Rational Numbers

Posted in Terms Defined

A number which can be written in the form \(\frac{p}{q}\), where p and q are integers and q ≠ 0 is called a rational number.

For e.g.

\(\frac{-2}{5}\), \(\frac{6}{7}\)

Roman Numerals

Posted in Terms Defined

The symbols used in Roman numerals are:

I ---> 1; V ---> 5; X ---> 10; L ---> 50; C ---> 100; D ---> 500; M ---> 1000

Rules for the system are :

a) If a symbol is repeated, its value is added as many times as it occurs:

i.e. II is equal to 2, XXX is equal to 30, CC is equal to 200

b) A symbol is not repeated more than three time. But the symbols V, L and D are never repeated.

c) If a symbol of smaller value is written to the left of a symbol of greater value, its value get added to the value of the greater symbol.

i.e. VI is equal to 5 + 1 = 6; CXX is equal to 100 + 10 + 10 = 120

d) If a symbol of smaller value is written to the left of a symbol of greater value, its value is subtracted from the value of the greater symbol.

i.e. IV is equal to 5 - 1 = 4; XC is equal to 100 - 10 = 90

e) The symbols V, L and D are never written to the left of a symbol of greater value, i.e. V, L and D are never subtracted. The symbol I can be subtracted from V and X only. The symbol X can be subtracted from L, M and C only. 

Pythagorean triplets

Posted in Terms Defined

Consider the following:

32 + 42 = 9 + 16 = 25 = 52

The collection of numbers 3, 4 and 5 is known as Pythagorean triplet. Similarly 6, 8 and 10 are also pythagorean triplets because

62 + 82 = 100 = 102

 

For any natural number m > 1, we have (2m)2 + (m2 - 1)2 = (m2 + 1)2. So, 2m, m- 1 and m+ 1 forms a Pythagorean triplet.

Whole Numbers

Posted in Terms Defined

Definition: The natural numbers (i.e. 1, 2, 3, 4, and so no) along with zero form the collection of whole numbers.

Therefore:-

  1. All natural numbers are whole numbers.

Properties of Whole Numbers:

  1. Closure property: Whole numbers are closed under addition and also under multiplication.

    It means that when you add two or more whole numbers, the result also be a whole number.
    Similarly, if you multiply two or more whole numbers, the result will be a whole number.

  2. Commutativity of addition and multipication: Two whole numbers can be added in any order. Also two whole numbers can be multiplied in any order.
  3. Associativity of addition and multiplication: If you solve 2 + ( 3 + 4 ) and ( 2 + 3 ) + 4, you will see that the result is same. Similary, if you solve 2 x ( 3 x 4 ) and ( 2 x 3 ) x 4, you will see that the result is same. This property is known as associativity of addition and multiplication.
  4. Distributivity of multiplication over addition: You can solve 2 x ( 3 + 5 ) in two ways:

    2 x ( 3 + 5 ) = 2 x 8 = 16

    or

    2 x ( 3 + 5 ) = 2 x 3 + 2 x 5 = 6 + 10 = 16

    The second way of solving the above equation is known as distributivity of multiplication over addition.