Is 345,740 divisible by 5? Letís check: Since 345,740 ends in 0 then it is divisible by 5.
Is 560,985 divisible by 5? Letís check: Since 560,985 ends in 5 then it is divisible by 5.
Divisibility by 6:
Is 5964 divisible by 6? Letís 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? Letís 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:
Is 217 divisible by 7? Letís check: 7*2 = 14. 21-14 = 7. Since 7 is divisible by 7 then 217 is also.
Is 896 divisible by 7? Letís check: 6*2 = 12. 89-12 = 77. Since 77 is divisible by 7 then 896 is also.
Divisibility by 8:
Letís try two examples:
Divisibility by 9:
Is 675 divisible by 9? Letís check: 6 + 7 + 5 = 18. Since 18 is divisible by 9, then so is 675.
Divisibility by 10:
Is 5430 divisible by 10? Letís check: Since 5430 ends in 10, it is divisible by 10.
Divisibility 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.
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:
Is 6756 divisible by 12? Letís 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:
Is 8294 divisible by 13? Letís 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:
Is 8294 divisible by 13? Letís check: 4*82 = 328 - 94 = 234 4* 2 = 8 - 34 = -26 Since -26 is divisible by 13, then so is 8294.
Divisibility by 17
Divisibility by 19
|Contributed by Linette Liby|