MyMaths MyLife

Pure mathematics is more like art. Pure mathematicians work on building a foundation for a theory. One nice feature about pure mathematics is that it is free from argument. When a mathematician makes a discovery there is no opposition, as in science. And his theory stands the test of time, unlike science where one law is shown to be wrong in special cases. But once a foundation is build (like complex analysis) applied mathematicians take its result and use it to solve important problems.

Pure math is much more difficult. Classes in applied math consist of memorizing the steps to solve problems. However, classes in pure math involve proofs, which implies a good understanding of the subject matter is required. In pure math you need to justify everything you do. Which can sometimes make a simple argument long and complicated. It is easier for someone in pure math to learn applied math rather than someone in applied math to learn pure math.

Determinants and matrices, they look alike. Their similarities caught many unaware and results in "excitements" and much interests.

Both contain numbers within. But ......

- determinants are bounded by two straights lines whereas matrices are by square braces

- determinant resulted in a single numerical value, whereas matrices are sets of numbers grouped within the braces

- determinant can be extracted from matrix, but not the other way round

- there are inverse matrix but not inverse determinant

- a scalar multiplier affects only a single row or single column of a determinant, but affects all the numbers within a matrix

The problems relating to geometry cover mostly triangles and circles. Even though polygons also are covered, the emphasis on polygons is not as much as on triangles and circles.

An angle of 90⁰ is a right angle; an angle less than 90⁰ is an acute angle; an angle between 90⁰ and 180⁰ is an obtuse angle; and angle between 180⁰ and 360⁰ is a reflex angle.
The sum of all angles on one side of a straight line AB at a point O by any number of lines joining the line AB at O is 180⁰. When any number of straight lines join at a point, the sum of all the angles around that point is 360⁰.
Two angles whose sum is 90⁰ are said to be complementary to each other and two angles whose sum is 180⁰ are said to be supplementary angles.



When two straight lines intersect, vertically opposite angles are equal. In the figure given alongside,

Facts And Formulae:

Highest Common Factor:(H.C.F) or Greatest Common Meaure(G.C.M) :
The H.C.F of two or more than two numbers is the greatest
number that divides each of them exactly.

There are two methods :

i.Factorization method: Express each one of the given numbers as
the product of prime factors. The product of least powers of common
prime factors gives HCF.

Example : Find HCF of 26 * 32*5*74 , 22 *35*52 * 76 ,
2*52 *72
Solution: The prime numbers given common numbers are 2,5,7
Therefore HCF is 22 * 5 *72 .

ii.Division Method : Divide the larger number by smaller one. Now
divide the divisor by remainder. Repeat the process of dividing
preceding number last obtained till zero is obtained as number. The
last divisor is HCF.


Least common multiple[LCM] : The least number which is
divisible by each one of given numbers is LCM.

There are two methods for this:

i.Factorization method : Resolve each one into product of prime
factors. Then LCM is product of highest powers of all factors.

ii.Common division method.

When a straight line XY cuts two parallel line PQ and RS [as shown in figure], the following are the relationships between various angles that are formed. [M and N are the points of intersection of XY with PQ and RS respectively].

a) alternate angles are equal
i.e, angle PMN = angle MNS
angle QMN = angle MNR
b) corresponding angles are equal
i.e, angle XMQ = angle MNS
angle QMN = angle SNY
angle XMP = angle MNR
angle PMN = angle RNY
c) Sum of interior angles on the same side of cutting line is equal to 180⁰
i.e, angle QMN + angle MNS = 180⁰
angle PMN + angle MNR = 180⁰
d) Sum of exterior angles on the same side of transversal line is equal to 180⁰
i.e, angle XMQ + angle SNY = 180⁰
angle XMP + angle RNY = 180⁰
If three or more parallel lines make equal intercepts on one transversal, they make equal intercepts on any other transversal as well.

Important Facts:

Direct Proportion: Two Quantities are said to be directly
proportional, if on the increase (or decrease) of the one, the
other increases(or decreases) to the same extent.

Ex:(i) Cost is directly proportional to the number of articles.
(More articles, More cost).

(ii) Work done is directly proportional to the number of men
working on it. (More men, more work).

Indirect Proportion: Two Quantities are said to be
indirectly proportional,if on the increase of the one , the other
decreases to the same extent and vice-versa.

Ex:(i) The time taken by a car covering a certain distance is
inversely proportional to the speed of the car.(More speed,
less is the time taken to cover the distance).

(ii) Time taken to finish a work is inversely proportional to
the number of persons working at it.
(More persons, less is the time taken to finish a job).

Note: In solving Questions by chain rule, we compare every
item with the term to be found out.

We all know about cm, mm, Km ..................., but we dont know much about inches, yard .............
you might have got a question that " what is the relationship between inch and cm, inch and yard"...........
This is going to list all the conversions of such units

LINEAR

Inch=Basic Unit

1 hand = 4 inches
1 link = 7.92 inches
1 span = 9 inches
1 foot = 12 inches
1 yard = 3 feet
1 fathom = 2 yards
1 rod = 5.5 yards
1 chain = 100 links=22 yards
1 furlong = 20 yards
1 mile = 1760 yards
1 knot mile= 6076.1155 feet
1 league = 3 miles
1 inch = 2.54 cm

Cartesian Product:

Let A and B be any two sets. Then the Cartesian product of A and B is the set of all ordered pairs of the form (a, b), where aЄA and bЄB
The product is denoted by A×B
A×B = {(a, b)/ aЄA, bЄB }

Example
A = {a, b, c} and B={1,2}, then
A×B = {(a, 1), (b, 1), (c, 1), (a, 2), (b, 2), (c, 2)}
B×A = {(1, a), (2, a), (1, b), (2, b), (1, c), (2, c)}

Mathematical concept means just about anything with a mathematical name. For example, some of the mathematical concepts we learn in high school are: constant, variable, polynomial, factor, factoring, equation, solving an equation, logarithm, sine, cosine, tangent, etc., point, line, triangle, square, and other geometric figures, area, perimeter of a geometric figure, etc., and many others. Among the mathematical concepts we learn in our first years of college mathematics are: set, operation, limit, function, and, specifically, continuous function, derivative, integral, theorem, proof, countable infinity, uncountable infinity, algebra, linear algebra, vector space, group, ring, field, and many others.

Now one thing that makes the understanding of these concepts difficult is that they are defined in terms of other concepts.

Thus, e.g., a vector space is defined in terms of the concepts of vector, set, function, abelian group, field, and others. How does the typical mathematics textbook, and mathematics course, deal with this fact? It attempts to teach the concepts in logical order, i.e., it assumes that, e.g., when you begin your study of vector spaces, you will already know — through having remembered what you learned in previous courses — the meaning of each of the concepts in terms of which a vector space is defined. And, indeed, one of the things that makes mathematics such a frightening subject to many students, is the grandiose manner with which these assumptions are set forth in the list of prerequisites for the course.

Combinations:
Each of different groups or selections which can be formed by
taking some or all of a number of objects,is called a combination.
eg:- Suppose we want to select two out of three boys A,B,C .
then ,possible selection are AB,BC & CA.
Note that AB and BA represent the same selection.

Number of Combination:
The number of all combination of n things taken r at a time is:
nCr = n! / (r!)(n-r)!
= n(n-1)(n-2). . . . . . . tor factors / r!
Note: nCn = 1 and nC0 =1

An Important Result:
nCr = nC(n-r)

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