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Mathematics Dictionary
Dr. K. G. Shih
Questions and Answers
Questions

Answers


Q01. #020703 : Pythagorean triples
  • Pythagorean triples are 3 numbers satisfying the Pythagorean Law.
  • Example 1
    • 3, 4, 5 are Pythagorean numbers. Since 5^2 = 3^2 + 4^2.
    • Multiples of these numbers are also Pythagorean numbers.
  • Example 2
    • 5, 12, 13 are Pythagorean numbers.
    • Multiples of these numbers are also Pythagorean numbers.
  • Example 3
    • 8, 15, 17 are Pythagorean numbers.
    • Multiples of these numbers are also Pythagorean numbers.
  • Reference : MD2002 11 19.
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Q02. #020704 : Abundunt numbers
  • Sum of the factors of a number is greater than the number itself.
  • The number is called abundunt number.
  • Note : The sum is not included the number itself.
  • Example 1 : Prove that 30 is abundunt number.
    • facors of 30 : 1, 2, 3, 5, 6, 10, 15.
    • Sum of factors = 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42.
  • Example 2 : Prove that 29 is not abundunt number.
    • facors of 29 : 1.
    • Sum of factors = 1.
    • Hence number 29 is a deficient number and it is also a prime number.
  • Example 3 : Prove that 28 is not abundunt number.
    • facors of 28 : 1, 2, 4, 7, 14.
    • Sum of factors = 1 + 2 + 4 + 7 + 14 = 28.
    • Hence 28 is not abundunt number and it is also a perfect number.
  • Example 4 : Prove that 27 is deficient number.
    • Factors of 27 : 1, 3, 9
    • Sum of factors = 1 + 3 + 9 = 13.
    • Hence 27 is a deficient number.
  • Reference : MD2002 01 14.
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Q03. #020822 Amicable number pairs
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Q04. #020823 What is perfect number ?

  • Study subject What is perfect number ?
  • Exercise 1 : Prove that 6 is the first perfect number.
  • Exercise 2 : Prove that 28 is the second perfect number.
  • Exercise 3 : How to find the 3rd perfect number ?
  • Exercise 4 : How to find the 4rd perfect number ?

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Q05. How to find Fibonacci's sequence in Pascal Triangle ?

  • Study subject Fibonacci's sequence
  • Recursion formula
    • T(0) = 0 and T(1) = 1.
    • T(k) = T(k-2) + T(k-1) for k > 1.
  • The sequence : 1, 1, 2, 3, 5, 8, 13, ......
  • Find T(8) = 1 + 6 + 10 + 4 = 21 in Pascal triangle as shown in diagram.
  • Demo are given in MD2002 program 04 07.

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Q06. Solve x^7 + x^6 -5*x^5 - 13*x^4 - 13*x^3 - 5*x^2 + x + 1 = 0
Hint
  • It is not easy to find the answer.
  • Unless we know (x+1) and (x^2+x+1) are factor of the equation
Solutions Go to Begin

Q07. How to find signs of y, y' and y" using diagram ?
Find zeros of y, y' and y"
  • 1. y' is zero if there is maximum or minimum point
  • 2. y" is zero if there is a point of reflexiom
Find signs of y, y' and y"
  • 1. If the curve is in 1st or 2nd quadrant, then y has postive sign
  • 2. If the curve is in 3rd or 4th quadrant, then y has negative sign
  • 3. If the curve is increasing, then y' has positive sign
  • 4. If the curve is decreasing, then y' has negative sign
  • 5. If the curve is cocave upward, then y" has positive sign
  • 6. If the curve is cocave downward, then y" has negative sign
Examples to get diagra
  • Study subject Diagram polynomial function in 01 01
  • Select section 01
  • Enter the application program
    • Select run at current location (no download)
    • Select yes to run
  • Select quartic function
    • Click menu
    • Select 01 in upper box
    • Click Demo command
    • Click 04 in lower box
    • Now we have a diagram of quartic function y = x^4 + x^3 - 4*x^2 + x + 1
  • Estimate the zeros of y, y' and y" and write down the domains
  • Estimate the signs of y, y' and y" and write down the domains
zeors and signs of y = x^4 + x^3 - 4*x^2 + x + 1, y' and y"
  • zeros of y
    • x = -2.7
    • x = -0.4
    • x = +1.1
  • zeros of y'
    • Minimum at x = -1.9
    • Maximum at x = +0.1
  • zeros of y" : None
  • Signs of y
    • y is positive when x = -infinit to x = -1.9
    • y is positive when x = -0.4 to x = +infinite
    • y is negative when x = -2.7 to x = -0.4
  • Sign of y'
    • y' is negative when x = -infinite to x = -1.9
    • y' is positive when x = -1.9 to x = 1.1
    • y' is negative when x = +0.1 to x = -1.9
    • y' is positive when x GT 1.1
  • Sign of y"
    • y" is positive when x = -infinite to x = -0.4
    • y" is negative when x = -0.4 to x = +1.1
    • y" is positive when x GT 1.1
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    Q08. Answer
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    Q09. Answer
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    Q10. Answer
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Copyright © Dr. K. G. Shih, Nova Scotia, Canada.
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