mathematics is fun: 2016

Thursday 8 December 2016

measure of central tendency

MEASURE OF CENTRAL TENDENCY

Single value in series of observation which indicate the characteristics of observation.

All data/values clustered around it and used to compare between one series to another.

Measures:

a. Mean (Arithmetic/ Geometric/Harmonic)

b. Median

c. Mode

MEDIAN

Median is what divides the scores in the distribution into two equal parts.

Fifty percent(50%) lies below the median value and 50% lies above the median value.

It is also known as the middle score or 50th percentile.

MEDIAN OF UNGROUPED DATA

•Arrange the scores(from lowest to highest or highest to lowest)

•Determine the middle score in a distribution if n is an odd number and get the average of the two middle most scores if n is an even number.

EXAMPLE

•Find the median score of 7 students in an English class.

x(score)

MEDIAN OF GROUPED DATA

Formula:

Median= l + ( (n/2-m)/f)*c

Where l=Lower limit of the median class

f=Frequency of the median class

c=Width of the median class

N=The total frequency

m=cumulative frequency of the class

preceeding the median class

PROPERTIES OF MEDIAN

•It may not be an actual observation in the data set.

•It can be applied in ordinal level.

•It is not affected by extreme values because median is a positional measure.

APPLICATION OF MEDIAN

•It is used to measure the distribution of the earning.

•Use to find the players height e.g. Football players.

•To find the middle age from the classroom students.

•Also used to find the poverty line.

MODE

The mode or the modal score is a score or scores that occurs in the distribution.

It is classified as unimodal, bimodal, trimodal or multimodal.

CLASSIFICATION OF MODE

•Unimodal is a distribution of scores that consists of only one mode.

•Bimodal is a distribution of scores that consist of two modes.

•Trimodal is a distribution of scores that consist of three modes or multimodal is a distribution of scores that consist of more than two modes.

MODE FOR GROUPED DATA

Formula:

Mode= l+ ( f-f1 /2f-f1-f2) *c

Where l=lower limit of the modal class

f=frequency of modal class

c=class width of the modal class

f =frequency of the class just preceeding the

modal class

f2=frequency of the class succedding the

modal class

PROPERTIES OF MODE

•It can be used when the data are qualitative as well as quantitative.

•It may not be unique.

•It is affected by extreme values.

It may not exist

APPLICATION OF MODE

•It is used to influx of the public transport.

•The number of games succeeded by any team of players.

•The frequency of the need of infants.

•Used to find the number of the mode is also seen in calculation of the wages, in the patients going to the hospitals, mode of travels, etc.,

CONCLUSION

• A measure of central tendency is a measure tell us where the middle of a bunch of data lies.

•Median is the number of present in the middle when the number in a set of data are arranged in ascending or descending order.

•Mode is the value that occurs most frequently in a set of data.

Thursday 1 December 2016

Different meanings of "algebra"

The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

As a single word without an article, "algebra" names a broad part of mathematics.
As a single word with an article or in plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication. When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations.
With a qualifier, there is the same distinction:

Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra(the study of the algebraic structures for themselves).
With an article, it means an instance of some abstract structure, like a Lie algebra, an associative algebra, or avertex operator algebra.
Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers.

Algebra as a branch of mathematics

Algebra began with computations similar to those of arithmetic, with letters standing for numbers.^[8] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation

ax^{2}+bx+c=0,

a,b,c

can be any numbers whatsoever (except that

a

cannot be

0

), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity

x

which satisfy the equation. That is to say, to find all the solutions of the equation.

Historically, and in current teaching, the study of algebra starts with the solving of equations such as the quadratic equationabove. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions lead to ideas of form, structure and symmetry.^[10] This development permitted algebra to be extended to consider non-numerical objects, such asvectors, matrices, and polynomials. The structural properties of these non-numerical objects were then abstracted to definealgebraic structures such as groups, rings, and fields.

Before the 16th century, mathematics was divided into only two subfields, arithmetic and geometry. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of infinitesimal calculus as subfields of mathematics only dates from the 16th or 17th century. From the second half of 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.

Today, algebra has grown until it includes many branches of mathematics, as can be seen in the Mathematics Subject Classification^[11] where none of the first level areas (two digit entries) is called algebra. Today algebra includes section 08-General algebraic systems, 12-Field theory and polynomials, 13-Commutative algebra, 15-Linear and multilinear algebra;matrix theory, 16-Associative rings and algebras, 17-Nonassociative rings and algebras, 18-Category theory; homological algebra, 19-K-theory and 20-Group theory. Algebra is also used extensively in 11-Number theory and 14-Algebraic geometry.

Wednesday 23 November 2016

AREA FORMULA

AREA OF PLANE FIGURE

DEFINITION:

The area of an object is the space occupied by it on a plane surface.

FORMULAE OF PLANE FIGURE:

1)AREA OF RECTANGLE=( l*b ) sq. units

where l is length and

b is breadth

2)AREA OF SQUARE= (a*a) sq. units

where a is the side

3)AREA OF RIGHT TRIANGLE= 1/2*(b*h) sq. units

where b is base and

h is height

4)AREA OF QUADRILATERAL=1/2*d*(h1*h2) sq. units

where d is the length of a diagonal and

h1 and h2 are perpendiculars drawn to the diagonal from the opposite vertives

5)AREA OF PARALLELOGRAM=bh sq. units

where b is the base and

h is the height

6)AREA OF RHOMBUS=1/2*(d1*d2) sq. units

where d1 and d2 are diagonals

7)AREA OF TRAPEZIUM=1/2*h(a+b) sq.units

where h is height and

a and b are sum of the parallel sides

8)AREA OF CIRCLE= $π(r*r)sq. units$

$where r is the radius of the semi-circle$

$9)AREA OF SEMICIRCLE=$ $π(r*r)/2 sq. units$

$where r is the radius of the circle$

$10)AREA OF A QUADRANT=1/4*$ $π*(r*r) sq. units$

$where r is the radius$

Thursday 17 November 2016

Formula of trignometry

TRIGNOMETRY

TRIGNOMETRIC FUNCTION:

1. Sine function(sin), defined as the ratio of the side of the opposite the angle to the hypotenuse.

$\sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.$

2. Cosine function(cos), defined as the ratio of the adjacent leg to the hypotenuse.

$\cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.$

3. Tangent function(tan), defined as the ratio of the opposite leg to the adjacent leg.

$\tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{a}{\,c\,}*\frac{c}{\,b\,}=\frac{a}{\,c\,} / \frac{b}{\,c\,}=\frac{\sin A}{\cos A}\,.$

The hypotenuse is the side opposite to the 90 degree angle in a right triangle, it is the longest side of the triangle and one of the two sides adjacent to angle A. The adjacent side is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A.

The reciprocals of these functions are named the cosecant(cosec), secant(sec) and cotangent(cot).

$\csc A={\frac {1}{\sin A}}={\frac {{\textrm {hypotenuse}}}{{\textrm {opposite}}}}={\frac {c}{a}},$

$\sec A={\frac {1}{\cos A}}={\frac {{\textrm {hypotenuse}}}{{\textrm {adjacent}}}}={\frac {c}{b}},$

$\cot A={\frac {1}{\tan A}}={\frac {{\textrm {adjacent}}}{{\textrm {opposite}}}}={\frac {\cos A}{\sin A}}={\frac {b}{a}}.$

USES OF TRIGNOMETRY:

The technique of triangulation is used in astronomy to meausre the distance to nearby stars.
In geography to measure distance between landmarks.
In satellite navigation system.
The sine and cosine functions are fundamental to the theory of periodic functions, such as those that describe sound and light waves.
Fields that use trignometric functions include music theory, audio synthesis, accoustics, optics, electronics, biology, pharmacy, chemistry, number theory, electrical engineering and game development.

PYTHAGOREAN IDENTITIES:

The identities are related to the Pythagorean theorem and hold for any values.

   $\sin^2 A + \cos^2 A = 1 \$
   $\tan ^{2}A+1=\sec ^{2}A\$
   $\cot ^{2}A+1=\csc ^{2}A\$

ANGLE TRANSFORMATION FORMULAE:

   $\sin (A \pm B) = \sin A \ \cos B \pm \cos A \ \sin B$
   $\cos (A \pm B) = \cos A \ \cos B \mp \sin A \ \sin B$
   $\tan(A\pm B)={\frac {\tan A\pm \tan B}{1\mp \tan A\ \tan B}}$
   $\cot (A \pm B) = \frac{ \cot A \ \cot B \mp 1}{ \cot B \pm \cot A }$

Thursday 10 November 2016

Mathematicians of India

GREAT MATHEMATICIANS OF INDIA

ARYABHATA:

Aryabhata was born in 476AD. He called himself a native of Kusumapura or Pataliputra(Patna). He was died on 550AD. His achievements in mathematics are place value system and zero, approximation of π, trigonometry , indeterminate equations algebra.

BRAHMAGUPTA:

Brahmagupta was born in 598CE in Bhinmal, a state of Rajasthan, in India. He was died between 660 to 670. His achievements in mathematics are linear equation, arthimaetic, sum of squares and cubes of first n integer, mentions zero as a number, Pythagorean triples, pell’s equation, brahmagupta formula for cyclic quadrilaterals, Brahmagupta’s theorem, sine table and interpolation formula.

SRINIVASA RAMANUJAN:

Srinivasa Ramanujan was born in 22nd december1887 in Erode, Madras Presidency. He was died on 26^th April 1920. His achievements in mathematics are Ramanujan stato-series, Ramanujan conjecture and Ramanujan’s notebooks.