History of mathematical functions
- In 17 century Rene Descartes (1596-1650) used this concept in his book La Geometrie (Geometry)(1637) to describe mathematical relations
- Gottfried Wilhelm Leibniz (1646-1716) introduced the term “function” about fifty years after the publication of Geometry in 1637.
- Johann Bernoulli used the greek letter phi to denote the character of the function in 1718. the terms lx rx sx (sinx) were prevalent in Bernoulli’s time after that. lx for logarithmic function, rx for root functions, and sx for sin function.
- Leonhard Euler (pronounced “oiler” 1707-1783) introduced the notation for a function, f x instead of phi of greek he used Latin f and thus formalized the concept of mathematical functions. Even he did not use f(x) with the parenthesis.
- Then the definition of a function went all over the place till in 1923 Goursat gave the definition: One says that y is a function of x if to a value of x corresponds a value of y. One indicates this correspondence by the equation y = f(x).
- Patrick Suppes in 1960:-
Definition. A is a relation ⇔ (∀x)(x ∈ A => (∃y)(∃z)(x = (y, z)). We write y A z if (y, z) ∈ A.
Definition. f is a function ⇔ f is a relation and (∀x)(∀y)(∀z)(x f y and x f z) => y = z.
Functions
- Introduction
- history
- etymology
- definition
- explanation
- Notation
- Representation
- Graph of a function
- Vertical line test
- Horizontal line test
- terminology
- domain
- codomain
- range
- General properties of functions
- Value of a function
- evaluation of a function
- function from equation
- Types of functions
- Injective
- Surjective
- Bijective
- Some basic functions
- Constant function
- Identity function
- Power functions
- Square function
- Cube function
- Polynomial functions
- Linear function
- Quadratic function
- Cubic function
- Radical functions
- Square root function
- Cube root function
- Rational function
- Exponential function
- Logarithmic function
- Trigonometric function
- Algebra of functions
- Inverse functions
- Piecewise defined functions
- Signum function
- Modulus function
- Step function
- Fractional part function
Circle
Topics on Circle
- Introduction
- history
- etymology
- definition
- explanation
- Terminology
- Lines
- curves
- sections
- Analytics
- Area of a circle
- Equations
- equations of a circle
- standard form
- center radius form
- diameter form
- general equation of a circle
- parametric form
- equations of a circle
- Tangent
- general equation of a tangent for a standard form equation of a circle at P(x1,y1)
- equation of a tangent in parametric form
- conditions of tangency
- tangents from a point to the circle
- Director circle
- Properties
- Chord
- Tangent
- Theorems
- Inscribed angles
- Sagitta
7. Director circle
8. Specially named circles
9. Construction of circles
Sets topics for std 11th
1.Introduction
- history
- etymology
- definition
- explanation
2. Representation
3. Types
- Empty set
- Finite set
- Infinite set
- Equal sets
- Equivalent sets
- Disjoint sets
- Subsets
- Proper subsets
- Subsets of R
- Intervals as subsets of R
- Power sets
- Universal sets
- Venn Diagrams
- Operations on sets
- Union
- Intersection
- Difference
- Complement
- Union & Intersection of 2 sets
List of topics in mathematics 11th std
11th std
—Pure Maths
I ) Structure
Algebra
[pre calculus]
- Sets
- Relations & Functions
- Trigonometric Functions
- Mathematical Induction
- Complex Numbers
- Quadratic Equations
- Linear Inequalities
Change
[pre-calculus]
- Limits
- Derivatives
Discrete Maths
- Permutation and Combinations
- Binomial theorem
- Sequences & Series
Space
- Straight lines
- Circle
- Conic Sections
- 3D Geometry
Philosophy
- Mathematical Reasoning
— Applied Maths
Decision Sciences
- Statistics
- Probability
Pythagoras’ theorem 1
This is a famous theorem and has many interesting proofs. It is known as Pythagoras’ theorem but it was known by the Greeks, Chinese, Indians, Arabs, Persians way before the Greek mathematician Pythagoras’ birth but it is his name which is carried by this theorem.
Wait, we learnt about notions, axioms, and postulates, now what is a theorem and proof? Let us figure out a few of these terms before proceeding with Pythagoras’ theorem also known as Pythagorean theorem.
Definition: A concise and precise explanation of a mathematical word.
Theorem: An important true statement.
Proof: An explanation of why a statement is true or false.
Lemma: A true statement used to prove another statement to be true.
Axiom: A basic assumption about a mathematical statement.
Converse: To switch the hypothesis with the conclusion.
And now back to ‘an important true statement’, known as the Pythagorean theorem.
Pythagorean theorem
In a right triangle, the sum of the squares of the legs (a & b) of the right angle is equal to the square of the hypotenuse (c).
a^2 + b^2= c^2
The Converse of Pythagorean theorem:
If the sum of the squares of the legs (a & b) of the right angle is equal to the square of the hypotenuse (c)in a triangle, then the triangle is a right angled triangle.
So the converse of Pythagorean theorem is also true.
A corollary to the converse of Pythagorean theorem:
Let c be chosen to be the longest of the three sides and a + b > c (otherwise there is no triangle according to the triangle inequality). The following statements apply:
- If a2 + b2 = c2, then the triangle is right.
- If a2 + b2 > c2, then the triangle is acute.
- If a2 + b2 < c2, then the triangle is obtuse.
Uses of Pythagorean theorem:
1> To find the length of hypotenuse of a right angled triangle when the length of the other two sides are known.
2> To find the length of any side of a right angled triangle when the length of the other two sides are known.
3> To find out if a triangle is a right angled triangle,acute or obtuse.
Apart from these uses this theorem has many uses in higher level maths, architecture, physics etc.
Triangle 1
We are in Euclidean geometry also called plane geometry. It has two dimensions like the plane. We are going to study a polygon called triangle. But wait what is a polygon.
Polygon
Poly = many , gon = sides in Greek.
Figures made by joining line segments end to end making corners (vertices) with enclosed space are called polygons. There are many types of polygons. One of them is a triangle.
Triangle
Tri = three, angle = corner
Definition 1> A polygon with three sides and three vertices.
Definition 2> Three line segments joined end to end making three vertices completing a closed figure.
Properties of a triangle
Base: Bottom side of a triangle.
notice: Any side can be chosen as the base. Usually the unequal side is chosen as the base in isosceles triangles. And in some convention the biggest side is taken as the base.
Altitude: A perpendicular from the base to the opposite vertex is called the altitude. Sometimes it is called the height of the triangle.
Notice: The base may need to be extended in some cases.
Any of the three sides can be taken as the base so three altitudes are possible.
Orthocenter : The point of concurrency made by three altitudes in a triangle is known as orthocenter of the triangle.
Median: A line from a vertex to the midpoint of the opposite side is called the median. There are three medians possible in a triangle.
Centroid: The point of concurrency made by three medians is called centroid of the triangle.
Interior Angles: Angles formed inside the triangle at the vertices.
Exterior Angles: Angles formed outside the triangle at the vertices.
Area of triangle: Space enclosed by the sides of the triangle.
Perimeter of triangle: Sum of the lengths of all three sides of the triangle.
notice: Shortest side is opposite to the smallest interior angle.
Longest side is opposite to the biggest interior angle.
Mid sized side is opposite to the mid sized interior angle.
notice 2:
1> The sum of all three interior angles is 180 degrees.
2> The sum of all exterior angles is 360 degrees.
Types of triangles
1>Equilateral triangle: All sides are of same length.
2>Isosceles: <i-sos-elles> Two sides have same length.
3> scalene: No sides have same length.
4> Right triangle: One angle is 90 degrees.
5> Acute triangle: all angles less than 90 degrees.
6> Obtuse triangle: One angle more than 90 degrees.
7> Equiangular triangle: All interior angles are same.
Notice: Equilateral and equiangular triangles are same triangles.
They have 60 degree interior angles.
Representation
Graphically a triangle is presented by drawing a triangle. The vertices are usually named by capital alphabets e.g. A, B etc. And sides are named by same alphabets corresponding to the opposite vertex but in small letters. e.g., a,b etc.
Euclidean Geometry 3
In this part we study the third ‘undefined term’ of EG.
PLANE
Imagine a flat surface which is infinitely large and has no thickness at all. A closest example is the top surface of a paper without the thickness of the paper and never ending on all four sides.
Now that we are moving in the fundamentals of maths and geometry it is time i think we talk about some notions ..
Notion
Point line and plane make up the ‘undefined terms’ of EG. Undefined terms are also known as ‘common notions’. Notion means an idea or a mental image. Notions are a general understanding.
Axioms
Axiom is the starting point of reasoning. A premise. Axiom is a premise so evident as to be accepted without controversy.
Postulates
Same as axiom but historically axiom was used for a premise across subjects and postulate was used for a requisite to use a premise in a specific topic or subject.
So coming back to the notion of plane let us attempt to define it.
Definition 1 > Infinite lines lying adjacently continuously and flatly making up a continuous flat surface is called a plane.
Definition 2> An infinite flat surface in two dimensions with zero thickness.
Determining plane using points and lines when the plane is embedded in 3d space .
1> Three non co linear points make a plane.
2> A line and a point not on that line make a plane.
3> When two distinct lines are perpendicular to the same plane then they are parallel to each other.
4> When two distinct planes are perpendicular to the same line then they are parallel to each other.
Intersections:
Intersecting planes have a line of intersection.
At the intersection of a line and a plane there will be a point of intersection.
Characteristics of Plane
1> A plane has two dimensions. Length and width.
2> A plane has no thickness (or height).
3> It spreads to infinity in both the dimensions and in all four directions.
Representation of plane.
Graphically it is represented by a square or a rectangle. Representation has edge lines but one is to think about it like it has no end edges.
Name wise it is usually represented by a capital alphabetical letter in cursive writing. A plane ℘.
And that brings the end of euclidean geometry’s introduction in three parts.
Elements of Geometry 2
In part one we covered spatial dimensions and ‘undefined term’ called ‘point’. Here we move on to second ‘undefined term’ in Euclidean Geometry called ‘line’.
Line
When we say line we mean a straight line and not a curved line at all.
Definition 1> A set of points joined adjacently continuously and in only one dimension but infinitely is called a line.
Definition 2> A set of all points equidistant from two separate points makes up a line.
Definition 3> A straight path connecting any two points and extending infinitely in both directions is called a line.
Definition 4> Shortest path between any two points and having infinite extension in both directions is a line.
Definition 5> When a point moves in one dimension but in both directions its path if extended to infinity in both directions will trace a line.
Characteristics of a line
1> A line has only one dimension. Length.
2> A line is infinite and has no end points.
3> Only one line can pass through any two points.
4> A line does not have any width or thickness.
Parallel postulate involving point and line
Through a point not on a line, there can be no more than one line parallel to that line.
Representation of line
Graphically a line is represented by a line with arrow heads at both ends.
Line is either named with a single small alphabetical letter like ‘l’, or by a pair of capital alphabetical letters like AB with a line above the letters with arrows on both ends.
Defining other geometric figures using point and line.
1> Vertical line: A line running up and down.
2> Horizontal line: A line running right and left.
Note: Vertical lines and horizontal lines meet at right angles.
3> Intersecting lines: Lines which cross each other.
4> Parallel lines: Lines which do not cross each other, ever.
5> Ray: A line with one end point. (It is no more a line.)
6> Segment/ Line segment: A line with two end point. (It is no more a line.)
7> Segment: A line with finite length. (it is no more a line)
8> Segment: A part of a line.
9> Perpendicular lines: Lines which meet or cross each other at right angles.
10>Normal: A line which is perpendicular to another line is normal to that line. (same as perpendicular.).
11> Mid point: A point which divides a line segment in two equal parts.
Notes: A line or a ray cannot have a midpoint.
12> Congruent lines: Line segments having same length.
13> Concurrent lines: When three or more lines intersect at the same point on a line they are called concurrent line.
14> Point of concurrency: The point where three or more lines intersect is called point of concurrency.
15> Bisector: A line, a line segment or a ray passing through the midpoint of a line segment is called a bisector.
16> Perpendicular bisector: A bisector forming a right angle with the line segment is called a bisector.
17> Vertex: Point where end points of two line segments meet.
So there we are at the end of line..
Next part 3 would cover Euclidean Geometry plane
mathelogy 