Classical Mathematics – Arithmetic and Numbers
In a strict sense, arithmetic, or number theory, primarily deals with calculations using numbers, leading to the examination and development of the principles of different types of numbers.
Arithmetic deals with calculations using basic rules derived from an intuitive and natural handling of objects. For example, it does not make a difference whether a person first adds two cows to four cows and then adds another three, or if he adds four cows to three and so on.
This is called the associative law: (a + b) + c = a + (b + c). It applies not only to natural numbers, but to all numbers in the number system.
Why are new numbers needed?
Generally, you cannot divide or subtract using only natural whole numbers. For example, dividing eight apples be- tween three people, or distributing three apples among five people, is not possible using only whole numbers.
These calculations can, however, be solved through the introduction of new kinds of numbers such as fractional and negative numbers. Eight apples di- vided by three people results in eight-thirds of an apple per person, and three subtracted by five is a negative two.
Number ranges
Natural numbers, N, allow addition and multiplication unconditionally. The set of integers, Z, includes negative whole numbers and allows for subtraction to occur unconditionally. The set of fractional numbers, Q+, allows for division. The set of rational numbers, Q, made up of N, Z, and Q+, allows all of the basic arithmetic operations to be carried out, except for dividing by zero.
The set of real numbers, R, includes irrational numbers and allows nonterminating nonrecurring decimal fractions, such as it and square roots, to be expressed. There are also the imaginary numbers, which are purely thought up. These may be meaningless in some contexts but not in others. Physics and engineering calculations are made easier through the application of imaginary numbers.
RAMANUJAN’S NOTEBOOKS
Srinivasa Ramanujan (1887–1920) grew up in a poor family in a small South Indian town. Self-taught, he gained most of his early knowledge from two books, which included over 6,000 theorems. Later, he was able to prepare his own set of propositions which included insights into pi, prime numbers, and partition functions.
He spent five years at Cambridge University’s Trinity College, but due to health problems he returned to India where he died one year later. He left behind over 600 formulas, without proofs, in a notebook that caused a great sensation when it was rediscovered in 1976
COMPUTING LAWS
Commutativity: the factors are exchangeable in multiplication, and the addends are exchangeable in addition.
Associativity: The order of addends or factors is immaterial, but the calculations that are within brackets are always done first.
Distributivity: When an equation uses both multiplication and addition, every addend in a bracket must be multiplied by the factor outside the bracket. Multiplication can take place after common factors are combined.
This principle forms the basis of the binomial theorem.