* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Title Goes Here
Survey
Document related concepts
Approximations of π wikipedia , lookup
Mathematical proof wikipedia , lookup
Philosophy of mathematics wikipedia , lookup
History of logarithms wikipedia , lookup
Law of large numbers wikipedia , lookup
Mathematical model wikipedia , lookup
Large numbers wikipedia , lookup
Laws of Form wikipedia , lookup
Mathematics and architecture wikipedia , lookup
Foundations of mathematics wikipedia , lookup
Ethnomathematics wikipedia , lookup
History of mathematics wikipedia , lookup
Mathematics and art wikipedia , lookup
Location arithmetic wikipedia , lookup
Transcript
MATHEMATICAL WATCHES Teresa Efigenia Alarcón Martínez*, Iván Guillén Escamilla ** *Departamento de Ciencias Computacionales e Ingenierías ** Departamento de Ciencias Naturales y Exactas Centro Universitario de los Valles Universidad de Guadalajara. This section is dedicated to answer the problems related with the watches, in which the numbers are expressed using a mathematical constant or a mathematical expression. From the pages http://simplementenumeros.blogspot.com/2011/07/733-relojesmatematicos.html and http://www.google.com.mx/search?q=relojes+matem%C3%A1ticos&hl=es419&client=firefox-a&hs=BLr&rls=org.mozilla:enUS:official&prmd=imvns&tbm=isch&tbo=u&source=univ&sa=X&ei=FNCNTpSBH MassAK2hMGIAQ&ved=0CEoQsAQ&biw=1831&bih=959 were selected 5 images with mathematical watches. Watch No. 1 FIGURE 1. Numbers are expressed using combinatorial expressions, factorial, rounding, mathematical constants, numerical representations using a base different to decimal and html code. http://simplementenumeros.blogspot.com/2011/07/733-relojes-matematicos.html The explanation for the watch in Figure 1 appears in the Table No. 1 Number TABLE 1. Answers for the watch No. 1. Mathematical expression 1 2 Meaning is the Legendre's constant that is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function Its value is now known to be exactly 1 [1]. ∑ This is a geometric serie of ratio , ∑ ∑( ) 3 ! This is the number 3 in HTML code. 33 is the hexadecimal representation of the number 51 and 51 is the ASCII code of number 3. 4 20 mod 8 4 is the rest of the division between 20 and 8. 5 ( 6 3! 7 ) is the golden ratio [2], and it is equal to ) . √ , so ( The factorial of 3 is equal 6. ̅ 7 is the number rounding. ̅ after 8 9 The first ball in the figure is on (active), the rest ones are off (inactive). The binary representation for 8 is 1000 in which only the first bit is equal to 1 (is active). 214 The number 9 using base 4 is 21 = 4*2+1. (( )) ( ) 11 0x0B The number 11 in hexadecimal representation is B. 12 √ 12 is the cube root of 1728. 10 Watch No. 2 FIGURE 2. Numbers are expressed using different algebraical expressions. The explanation for the watch in Figure 2 appears in the Table No. 2 Number Mathematical expression 1 2 TABLE 2. Answers to the watch No 2. Meaning The rest of the numbers is equal 1. √ The square root of 4 is 2. 3 3 is quotient . 4 4 is the x value. 5 5 is the quotient. 6 6 is the final value of the expression. 7 52-x2+x=10 The roots of the polynomial -x2+x+42 are 7 and -6. For this case only one solution is considered: 7. 8 √ 8 is the square root of 64. 9 ( ) The result of the arithmetic operation ( Finally the product is 9. 10 -8=2-x 10 is the value of x in the equation. 11 1221:11121 11 is the quotient . 12 6*2 12 is the product of 6 and 2. ) is rounded to 3. Watch No. 3 FIGURE 3. Numbers are expressed using different mathematical expressions. The explanation for the watch in Figure 3 appears in the Table No. 3. Number Mathematical expression TABLE 3. Answers to the watch No 3. Meaning 1 . The value of the expression is -1. The appositive value is 1. 2 ∑ This is a geometric serie ,of ratio ∑ ∑( ) , 3 ⌊ ⌋ 3 nearest integer . 4 5 4 is the value of 2+2. ( ) 6 ( ) 7 ̅ is the golden ratio, and it is equal to √ , so ( ) . The recursion formula for the Gamma function is: ( ) ( ). If n is nonnegative integer, i. e, 0,1, 2, …, then: ( ) . For n = 3. The factorial of 3 is 6. 7 is the rounded number. 8 is the Legendre's constant that is a mathematical constant. Its value is known to be exactly 1 [1]. After adding 1 we obtain 8. 9 | 10 119 10 on base 9 is represented by 11. 11 √ The cube root of 1331 is 11. | The expression denotes the determinant of the matrix ( 12 The ( ) ( ) ). . The final values of the expression is 12. Watch No. 4 FIGURE 4. Numbers are expressed using binary numeral system. All numbers in the watch No. 4 are represented using binary numeral system. Watch No. 5 FIGURE 5. Numbers are expressed using different mathematical expressions and constants. The explanation for the watch in Figure 5 appears in the Table No. 4 Number Mathematical expression 1 TABLE 4. Answers to the watch No 5. Meaning The tangent of 45 grades is equal to 1. 2 √ The square root of 4 is 2. 3 Round( ) 3 is the nearest integer to . 4 ( ) The logarithm of 55 to base e is equal to 4.0073331852…, 4 is the rounded number. 5 The value of x is 5. 6 The factorial of 3 is 6. 7 ̅ The nearest integer to ̅ is 7. 8 The quotient is 8. 9 3x3 10 gn 11 12 The product is 9. The standard acceleration of gravity [3] is 9.806 65 m·s−2 . The rounded value is 10. is the number 11 using hexadecimal numeral representation. is the number 12 using binary numeral representation. REFERENCES [1]. Legendre, A. M. Essai sur la Théorie des Nombres. Paris: Duprat, 1808. [2]. Livio, Mario. The Golden Ratio: The Story of Phi, The World´s Most Astonishing Number, New York: Broadway Books, 2002. [3]. Métodos matemáticos para físicos, George Brown Arfken, Hans J. Weber; 5ta Edicion, Harcourt Academic Press, 2001.