important maths formulas

Average formula: 

Let a₁,a₂,a₃,......,a_n be a set of numbers, average = (a₁ + a₂ + a₃,+......+ a_n)/n

Fractions formulas: 

Converting an improper fraction to a mixed number:

Formula for a proportion: 

In a proportion, the product of the extremes (ad) equal the product of the means(bc), 

Thus, ad = bc

Percent: 

Percent to fraction: x% = x/100

Percentage formula: Rate/100 = Percentage/base

Rate: The percent. 
Base: The amount you are taking the percent of.
Percentage: The answer obtained by multiplying the base by the rate

Consumer math formulas: 

Discount = list price × discount rate

Sale price = list price − discount

Discount rate = discount ÷ list price

Sales tax = price of item × tax rate

Interest = principal × rate of interest × time

Tips = cost of meals × tip rate

Commission = cost of service × commission rate

Geometry formulas: 

Perimeter:

Perimeter of a square: s + s + s + s 
s:length of one side

Perimeter of a rectangle: l + w + l + w
l: length
w: width

Perimeter of a triangle: a + b + c
a, b, and c: lengths of the 3 sides

Area:

Area of a square: s × s 
s: length of one side

Area of a rectangle: l × w
l: length
w: width

Area of a triangle: (b × h)/2
b: length of base
h: length of height

Area of a trapezoid: (b₁ + b₂) × h/2
b₁ and b₂: parallel sides or the bases
h: length of height

volume:

Volume of a cube: s × s × s 
s: length of one side

Volume of a box: l × w × h
l: length
w: width
h: height

Volume of a sphere: (4/3) × pi × r³
pi: 3.14
r: radius of sphere

Volume of a triangular prism: area of triangle × Height = (1/2 base × height) × Height
base: length of the base of the triangle
height: height of the triangle
Height: height of the triangular prism

Volume of a cylinder:pi × r² × Height
pi: 3.14
r: radius of the circle of the base
Height: height of the cylinder

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History of angles

History

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles.

Measuring angles

The angle θ is the quotient of s and r.

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g., with a pair of compasses. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a scaling constant k (which depends on the units of measurement that are chosen):

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.

In many geometrical situations, angles that differ by an exact multiple of a full circle are effectively equivalent (it makes no difference how many times a line is rotated through a full circle because it always ends up in the same place). However, this is not always the case. For example, when tracing a curve such as a spiral using polar coordinates, an extra full turn gives rise to a quite different point on the curve.

Units

Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k in the formula above.

With the notable exception of the radian, most units of angular measurement are defined such that one full circle (i.e. one revolution) is equal to n units, for some whole number n (for example, in the case of degrees, n = 360). This is equivalent to setting k = n/2π in the formula above. (To see why, note that one full circle corresponds to an arc equal in length to the circle's circumference, which is 2πr, so s = 2πr. Substituting, we get θ = ks/r = 2πk. But if one complete circle is to have a numerical angular value of n, then we need θ = n. This is achieved by setting k = n/2π.)

• The degree, denoted by a small superscript circle (°) is 1/360 of a full circle, so one full circle is 360°. One advantage of this old sexagesimal subunit is that many angles common in simple geometry are measured as a whole number of degrees. (The problem of having all "interesting" angles measured as whole numbers is of course insolvable.) Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the following sexagesimal subunits of the "degree-minute-second" system are also in use, especially for geographical coordinates and in astronomy and ballistics:
  • The minute of arc (or MOA, arcminute, or just minute) is 1/60 of a degree. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 + 30/60 degrees, or 3.5 degrees. A mixed format with decimal fractions is also sometimes used, e.g., 3° 5.72′ = 3 + 5.72/60 degrees. A nautical mile was historically defined as a minute of arc along a great circle of the Earth.
  • The second of arc (or arcsecond, or just second) is 1/60 of a minute of arc and 1/3600 of a degree. It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 7/60 + 30/3600 degrees, or 3.125 degrees.

θ = s/r rad = 1 rad.

• The radian is the angle subtended by an arc of a circle that has the same length as the circle's radius (k = 1 in the formula given earlier). One full circle is 2π radians, and one radian is 180/π degrees, or about 57.2958 degrees. The radian is abbreviated rad, though this symbol is often omitted in mathematical texts, where radians are assumed unless specified otherwise. The radian is used in virtually all mathematical work beyond simple practical geometry, due, for example, to the pleasing and "natural" properties that the trigonometric functions display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the SI system.

• The mil is approximately equal to a milliradian. There are several definitions.

• The full circle (or revolution, rotation, full turn or cycle) is one complete revolution. The revolution and rotation are abbreviated rev and rot, respectively, but just r in rpm (revolutions per minute). 1 full circle = 360° = 2π rad = 400 gon = 4 right angles.

• The right angle is 1/4 of a full circle. It is the unit used in Euclid's Elements. 1 right angle = 90° = π/2 rad = 100 gon.

• The angle of the equilateral triangle is 1/6 of a full circle. It was the unit used by the Babylonians, and is especially easy to construct with ruler and compasses. The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. One Babylonian unit = 60° = π/3 rad ≈ 1.047197551 rad.

• The grad, also called grade, gradian, or gon is 1/400 of a full circle, so one full circle is 400 grads and a right angle is 100 grads. It is a decimal subunit of the right angle. A kilometer was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon is used mostly in triangulation.

• The point, used in navigation, is 1/32 of a full circle. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is called "boxing the compass." 1 point = 1/8 of a right angle = 11.25° = 12.5 gon.

• The astronomical hour angle is 1/24 of a full circle. The sexagesimal subunits were called minute of time and second of time (even though they are units of angle). 1 hour = 15° = π/12 rad = 1/6 right angle ≈ 16.667 gon.

• The binary degree, also known as the binary radian (or brad), is 1/256 of a full circle. The binary degree is used in computing so that an angle can be efficiently represented in a single byte.

• The grade of a slope, or gradient, is not truly an angle measure (unless it is explicitly given in degrees, as is occasionally the case). Instead it is equal to the tangent of the angle, or sometimes the sine. Gradients are often expressed as a percentage. For the usual small values encountered (less than 5%), the grade of a slope is approximately the measure of an angle in radians.

some shapes

Equilateral Triangle	Equilateral triangles have all angles equal to 60° and all sides equal length. All equilateral triangles have 3 lines of symmetry.
Isoscles Triangle	Isosceles triangles have 2 angles equal and 2 sides of equal length. All isosceles triangles have a line of symmetry.
Scalene Triangle	Scalene triangles have no angles equal, and no sides of equal length.
Right Triangle	Right triangles (or right angled triangles) have one right angle (equal to 90° ).
Obtuse Triangle	Obtuse triangles have one obtuse angle (an angle greater than 90° ). The other two angles are acute (less than 90° ).
Acute Triangle	Acute triangles have all angles acute.

Is an Equilateral triangle a special case of an Isosceles triangle?

This means that there is some dispute as to whether an equilateral triangle is a special case of an isosceles triangle or not!

Most modern textbooks include use the 'at least' definition for isosceles triangles.

List of Geometric Shapes - Quadrilaterals

A quadrilateral is a polygon with 4 sides.

Quadrilaterals are also sometimes called quadrangles or tetragons.

There are quite a few members of the quadrilateral family. There are also some members which are a subset of other members of this family!
See below if this confuses you!

Square	Squares have 4 equal sides and 4 right angles. They have 4 lines of symmetry. All squares belong to the rectangle family. All squares belong to the rhombus family. All squares are also parallelograms.
Rectangle	Rectangles have 4 sides and 4 right angles. They all have 2 lines of symmetry (4 lines if they are also a square!) All rectangles belong to the parallelogram family.
Rhombus	Rhombuses (rhombii) have 4 equal sides. Both pairs of opposite sides are parallel. They all have 2 lines of symmetry (4 lines if they are a square!) All rhombuses belong to the parallelogram family.
Parallelogram	Parallelograms have 2 pairs of parallel sides. Some parallelograms have lines of symmetry (depending on whether they are also squares, rectangles or rhombuses), but most do not.
Trapezoid US (Trapezium UK)	Trapezoids US (Trapeziums UK) have one pair of parallel sides. Some trapezoids have a line of symmetry. Please note the differences between the definitions for US and UK.
Kite	Kites have 2 pairs of equal sides which are adjacent to each other.
Trapezium US (Trapezoid UK)	Trapeziums US (Trapezoids UK) are quadrilaterals with no parallel sides. Please note the differences between the definitions for US and UK.

how to use maths in daily life

At Home

Some people aren't even out of bed before encountering math. When setting an alarm or hitting snooze, they may quickly need to calculate the new time they will rise. Or they might step on a bathroom scale and decide that they’ll skip those extra calories at lunch. People on medication need to understand different dosages, whether in grams or milliliters. Recipes call for ounces and cups and teaspoons — all measurements, all math. And decorators need to know that the dimensions of their furnishings and rugs will match the area of their rooms.

In Travel

Travelers often consider their miles per gallon when fueling up for daily trips, but they might need to calculate anew when faced with obstructionist detours and consider the additional cost in miles, time and money. Air travelers need to know departure times and arrival schedules. They also need to know the weight of their luggage, unless they want to risk some hefty baggage surcharges. Once on board, they might enjoy some common aviation-related math such as speed, altitude and flying time.

At School and Work

Students can’t avoid math. Most take it every day. However, even in history and English classes they may need to know a little math. Whether looking at time expanses of decades, centuries or eras or calculating how they’ll bring that B in English to an A, they’ll need some basic math skills. Jobs in business and finance may require sophisticated knowledge of how to read profit and earning statements or how to decipher graph analyses. However, even hourly earners will need to know if their working hours multiplied by their rate of pay accurately reflects their paychecks.

At the Store

Whether buying coffee or a car, basic principles of math are in play. Purchasing decisions require some understanding of budgets and the cost and affordability of items from groceries to houses. Short-term decisions may mean only needing to know cash on hand, but bigger purchases may require knowledge of interest rates and amortization charts. Finding a mortgage may be much different from choosing a place to have lunch, but they both cost money and require math.

measurement history

Length

Length is the most necessary measurement in everyday life, and units of length in many countries still reflect humanity's first elementary methods. 

The inch is a thumb. The foot speaks for itself. The yard relates closely to a human pace, but also derives from two cubits (the measure of the forearm). The mile is in origin the Roman mille passus - a 'thousand paces', approximating to a mile because the Romans define a pace as two steps, bringing the walker back to the same foot. With measurements such as these, it is easy to explain how far away the next village is and to work out whether an object will get through a doorway. 

For the complex measuring problems of civilization - surveying land to register property rights, or selling a commodity by length - a more precise unit is required. 

The solution is a rod or bar, of an exact length, kept in a central public place. From this 'standard' other identical rods can be copied and distributed through the community. In Egypt and Mesopotamia these standards are kept in temples. The basic unit of length in both civilizations is the cubit, based on a forearm measured from elbow to tip of middle finger. When a length such as this is standardized, it is usually the king's dimension which is first taken as the norm.

      
Weight

For measurements of weight, the human body provides no such easy approximations as for length. But nature steps in. Grains of wheat are reasonably standard in size. Weight can be expressed with some degree of accuracy in terms of a number of grains - a measure still used by jewellers. 

As with measurements of length, a lump of metal can be kept in the temples as an official standard for a given number of grains. Copies of this can be cast and weighed in the balance for perfect accuracy. But it is easier to deceive a customer about weight, and metal can all too easily be removed to distort the scales. An inspectorate of weights and measures is from the start a practical necessity, and has remained so. 

      
Volume

Among the requirements of traders or tax collectors, a reliable standard of volume is the hardest to achieve. Nature provides some very rough averages, such as goatskins. Baskets, sacks or pottery jars can be made to approximately consistent sizes, sufficient perhaps for many everyday transactions. 

But where the exact amount of any commodity needs to be known, weight is the measure more likely to be relied upon than volume. 

      
Time

Time, a central theme in modern life, has for most of human history been thought of in very imprecise terms. 

The day and the week are easily recognized and recorded - though an accurate calendar for the year is hard to achieve. The forenoon is easily distinguishable from the afternoon, provided the sun is shining, and the position of the sun in the landscape can reveal roughly how much of the day has passed. By contrast the smaller parcels of time - hours, minutes and seconds - have until recent centuries been both unmeasurable and unneeded. 

trignometry history

History of Trigonometry Outline

Trigonometry is, of course, a branch of geometry, but it differs from the synthetic geometry of Euclid and the ancient Greeks by being computational in nature. For instance, Proposition I.4 of the Elements is the angle-side-angle congruence theorem which states that a triangle is determined by any two angles and the side between them. That is, if you want to know the remaining angle and the remaining two sides, all you have to do is lay out the given side and the two angles at its ends, extend the other two sides until they meet, and you've got the triangle. No numerical computations involved.

But the trigonometrical version is different. If you have the measurements of the two angles and the length of the side between them, then the problem is to compute the remaining angle (which is easy, just subtract the sum of the two angles from two right angles) and the remaining two sides (which is difficult). The modern solution to the last computation is by means of the law of sines. Details are at Dave's Short Trig Course, Oblique Triangles.

All trigonometrical computations require measurement of angles and computation of some trigonometrical function. The modern trigonometrical functions are sine, cosine, tangent, and their reciprocals, but in ancient Greek trigonometry, the chord, a more intuitive function, was used.

Trigonometry, of course, depends on geometry. The law of cosines, for instance, follows from a proposition of synthetic geometry, namely propositions II.12 and II.13 of the Elements. And so, problems in trigonometry have required new developments in synthetic geometry. An example is Ptolemy's theorem which gives rules for the chords of the sum and difference of angles, which correspond to the sum and difference formulas for sines and cosines.

The prime application of trigonometry in past cultures, not just ancient Greek, is to astronomy. Computation of angles in the celestial sphere requires a different kind of geometry and trigonometry than that in the plane. The geometry of the sphere was called "spherics" and formed one part of the quadrivium of study. Various authors, including Euclid, wrote books on spherics. The current name for the subject is "elliptic geometry." Trigonometry apparently arose to solve problems posed in spherics rather than problems posed in plane geometry. Thus, spherical trigonometry is as old as plane trigonometry.

The Babylonians and angle measurement

The Babylonians, sometime before 300 B.C.E. were using degree measurement for angles. The Babylonian numerals were based on the number 60, so it may be conjectured that they took the unit measure to be what we call 60°, then divided that into 60 degrees. Perhaps 60° was taken as the unit because the chord of 60° equals the radius of the circle, see below about chords. Degree measurement was later adopted by Hipparchus.

The Babylonians were the first to give coordinates for stars. They used the ecliptic as their base circle in the celestial sphere, that is, the crystal sphere of stars. The sun travels the ecliptic, the planets travel near the ecliptic, the constellations of the zodiac are arranged around the ecliptic, and the north star, Polaris, is 90° from the ecliptic. The celestial sphere rotates around the axis through the north and south poles. The Babylonians measured the longitude in degrees counterclockwise from the vernal point as seen from the north pole, and they measured the latitude in degrees north or south from the ecliptic.

Algebra history

The history of algebra began in ancient Egypt and Babylon, where people learned to solve linear (ax = b) and quadratic (ax² + bx = c) equations, as well as indeterminate equations such as x² + y² = z², whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equationsby essentially the same procedures taught today. They also could solve  some indeterminate equations.

The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus's book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the "science of restoration and balancing." (The Arabic word for restoration, al-jabru, is the root of the word algebra.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x² + y² = z², and xz =y².

Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and work out the basic algebra of polynomials (without yet using modern symbolism). This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of Al-Khwarizmi's Algebra appeared in the 12th century. In the early 13th century, the great Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the cubic equation x³ + 2x² + cx = d. Because Fibonacci had traveled in Islamic lands, he probably used an Arabic method of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano solved the general cubic equation in terms of the constants appearing in the equation. Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree (see quartic equation), and as a result, mathematicians for the next several centuries tried to find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician Evariste Galois proved that no such formula exists.

An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. As a result of this development, Book III of La géometrie (1637), written by the French philosopher and mathematician René Descartes, looks much like a modern algebra text. Descartes's most significant contribution to mathematics, however, was his discovery of analytic geometry, which reduces the solution of geometric problems to the solution of algebraic ones. His geometry text also contained the essentials of a course on the theory of equations, including his so-called rule of signs for counting the number of what Descartes called the "true" (positive) and "false" (negative) roots of an equation. Work continued through the 18th century on the theory of equations, but not until 1799 was the proof published, by the German mathematician Carl Friedrich Gauss, showing that every polynomial equation has at least one root in the complex plane (see  Number: Complex Numbers).

By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behavior of mathematical objects, such as complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are algebraic groups (see  Group) and quaternions, which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but they became one of the chief unifying concepts of 19th-century mathematics. Important contributions to their study were made by the French mathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, who extended the arithmetic of complex numbers to quaternions while complex numbers are of the form a + bi, quaternions are of the form a + bi + cj + dk.

Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra—also called abstract algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well.