Start Here the story of the history of 'arithmetic of number systems, and methods of calculation: the "Calculus" to count, the abacus, the "Napier's rods, the slide rule. It is a story that goes along centuries and centuries, in which the surprises never end: enjoy!
Summary:
- Count
The numbers "Primitives"
Count
; The "Calculations"
Addition
subtraction
Multiplication
Division
; root extraction
a provocation!
+ Numbering system
+ Addition and Subtraction Multiplication
+
Division
+ + Square Root
+ Exponentiation
+ + Logarithms
slide rule
The numbers "Primitives"
We are all used to talking about the "Numbers Natural ... but this definition is now mature, identified by mathematicians / philosophers who knew exactly what they were talking about!
History of arithmetic do not know exactly when it began: the first historical evidence of some way to count back to the Palaeolithic period, ie between 20,000 BC and 18,000 BC is the ' Bone of Ishango , a bone covered with a series of nicks grouped in various ways, and with a sharp sliver of quartz probably engaged at one end used to practice the incisions. According to some scientists may have played an even more complex than the simple count for something.
"Count" together with the tool and the use of the word, is certainly one of those activities that distinguishes humans from other animals. Now, you could argue that some animals can distinguish groups of objects more numerous than those less numerous, but do not count at all: the operation of counting a means to associate the generic names, two, three ... objects are always different, and it takes a great capacity for abstraction to understand that the series of "big names" one, two, three ... okay to count anything: do today as "a" on an apple, I'll give it tomorrow ... a wife, and so on.
Men have learned to communicate with each other things like "three goats" and "three apples" are concepts that represent groups of different things, they share a very precise indication of quantity. But it took a lot of work to dominate the intellectual concept of number, moving from idea to finished early understanding of numbers, and especially dell'infinito: quest'ultimo concetto è stato affrontato con successo solo verso la metà del XIX secolo! ▲
Contare
L'operazione del contare non è un'operazione banale neanche per noi "moderni", infatti quando ci troviamo in difficoltà... usiamo le dita delle mani:
ecco perché quasi tutti i sistemi di numerazione sono sempre stati quinari o decimali. Di questo si ha evidenza perché, ovviamente, a un certo punto l'uomo ha cercato il modo di trascrivere il risultato their calculations, and found various ways to write the numbers: all based on 5 and 10 (and also on 20 ... that someone has also used the toes?). For us it is usual to keep score, count or other small numbers, like this:
usually grouped for five points, so at a glance we know that we have counted up to above 12.
The ability to count is related to the ability to transcribe the results obtained, and in this sense the human being has indulged in an endless series of different systems, from the roughest to the most surprising, as the Babylonian: I will talk about in the near chapter. ▲
The "Calculations"
The operation of counting is not sufficient to solve all the arithmetic problems, however, just counting, it can perform several calculations.
The term "computation" is derived from the Latin calculus , which means little stone: surely pebbles, or seeds, or other small objects were the first system to account. So muniamoci a bowl and a sufficient number of pebbles, and we begin to get to work! ▲
Addition
to have to add the numbers 32 and 25: 32 stones by putting them into account in the bowl, it has 25 other; account at the end how many pebbles are there in the bowl : If I made no mistakes, I can only count 57: ▲
Stealing
If I subtract 17 from 65 just in my bowl that I pebbles, 65, and then they take off 17, counting finally realize how stones were:
A more "creative", but still count it, is this: I put the pebbles in the bowl counting from 18. If I put the first pebble count of 18, then the second has 19 and so on up to 65 at the end of the pebbles in the bowl will only be 48. ▲
Multiplication
Having to multiply 4 x 18, I put in the bowl 4 times eighteen stones, and finally into account the content (72). ▲
Calculation of the square.
As for multiplication! An example: If you want to know how many eggs are there in a carton of 12 x 12, just count them! ▲
Calculation of the cube.
Let's try the number 6. Thanks to the calculation of the square, I find that 6 x 6 is 36. 6 times I put 36 stones in the bowl, and eventually account! (216). Returning to eggs: if they buy a box of 12 cartons of 12 x 12 (total 1728), there is no need to do many calculations: just count them! ▲
Division
Let's say we want to divide 70 to 15. I put 70 stones in the bowl, then I take off my 15 time making small heaps at the end I find myself with four piles (quotient) and a surplus of ten stones (rest). ▲
root extraction
I can do trial and error. Let's say we want to compute the root of 55: I try to calculate 5 ², and I see that the result is lower (25) of the given number, then with 6 ² (36), with 7 ² (49) and finally 8 ²: the result is 64, then increased by 55 I wanted to calculate the root, then the root of 55 is 7 (at least for the integer part). ▲
a provocation!
So we found that just counting ... covers the curriculum of virtually all elementary arithmetic! ▲
Next Chapter: Numbering System
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