Special Numbers

• see also:
  • What's a number?
  • Algebra
  • Mensuration
  • Calculus
  • Circular functions
  • Coordinate geometry, parabolae, etc
  • Finance equations
  • Logarithmic, exponential and hyperbolic functions

Pi:

• Rhind papyrus (1650BC) approximates pi = 3.16
• Archimedes (250BC) approximates pi = 3.1418
• Tsu Ch'ung Chi (480AD) approximates pi to 7 places
•  

The number systems:

• real numbers (set of R):
  • trichotomy law: set of R = set of R⁺ (positive real numbers) union set of R^- (negative real numbers) union {0}
  • real numbers consist of the union of:
    • rational numbers (set of Q; numbers that can be expressed as a ratio of numbers): 
      • integers (set of J):
        • these were the initial numbers of the ancient Greeks, initially consisting of 1,2,3,4...etc
        • later zero was added
        • but it was not until the Renaissance that negative numbers were added as it was then that they were used to interpret business debt
        • with the coming of computers, the need to restrict the range of numbers to minimise memory requirements, resulted in the introduction of sub-types of integers:
          • byte
          • small integer
          • long integer
      • fractions:
    • irrational numbers:
      • numbers that can be expressed as a ratio of numbers
      • irrational numbers were not accepted until the theory of Eudoxus (c. 370 BC) to allow for the determination of sides of a square whose area is 2 units (ie. square root of 2), such irrational numbers are called surds
  • real numbers can also be seen as being a union of:
    • algebraic numbers
      • the set of all numbers that satisfy some polynomial equation of order n is countable and thus is "algebraic" although not always real?
    • transcendental numbers
      • the set of real numbers which are not countable
• imaginary numbers:
  • complex numbers:
    • a complex number c is a pair (x, y) of two real numbers. x in the pair is called the real part and y is the imaginary part of the complex number c
    • a complex number is usually represented as = x + yi where i² = -1
    • complex numbers have unusual properties especially when it comes to differentiation. With real numbers differentiation is a starting point of Calculus and Real Analysis. With complex numbers it leads to the Analytic Function Theory.
  • hypercomplex numbers:
    • a hypercomplex number c is a pair (x, y) of two complex numbers, giving a quadruple of reals (a,b,c,d) called a quaternion.

Palindromic number:

• a palindromic number is one which remains the same when the digits are reversed. eg 14941

Factors:

• a natural number, a, is a factor of a natural number, b, if there exists a natural number, k, such that b = ak

Perfect numbers:

• a perfect number is one that equals the sum of its factors, excluding the number itself. eg. 6, 28, 496, 8128, 33550336
• NB. this is different from a perfect square/cube/nth power as these are natural numbers which are the square/cub/nth power of another natural number

Prime numbers:

• a prime number is one that is greater than 1 and has factors 1 and itself, but no other
• there are 25 prime numbers < 100
• there are infinite number of prime numbers
• there is only 1 even prime = 2
• all odd primes are of the form 4n +1 or 4n + 3
• can determine if a number is prime by either:
  • consider all prime numbers less than square root of the number, and see if any are factors of the number. If there are no such factors, then it is a prime number.

Composite numbers:

• a number, m, is a composite number if it can be written as a product m = a x b, where a & b are numbers greater than 1 and less than m

Prime decomposition:

• by breaking a composite number into product of prime numbers only, this is called prime decomposition or prime factorisation
• each number has a unique prime decomposition
• eg. 24192 = 2⁷ x 3³ x 5⁰ x 7¹
• the prime decomposition of a number can be determined either by:
  • repeated division:
    • starting with the prime number, 2, keep dividing the number by the prime number until you need to step to the next prime number, and keep repeating until you end up with a prime number
• the prime decomposition makes it easy to determine how many factors a number has by:
  • if prime decomposition = 2^a x 5^b x 7^c, then no factors = (a+1) x (b+1) x (c+1) (ie. product of all [exponentials plus 1])

Perfect squares:

• perfect squares are natural numbers which are the square of another natural number
• a perfect square's prime decomposition has all even exponents (ie. all multiples of 2)
• perfect squares have an odd number of factors eg. factors of 9 are 1,3 & 9

Perfect cubes:

• perfect cubes are natural numbers which are the cube of another natural number
• a perfect cube's prime decomposition has exponents which are all multiples of 3
• eg. 27 = 3³ => is a perfect cube

Perfect nth power:

• perfect nth power is a natural number which is the nth power of another natural number
• a perfect nth power's prime decomposition has exponents which are all multiples of n
• eg. 32 = 2⁵ => is a perfect fifth power

Least common multiple (LCM):

• the LCM of natural numbers l,m and n is the smallest natural number which is a multiple of both m and n
• LCM can be determined either by:
  • prime factorisation of l, m & n:
    • determine the prime decomposition of l, m & of n
    • the LCM is found by choosing the highest power of each prime occurring in the prime decompositions & then taking the product of these
    • eg: determine LCM of 20, 30, 35:
      • 20 = 2² x 5; 30 = 2 x 3 x 5; 35 = 5 x 7;
      • thus LCM = 2² x 3 x 5 x 7 = 420

Highest common factor (HCF):

• the HCF of 2 natural numbers a & b is the largest number which divides both a and b
• eg. the HCF of 8 and 12 is 4
• NB. the HCF of 0 and n is n
• the HCF can be determined either by:
  • prime factorisation of a & b:
    • determine the prime decomposition of a & b
    • the HCF is found by choosing which prime numbers are in common and then choosing the lowest exponent of these, then finding the product of these chosen prime numbers
    • eg. HCF of 110 and 140:
      • 110 = 2 x 5 x 11; 140 = 2² x 5 x 7;
      • HCF (110,140) = 2 x 5 = 10
  • Euclidean algorithm:
    • this uses the division algorithm repeatedly until there is no remainder
    • eg. HCF of m & n:
      • m = an + R1;
      • n = bR1 + R2;
      • R1 = cR2 + R3;
      • R2 = dR3 + R4;...and so on until remainder = 0, then the previous remainder will be the HCF
• two numbers are said to be relatively prime or coprime if their HCF = 1
• also, (LCM of m,n) x (HCF of m,n) = product of the m & n

Division algorithm:

• for a number n and a smaller number q, we can express n in the form of n = aq + r, where a is a positive integer and r is the remainder and has a possible range of 0 to q-1 (this range is called the least residues modulo q)
• thus one can express any number as a multiple of another plus a remainder
  • eg. if q = 3, any number n can be expressed as either 3a, 3a + 1, or, 3a + 2
• this can also be represented as r = n modulo q

Figurate numbers:

• the number of dots spaced a unit apart around the perimeter and within a polygon is called a figurate number
• this results in the following patterns of numbers:
  • linear (counting numbers): 1, 2, 3, 4, 5,...
  • triangular numbers: 1, 3, 6, 10, 15, ....
    • this is a sequence of sums of arithmetic sequence: 1, 1+2, 1+2+3, ... (n² + n)/2
    • the sum of the squares of any two consecutive triangular numbers is the nth triangular number, where n is a square
    • the difference between the squares of two consecutive triangular numbers is a cube number
  • square numbers: 1, 4, 9, 16, 25, ....n²
    •  
  • pentagonal numbers: 1, 5, 12, 22, ... n(3n-1)/2
    • this is formed from the sums of the arithmetic sequence with difference = 3 (ie. 1,4,7,10)
    • the nth pentagonal number  = nth square number + (n-1)th triangular number
  • hexagonal numbers: 1, 6, 15, 28, ... n(2n-1)
    • this is formed from the sums of the arithmetic sequence with difference = 4 (ie. 1,5,9,13..)
    • the nth hexagonal number = (2n-1)th triangular number

Congruent numbers and modulo:

• if a and b are integers, and their difference a-b is divisible by m, a is said to be congruent to b, modulo m
• some properties of congruence:
  • if a is congruent to b (mod m) this implies b is congruent to a (mod m)
  • if a is congruent to b (mod m) and b is congruent to c (mod m) this implies a is congruent to c (mod m)
  • congruence a congruent to b (mod m) holds if and only if a and b are numbers which have the same remainder when divided by m
  • if  a is congruent to b (mod m) and c is congruent to d (mod m) then:
    • a+c is congruent to b+d (mod m), and,
    • a-c is congruent to b-d (mod m), and,
    • ac is congruent to bd (mod m)
  • if a is congruent to b (mod m), then if aⁿ is congruent to bⁿ (mod m)
• any number can be written as being congruent  to one of a least residues modulo m (mod m)
• to find the final digit of a number, work in modulo 10
• example:
  • what is the remainder when 3²⁰⁰² +(29 x 5²⁰⁰²) is divided by 26?
    • ie. what is 3²⁰⁰² +(29 x 5²⁰⁰²) modulo 26
    • 1st need to reduce the exponential components to simpler exponents by finding congruences between:
      • 1 mod 26 and 3ⁿ mod 26: in this case, 3³ = 27 which is congruent with 1 mod 26, thus 3²⁰⁰² can be replaced by 3^{(3 x 667 + 1)} and by replacing 3³ by 1, will equal 1⁶⁶⁷ x 3¹ mod 26 which can be further simplified to 3 mod 26
      • 29 can be replaced by 3 mod 26
      • 1 mod 26 and 5ⁿ mod 26: in this case, 5⁴ = 625 which is congruent with 1 mod 26, thus 5²⁰⁰² can be replaced by 5^{(4 x 500 + 2)} and by replacing 5⁴ by 1, will equal 1⁵⁰⁰ x 5² mod 26 which can be further simplified to 25 mod 26
    • thus result is [3 + (3x25)] mod 26  which can be simplified to 78 mod 26 which is congruent to 0 mod 26 as 78/26 = 3 and no remainder
    • thus the remainder is zero!

Unit fraction:

• a fraction with numerator = 1
• all unit fractions can be written as the sum of two unequal unit fractions:
  • 1/n = 1/(n+1) + 1/n(n+1)

see www.nzamt.org.nz/nzamt9/to/MathematicalCuriositiesBThomas.ppt for an interesting Powerpoint of mathematical curiosities.