الرئيسية
Extract athletes Arabs and Muslims unknowns numerical analysis by two other ways seldom defined at the person in the modern era, only specialists in mathematics. These two approaches are two mistakes account, analysis and inversion. And had their works including a book that two mistakes the entire computer for my father and the Egyptian Book of the expense of two mistakes to Jacob ibn Muhammad al-Razi and others. Both of these methods are the Arabs became public, and more commonly used than others. Here are two examples: the first illustrates the method of calculation and error, and the second shows how to access the anonymous way of analysis and inversion.
1) Find the number which, when added to two thirds and three output was 18.
First Step: suppose the unknown and what you should feature the first, and then dispose of it according to the question, it conforms is required, although not as well as the error of plus or minus is the first error.
Second step: Suppose another unknown feature and imposed the second, the mistake happened the second error.
Step Three: Multiply the first imposed in the second error, and saved the first feature.
Step Four: Multiply the second imposed in the first error, and saved the second feature.
Step five: If the lines of redundant or incomplete Voksm the difference between the saved on the difference between the two mistakes, although they differed on the total The total saved two mistakes to get to the unknown.
Take supposed to resolve the issue I: 3 0 if we act according to which the question is:
3 + 3 × 2 / 3 + 3 = 3 + 2 + 3 = 8 ... .... .... The first error is 18-8 = 10 minus
Take imposed II: 6 0 if we act according to which the question is:
6 + 6 × 2 / 3 + 3 = 13 ... ... ... The second error is 18-13 = 5 minus
Permission to be saved I = 3 × 5 = 15 and have saved the second = 6 × 10 = 60
The difference between 60 and 15 = 45 The difference between the two mistakes is 10-5 = 5
... ... ... ... ... ... ... ... ... ... ... .... The answer is 45 / 5 = 9
The extraction of unknowns in a manner analysis and inversion are based on the work the opposite of what gave him half of the liquid, a weak, although increased Vangs, although beaten Voksm or the root of one quarter or reverse Vaeks beginning of the last question. Have been received this question in the book Baha'eddin Global: ¸ beaten in the same number was increased to winning two and weakness, and increased the sum of three dirhams and a section of society (total) on five and struck out in ten got fifty •. (What is the number?)
Start another question Venksm 50-10 and then multiply in like 5; any 5 × 5 = 25 and from 25 Nnqs No. 3 Vicu rest 22 and half that number Nnqs 2; any 11-2 = 9 The answer is the square root of 9, or 3.
Contributions in mathematics
First: In the area of account:
The Arab scientists first developed four arithmetic operations, collection and multiple reward, half-life, differentiation, multiplication and division, and credit them to extract the roots. They have numbers divided into three types:
1 - Prepare the full: that Abu al-Banaa rationed by saying that the total number is the number that equals the sum of its parts (Qawasmeh). Issue 6 Number of full because 6 = 1 + 2 + 3
2 - the number of excess: the excessive number is what is less than the sum of its parts (Qawasmeh).
No. 12 plus the number of that 12 <1 +2 +3 +4 +6
3 - deficient number: a number that is greater than the sum of its parts.
Such as the number 10> 1 +2 +5
Ben also created a fixed base numbers Kara Altabp is that the total denominator Sun verses equal to the last example:
(220 284) love each other because the two issues:
Total denominators 220: 1 +2 +4 +5 +10 +11 +20 +22 +44 +55 +110 = 284
Total 284 denominators: 1 +2 +4 +71 +142 = 220
Kashi has also put the decimal point in the message book ocean for the first time in history, where he voiced: 2 i = 6.283185.7179865
Second: In the area of Reparation:
The first book known in algebra is a book-Khwarizmi: Algebra and the interview, which ranked him as the equations in the shape of return.
It was reported that the algebra algorithm based on three forms: the roots and the number of funds.
Money corresponds to Q 2, and the root called something, and something that characterized the number and money calling it a draw, and he said the root of money, equivalent to two dirhams.
When algebra equation by removing the negative border, and when the interview is similar deletion of the border from both sides.
Also reached the Arabs to solve equations of the forces on the image above:
M x 2 n + b n = c o
The Arabs offered solutions to the equations of the third and fourth degree and Akchwo theory, which says:
Nokia total number of cubes cubic, and this is the basis of the theory of Fermat's famous:
A n + b n = c n, which can not be solved when n> 2
Third: In the field of engineering and trigonometry:
Arabs have translated the book assets of Euclid, and increased it, as Ibn al-Haytham made theories and issues, including "How to draw straight lines from points within the circle Vdtin information to any point imposed on the surrounding area so that the tangent with the manufacture of the decree that point angles equal."
Al-Biruni also provided proof to the area of the triangle in terms of the ribs. The West is also known for engineering Euclid by the Arabs.
It exploits the Arabs in trigonometry is to use the six trigonometric ratios revealed Battani relationship:
Jtao = Jtab Jtaj + GAP GAGES Jtao, special spherical triangle italics where a, b, c represent the sides of the triangle, a corner of a triangle.
The first discovered by Alovlh relationship: Jtab = Jtab is then removed, for spherical right-angled triangle in c.
Altabani also discovered the law of a rising sun:
X = AGA (90 - a) \ is then removed
It was discovered the relations between the Arab enclave, tangent, secant, and elite counterparts, and to know the basic rule to the area of spherical trigonometry and work tables, sports tangent, secant, and cosecant.
That:
Gas = x \ (the root of x 2 + 1).
Ibn Yunus and reached to the law:
Jtas Jtas = 1 \ 2 cos (x + y) + 1 \ 2 cos (X - Y).
Of the wonders of numbers
Wonders of the figure (5)
8 × 5 = 40
88 × 5 = 440
888 × 5 = 4440
8888 × 5 = 44440
88888 × 5 = 444440
888888 × 5 = 4444440
8888888 × 5 = 44444440
88888888 × 5 = 444444440
888888888 × 5 = 4444444440
8888888888 × 5 = 44444444440
Wonders of the number (9) Wonders of the number (7)
9 × 345679 × 1 = 111111111 1 × 7 × 15873 = 111111
9 × 345679 × 2 = 222222222 2 × 7 × 15873 = 222222
9 × 345679 × 3 = 333333333 3 × 7 × 15873 = 333333
9 × 345679 × 4 = 444444444 4 × 7 × 15873 = 444444
9 × 345679 × 5 = 555555555 5 × 7 × 15873 = 555555
9 × 345679 × 6 = 666666666 6 × 7 × 15873 = 666666
9 × 345679 × 7 = 777777777 7 × 7 × 15873 = 777777
9 × 345679 × 8 = 888888888 8 × 7 × 15873 = 888888
9 × 345679 × 9 = 999999999 9 × 7 × 15873 = 999999
Wonders of the number (
1 × 8 +1 = 9
12 × 8 +2 = 98
123 × 8 +3 = 987
1234 × 8 +4 = 9876
12345 × 8 +5 = 98765
123456 × 8 +6 = 987654
1234567 × 8 +7 = 9876543
12345678 × 8 +8 = 98765432
123456789 × 9 +9 = 987654321
Of the wonders of Number 9 also
987654321 × 9 = 8888888889
98765432 × 9 = 888888888
9876543 × 9 = 88888887
987654 × 9 = 8888886
98765 × 9 = 888885
9876 × 9 = 88884
987 × 9 = 8883
98 × 9 = 882
9 × 9 = 81
1) Find the number which, when added to two thirds and three output was 18.
First Step: suppose the unknown and what you should feature the first, and then dispose of it according to the question, it conforms is required, although not as well as the error of plus or minus is the first error.
Second step: Suppose another unknown feature and imposed the second, the mistake happened the second error.
Step Three: Multiply the first imposed in the second error, and saved the first feature.
Step Four: Multiply the second imposed in the first error, and saved the second feature.
Step five: If the lines of redundant or incomplete Voksm the difference between the saved on the difference between the two mistakes, although they differed on the total The total saved two mistakes to get to the unknown.
Take supposed to resolve the issue I: 3 0 if we act according to which the question is:
3 + 3 × 2 / 3 + 3 = 3 + 2 + 3 = 8 ... .... .... The first error is 18-8 = 10 minus
Take imposed II: 6 0 if we act according to which the question is:
6 + 6 × 2 / 3 + 3 = 13 ... ... ... The second error is 18-13 = 5 minus
Permission to be saved I = 3 × 5 = 15 and have saved the second = 6 × 10 = 60
The difference between 60 and 15 = 45 The difference between the two mistakes is 10-5 = 5
... ... ... ... ... ... ... ... ... ... ... .... The answer is 45 / 5 = 9
The extraction of unknowns in a manner analysis and inversion are based on the work the opposite of what gave him half of the liquid, a weak, although increased Vangs, although beaten Voksm or the root of one quarter or reverse Vaeks beginning of the last question. Have been received this question in the book Baha'eddin Global: ¸ beaten in the same number was increased to winning two and weakness, and increased the sum of three dirhams and a section of society (total) on five and struck out in ten got fifty •. (What is the number?)
Start another question Venksm 50-10 and then multiply in like 5; any 5 × 5 = 25 and from 25 Nnqs No. 3 Vicu rest 22 and half that number Nnqs 2; any 11-2 = 9 The answer is the square root of 9, or 3.
Contributions in mathematics
First: In the area of account:
The Arab scientists first developed four arithmetic operations, collection and multiple reward, half-life, differentiation, multiplication and division, and credit them to extract the roots. They have numbers divided into three types:
1 - Prepare the full: that Abu al-Banaa rationed by saying that the total number is the number that equals the sum of its parts (Qawasmeh). Issue 6 Number of full because 6 = 1 + 2 + 3
2 - the number of excess: the excessive number is what is less than the sum of its parts (Qawasmeh).
No. 12 plus the number of that 12 <1 +2 +3 +4 +6
3 - deficient number: a number that is greater than the sum of its parts.
Such as the number 10> 1 +2 +5
Ben also created a fixed base numbers Kara Altabp is that the total denominator Sun verses equal to the last example:
(220 284) love each other because the two issues:
Total denominators 220: 1 +2 +4 +5 +10 +11 +20 +22 +44 +55 +110 = 284
Total 284 denominators: 1 +2 +4 +71 +142 = 220
Kashi has also put the decimal point in the message book ocean for the first time in history, where he voiced: 2 i = 6.283185.7179865
Second: In the area of Reparation:
The first book known in algebra is a book-Khwarizmi: Algebra and the interview, which ranked him as the equations in the shape of return.
It was reported that the algebra algorithm based on three forms: the roots and the number of funds.
Money corresponds to Q 2, and the root called something, and something that characterized the number and money calling it a draw, and he said the root of money, equivalent to two dirhams.
When algebra equation by removing the negative border, and when the interview is similar deletion of the border from both sides.
Also reached the Arabs to solve equations of the forces on the image above:
M x 2 n + b n = c o
The Arabs offered solutions to the equations of the third and fourth degree and Akchwo theory, which says:
Nokia total number of cubes cubic, and this is the basis of the theory of Fermat's famous:
A n + b n = c n, which can not be solved when n> 2
Third: In the field of engineering and trigonometry:
Arabs have translated the book assets of Euclid, and increased it, as Ibn al-Haytham made theories and issues, including "How to draw straight lines from points within the circle Vdtin information to any point imposed on the surrounding area so that the tangent with the manufacture of the decree that point angles equal."
Al-Biruni also provided proof to the area of the triangle in terms of the ribs. The West is also known for engineering Euclid by the Arabs.
It exploits the Arabs in trigonometry is to use the six trigonometric ratios revealed Battani relationship:
Jtao = Jtab Jtaj + GAP GAGES Jtao, special spherical triangle italics where a, b, c represent the sides of the triangle, a corner of a triangle.
The first discovered by Alovlh relationship: Jtab = Jtab is then removed, for spherical right-angled triangle in c.
Altabani also discovered the law of a rising sun:
X = AGA (90 - a) \ is then removed
It was discovered the relations between the Arab enclave, tangent, secant, and elite counterparts, and to know the basic rule to the area of spherical trigonometry and work tables, sports tangent, secant, and cosecant.
That:
Gas = x \ (the root of x 2 + 1).
Ibn Yunus and reached to the law:
Jtas Jtas = 1 \ 2 cos (x + y) + 1 \ 2 cos (X - Y).
Of the wonders of numbers
Wonders of the figure (5)
8 × 5 = 40
88 × 5 = 440
888 × 5 = 4440
8888 × 5 = 44440
88888 × 5 = 444440
888888 × 5 = 4444440
8888888 × 5 = 44444440
88888888 × 5 = 444444440
888888888 × 5 = 4444444440
8888888888 × 5 = 44444444440
Wonders of the number (9) Wonders of the number (7)
9 × 345679 × 1 = 111111111 1 × 7 × 15873 = 111111
9 × 345679 × 2 = 222222222 2 × 7 × 15873 = 222222
9 × 345679 × 3 = 333333333 3 × 7 × 15873 = 333333
9 × 345679 × 4 = 444444444 4 × 7 × 15873 = 444444
9 × 345679 × 5 = 555555555 5 × 7 × 15873 = 555555
9 × 345679 × 6 = 666666666 6 × 7 × 15873 = 666666
9 × 345679 × 7 = 777777777 7 × 7 × 15873 = 777777
9 × 345679 × 8 = 888888888 8 × 7 × 15873 = 888888
9 × 345679 × 9 = 999999999 9 × 7 × 15873 = 999999
Wonders of the number (
1 × 8 +1 = 9
12 × 8 +2 = 98
123 × 8 +3 = 987
1234 × 8 +4 = 9876
12345 × 8 +5 = 98765
123456 × 8 +6 = 987654
1234567 × 8 +7 = 9876543
12345678 × 8 +8 = 98765432
123456789 × 9 +9 = 987654321
Of the wonders of Number 9 also
987654321 × 9 = 8888888889
98765432 × 9 = 888888888
9876543 × 9 = 88888887
987654 × 9 = 8888886
98765 × 9 = 888885
9876 × 9 = 88884
987 × 9 = 8883
98 × 9 = 882
9 × 9 = 81