Precalculus is the basic introduction about mathematical analysis. First we will know about Pre-calculus. Pre-calculus is required to understand concepts which are used in calculus.
Precalculus covers large number of concepts which are mentioned below.
Sets, real numbers, complex numbers, composite and polynomial Functions, trigonometry, limits, vectors, metrics, binomial theorem, mathematical induction, sequence, series, conic section, logarithm function, basic graphs, roots and rational exponents, quadratic functions, inverse functions, inequality involving Rational functions, Algebraic simplification and many more. Let's discuss some of them here.
Set: - It is a collection of different types of objects in a well defined manner. Sets involve basic operations like union, intersection, complements and cardinality product. In sets we also study cardinality of a set. As we know that if set 'P' has 'n' distinct elements for Natural Numbers 'n', then 'n' is the cardinality of set ‘P’ and ‘P’ is also a finite set. Cardinality of a set is denoted by |P|.
For example: Cardinality of set 4, 3, 2 and 1 is 4.
There are different types of sets which are shown below:
Empty set, Universal set etc:
If no element is present in a set then it is called as empty set, and a set in which all elements are present in universe of discussion is known as universal set.
Empty set is represented by symbol ‘∅’ and universal set is represented by ‘U’.
Real number:- Real numbers involve all numbers that may be Rational and Irrational Numbers, whole numbers etc.
For example: 1, 2, 5, 8, 9 these are Whole Numbers. Sometimes number 0 is also included in whole numbers.
Rational numbers – 4/5, 0.126, 0.4444…2.2 etc are Rational Numbers.
Irrational numbers – (⊼, √ 3, etc are irrational number).
If we have √ -3, this given number is not real because it contains imaginary value and ‘∞’ is also not a Real Number.
Function: - Function is used to illustrate the property between input and respective outputs.
Conic sections: - Conic section is a special type of curve, which is of three types that are hyperbola, parabola and ellipse, and circle. Circle is considered as special type of ellipse.
Limits: - In mathematics limits are initial part of calculus and limits are helpful when we define continuity, derivatives and integrals.
Mathematical induction: - Mathematical induction is used for mathematical proofs, by use of mathematical induction we can determine whether given statement is true or false for given Natural Numbers.
Trigonometry: - It deals with study of angles of triangle and its sides, trigonometry represents relation between sides and angles of a triangle.
This is all about precalculus.
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