Formulas and Tables

Number Sets

Natural Numbers

Set of natural numbers: \(\mathbb{N}\)
Set of natural numbers with zero: \(\mathbb{N_0}\)
Natural numbers: \(n\), \(k\), \(a\), \(b\), \(c\)

Arabic numerals: \(0\), \(1\), \(2\), \(3\), \(4\), \(5\), \(6\), \(7\), \(8\), \(9\)
Roman numerals: \(I\), \(V\), \(X\), \(L\), \(C\), \(D\), \(M\)

  1. Natural numbers \(\mathbb{N} = \left\{ {1,2,3, \ldots } \right\}\)
    These numbers can be used for counting and ordering.
  2. Natural numbers including zero \(\mathbb{N_0} = \left\{ {0,1,2,3, \ldots } \right\}\)
    These numbers can be used for indicating the number of objects.
  3. Commutativity of addition \(a + b = b + a\)
  4. Associativity of addition \(a + \left( {b + c} \right) = \left( {a + b} \right) + c\)
  5. \(a + 0 = a\)
  6. Commutativity of multiplication \(a \cdot b = b \cdot a\)
  7. Associativity of multiplication \(a \cdot \left( {b \cdot c} \right) =\) \( \left( {a \cdot b} \right) \cdot c\)
  8. Distributivity of multiplication over addition \(a \cdot \left( {b + c} \right) =\) \( a \cdot b + a \cdot c\)
  9. \(a \cdot 0 = 0\)
  10. \(a \cdot 1 = a\)
  11. Even numbers \(n = 2k,\;\left( {n, k \in \mathbb{N}} \right)\)
  12. Odd numbers \(n = 2k + 1,\;\left( {n, k \in \mathbb{N}} \right)\)
  13. Roman numerals
Roman numerals
  1. Roman numerals \(1\) to \(100\)
Roman numerals 1 to 100
  1. Divisibility by \(2\)
    A number is divisible by \(2\) if its last digit is divisible by \(2,\) i.e. if the number is even.
  2. Divisibility by \(3\)
    A number is divisible by \(3\) if the sum of its digits is divisible by \(3.\)
  3. Divisibility by \(4\) A number is divisible by \(4\) if its last two digits are zero or form a number divisible by \(4.\)
  4. Divisibility by \(5\)
    A number is divisible by \(5\) if its last digit is divisible by \(5,\) i.e. equal to \(0\) or \(5.\)
  5. Divisibility by \(6\)
    A number is divisible by \(6\) if it is divisible by \(2\) and by \(3,\) i.e. if the number is even and the sum of its digits is divisible by \(3.\)
  6. Divisibility by \(7\)
    Sum 3 times the tens digit and the ones digit. If the result is divisible by \(7\) then the number is also divisible by \(7.\)
  7. Divisibility by \(8\)
    A number is divisible by \(8\) if its last three digits are zero or form a number divisible by \(8.\)
  8. Divisibility by \(9\)
    A number is divisible by \(9\) if the sum of its digits is divisible by \(9.\)
  9. Divisibility by \(10\)
    A number is divisible by \(10\) if its last digit is zero.

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