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Mathematics Dictionary
Dr. K. G. Shih
Numbers : Defintion
Subjects


  • Q01 | - Real number system
  • Q02 | - Complex number system
  • Q03 | - Special properties
  • Q04 | - Magic number circles
  • Q05 | - Numbers arranged in matrix pattern
  • Q06 | - Number sequences
  • Q07 | - Perfect square numbers and square free number
  • Q08 | - Find square root of a number
  • Q09 | - Find square root using arithmatic method
  • Q10 | - Mutiple of an integer whose digits are same
  • Q11 | - Amicable number pairs
  • Q12 | - Perfect numbers
    • Answers


    Q01. Real Number system
    Real Numbers
    • Definition - A real number has a postition on the number lines.
    • Real numbers :
      • Integer - Whole numbers 0, 1, 2, 3, 4, 5, ......
        • Odd number :
          • Integers have no factor of 2.
          • Examples : 1, 3, 5, ... are odd numbers.
        • Even number
          • Integers have factor of 2.
          • 2, 4, 6, ... are even numbers.
      • Rational number
        • It can be expressed as ratio of two integers.
        • Example : 0.3 = 3/10.
      • Irrational number
        • Defintion : It can't be expressed as ratio of two integers.
        • It has position on the number line. e.g. pi is between 3 and 4.
        • Example : pi = 3.141592...., Sqr(2) = 1.41....
    • Properties of real numbers
      • Positive number : It is on the Right of zero of a number line.
      • Negative number : It is on the left of zero of a number line.
    • Decimal number and fraction
      • Repeating decimals - Example : 0.33333... is repeating which is 1/3.
        • Let x = 0.33333....
        • Then 10*x = 3.3333...
        • Hence 10*x - x = 3.
        • Hence x = 1/3.
      • Fraction = numerator/denominator.
        • Proper fraction - denominator is greater than numerator.
        • Improper fraction - denominator is less than numerator.
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    Q02. Complex number system
    Defintion
    • Imaginary number : i = Sqr(-1).
      • i^2 = -1.
      • i^3 = (i^2)*i = -i
      • i^4 = (i^3)*i = -i^2 = -(-1) = 1.
      • i^5 = (i^4)*i = i
    • Complex number : z = a + b*i.
      • Example : (1 - i)*(1 + i) = 1^2 - i^2 = 1 -(-1) = 2.
      • Example : (1 + i)*(1 + i) = 1^2 + 2*1*i + i^2 = 1 + 2*i +(-1) = 2*i.
    Complex number coordinate system
    • It is similar as the rectangular coordinate system.
    • The x-axis is the real number.
    • The y-axis is then imaginary number.
    • z = a + b*i.
      • Coordiante at (a,b).
      • Make angle with x-axis is A = arctan(b/a).
      • Distance to origin is r = Sqr(a^2 + b^2).
      • Hence a = r*cos(A) and b = r*sin(A).
    • Example : z = 1 + 2*i is same as point (1,2).
    • Example : (1 + i)*(1 + i) = 2*i
      • (1 + i) = coordiante (1,1) or (1+i) makes 45 degrees angle with x-axis.
      • (1 + i)*(1 + i) = 2*i means makes angle 90 degree with x-axis.
    Complex number in polar form
    • z = a + b*i = r*(cos(A) + i*sin(A)).
    • Where r = Sqr(a^2 + b^2).
    • a = r*cos(A) and b = r*sin(A).
    • A = arctan(b/a)
    • Example : Convert r = 1/(1+i) into polar form.
      • r = 1/(1+i) = (1-i)/((1-i)*(1+i)) = (1-i)/2 = a + b*i.
      • Hence a = 1/2 and b = -1/2. r = 1/Sqr(2) = Sqr(2)/2.
      • A = arctan(b/a) = arctan(-1) = 225 degrees.
      • Hence r = Sqr(2)*(cos(225) + i*sin(225))/2
    Conjugate numbers
    • (a + b*i) and (a - b*i) are conjugate complex numbers
    • Sum of conjugate numbers = real number. Example (a + b*i) + (a - b*i) = 2*a.
    • Product of conjugate numbers = real number. Example (a+b*i)*(a-b*i) = a^2 + b^2.
    • Example : Prove that 1/(1-i) = (1+i)/2
      • Change denominator to real number.
      • Hence multiply numerator and denominator by (1+i).
      • (1+i)*(1-i) = 1^2 - i^2 = 2.
      • 1/(1-i) = (1+i)/((1+i)*(1-i)) = (1+i)/2.
    Study | Definiotn and application examples of the complex number system.

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    Q03. Special properties of real numbers
    • Study | Amicable number pairs
      • Example 1. What are amicable number pairs ?
      • Example 2. Prove that 220 and 284 are amicable pairs.
    • Study | Perfect numbers.
      • Example 1. What is perfect number ?
      • Example 2. Prove that 28 is a perfect number.
    Properties of numbers
    • Abundunt numbers : Sum of their factors is greater than number itself.
    • Deficient numbers : Sum of their factors is less than number itself.
    • Perfect numbers : Sum of their factors is equal to than number itself.
    • Prime numbers : Sum of their factors is 1.
    • Note : Sum of factors is not included number itself.
    • Example.
      • Sum of factors of 27 = 1 + 3 + 9 = 13. Hence 27 is deficient number.
      • Sum of factors of 28 = 1 + 2 + 4 + 7 + 14 = 28. Hence 28 is a perfect number.
      • Sum of factors of 29 = 1.
      • Sum of factors of 30 = 1 + 2 + 3 + 5 + 6 + 10 + 15 = 42. It is abundunt.
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    Q04. Magic number patterns
    Circle patterns
    Square patterns
    • Use integers 1 to 16 to make magic number circle.
    • Find the sum of the numbers for each row.
    • Find the sum of the numbers for each column.
    • Find the sum of the numbers for each diagonal.
    • Study | Magic numbers circle pattern is given here.
    Square patterns
    • Use integers 1 to 9 to make magic square.
    • Use integers 1 to 25 to make magic square.
    • Use integers 1 to 49 to make magic square.
    • Use integers 1 to 121 to make magic square.
    • Reference.
      • On PC computer : MD2002 program 01 10.
      • On Book : Chapter 11 of Computer mathematics by Dr. K. G. Shih.

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    Q05. Numbers arranged in matrix pattern
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    Q06. Number sequences
    Numbers in square patterns
    • Study | How many squares in squares ?.
    • What are the square number sequences ?
      • They are 1, 4, 9, 16, .....
      • What is the 10th number ?
    • Sketch square patterns for 1^2 + 2^2 = 5.
    • Sketch square patterns for 1^2 + 2^2 + 3^2 = 14.
    Numbers in cubuc patterns
    • Study | How many cubes in cubes ?.
    • What are the cubic number sequences ?
      • They are 1, 8, 27, 64, .....
      • What is the 10th number ?
    • Sketch cubic patterns for 1^3 + 2^3 = 9.
    • Sketch cubic patterns for 1^3 + 2^3 + 3^3 = 36.
    Numbers in triangular patterns
    • Study | What is triangular number sequence ?.
    • The sequence is 1, 3, 6, 10, 15, .....
    • Sketch triangular patterns for 1 + 2 + 3 + 4 = 10.
    • Sketch triangular patterns for 1 + 2 + 3 + 4 + 5 = 15.
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    Q07. Perfect square numbers and square free numbers
    • Perfect square numbers.
      • 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, ...
      • Its square roots are integers.
    • Square free numbers
      • 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 13, 15, 17, 18, .....
      • Its square roots are not integers.
    • Simplest form
      • For square root - It has only one factor is square free.
        • Example Sqr(180) has factors 4, 5, 9 and 5 is square free.
        • Hence simplest form of Sqr(180) = 36*Sqr(5).
      • Fraction - Numerator and denominator has no common factors.
        • 16/20 = 4*4/4*5 = 4/5.
    Example : Show that 1.41 < Sqr(2) < 1.42
    • Method : Every real number other than integers lies between two consecutive integers.
      • Sqr(2) between 1 and 2.
      • (10*Sqr(2))^2 is between 14^2 and 15^2
        • Hence 10*Sqr(2) is between 14 and 15.
        • Divide by 10, we have Sqr(2) between 1.4 and 1.5.
      • (100*Sqr(2))^2 is between 141^2 and 142^2
        • Hence 100*Sqr(2) is between 141 and 141.
        • Divide by 100, we have Sqr(2) between 1.41 and 1.42.
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    Q08. Find square root of a real number
    Example 1 : Find square root of 2
    • Method 1 : Use calculator. Sqr(2) = 1.41421356....
    • Method 2 : Every real number other than integers lies between two consecutive integers.
      • Sqr(2) between 1 and 2.
      • (10*Sqr(2))^2 is between 14^2 and 15^2
        • Hence 10*Sqr(2) is between 14 and 15.
        • Divide by 10, we have Sqr(2) between 1.4 and 1.5.
      • (100*Sqr(2))^2 is between 141^2 and 142^2
        • Hence 100*Sqr(2) is between 141 and 141.
        • Divide by 100, we have Sqr(2) between 1.41 and 1.42.
    • Method 3 : Approximate method
      • 1st approximate : A1 = 2.
      • 2nd approximate : A2 = 0.5*(A1 + A1/A1) = 1.5.
      • 3nd approximate : A3 = 0.5*(A2 + A1/A2) = 1.41666667
      • 4th approximate : A4 = 0.5*(A3 + A1/A3) = 1.41421568
      • 5th approximate : A5 = 0.5*(A4 + A1/A4) = 1.41421356
    • Reference :See chapter 7 in compter methematics by Dr. K. G. Shih

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    Q09. Find square root using arithmatic method
    • Step 1 : Arrange digits of integer number in pairs from right
      • Example : 14641 is arranged as 1 46 41
    • Step 2 : Divide the 1 or 2 digits at left by a and get quotient a.
      • Example : Find Sqr(14641)
        • Arranged as 1 46 41.
        • Divide 1 by 1 and 1st digit of quotient is 1 and remainder is 46 41
        • 1 ..... 1st digit of quotient if 1 divide by 1
        • -------
        • 1 46 41
        • 1
        • -------
        • 0 46 41
        • Hence 1st digit of quotient from left is 1.
    • step 3 : 46 divide by (20*1 + n) where n = 2nd digit of quotient
      • Let n = 1 and 46 divide by 21. quotient is 1 and Remainder is 25.
      • Let n = 2 and 46 divide by 22. quotient is 2 and Remainder is 2.
      • Let n = 3 and 46 divide by 23. quotient is 3 and Remainder is negative.
      • The 2nd digit of quotient is 2. Remainder is 241
        • 2 ..... 2nd digit of quotient if 46 divide by 22
        • -------
        • 46 41
        • 44
        • ------- 2 41
        • Hence 1st and 2nd digits of quotient from left is 12
    • step 4 : 241 divide by (20*12 + n) where n = 3rd digit of quotient
      • Let n = 1 and 241 divide by 241. quotient is 1 and Remainder is 0.
      • Let n = 2 and 241 divide by 242. quotient is 2 and Remainder is negative.
      • The 3rd digit of quotient is 1. and remainder is zero.
        • 1 ..... 3rd digit of quotient if 241 divide by 241
        • -------
        • 241
        • 241
        • ------- 0
        • Hence 1st, 2nd and 3rd digits of quotient from left is 121
    • Hence Sqr(14641) = 121.
    • If remainder is not 0, we divide the remainder by (20*121+n).
    Example : Find square root of 92.
    • Divide 92 by 9 and quotient is 9. Remainder is 11.
      • 9.591
      • ----
      • 92
      • 81
      • ----
      • 11 00 Now 11.00 divide by (20*9 + n) where n is the 2nd digit.
      • 09 25
      • --------
      • 01 75 00 divide by (20*95 + m). m = 9
      • 01 71 81
      • --------
      • 00 03 19 00 divide by (20*959 + r). r = 1
      • 00 01 91 80
      • -----------
      • 00 01 27 80 00 divide by (20*9591 + s) s = 6 (5th digit of quotient)

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    Q10. Mutiple of an integer whose digits are same
    • Example : Find Multiple of 49 whose digits are same. 777777 = 49*15873.
    • Reference : MD2002 program 01 17.
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    Q11. Amicable number pairs
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    Q12. Perfect numbers
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