Mathematics Worksheets

Mathematics Worksheets

• Math-Drills
• Math-Aids

Quadratic Equation Application '-Distance Speed Time

Quotes on Mathematics

Quotes on Mathematics

True beauty is a matter of maths

MATHEMATICS – The music of reason

As G H Hardy had aptly put it: “a mathematician, like a poet or a painter is a maker of patterns and that if his patterns are more permanent than theirs, it is because they are made with ideas.” 

Expectation and Fair price

Expectation and Fair price

A person spins the pointer and is awarded the amount indicated by the pointer. 



It costs $8 to play the game. Determine: 


· The expectation of a person who plays the game.


· The fair price to play the game.

My work:
E=P(wins)(amount won)+P(loss)(amount lost)
E=1/3(-6)+2/3(-8)
E=1/3*-6/1 + 2/3*-8/1
E=-2/1+-16/3
E=-6/3+-16/3
E=-22/3=7.333
E=$7.33

Fair price=expectation + cost of play
Fair price= 7.33 +8.00
Fair price= $15.33

This just doesn't seem right to me, can someone tell me where I went wrong?

Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of theintegral of a function on an interval.^[1] For many functions and practical applications, the Riemann integral can be evaluated by using the fundamental theorem of calculus or (approximately) by numerical integration.

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with theRiemann–Stieltjes integral, and most disappear with the Lebesgue integral

The integral as the area of a region under a curve.

A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function.

The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero.

Average

In colloquial language average usually means the sum of a list of numbers divided by the size of the list, in other words the arithmetic mean. However it can alternatively mean the median, themode, or some other central or typical value. In statistics, these are all known as measures of central tendency.

Pythagorean means

The three most common averages are the Pythagorean means – the arithmetic mean, the geometric mean, and the harmonic mean.

Geometric mean can be thought of as the antilog of the arithmetic mean of the logs of the numbers.

Harmonic mean

reciprocal of the arithmetic mean of the reciprocals of the a_i's:

The mode has the advantage that it can be used with non-numerical data (e.g., red cars are most frequent), while other averages cannot.


Comparison of arithmetic mean, median and mode of twolog-normal distributions with different skewness.

Mathematics play a large role in our lives

“All things are numbers,” said Pythagoras, the Greek mathematician, philosopher and mystic 2600 years ago. To him, numbers brought order and harmony to everything, from cosmos to life to music.

He was, of course, right. We see numbers all around us, and we live our daily lives in numbers: Our medical reports and vital signs come in numbers, as do our businesses and finances. We check our calories, weights, driving speed, time, temperature and weather in numbers.

The West learned algebra and algorithm from the 1200-year old work of al-Khwarizmi, a Muslim mathematician. (Algebra comes from al-jabr in Arabic, and algorithm is the misspelled name of Khwarizmi.)

Pythagorean Triples

 If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

A "Pythagorean Triple" is a set of positive integers, a, b and c that fits the rule:

a² + b² = c²

Example: The smallest Pythagorean Triple is 3, 4 and 5.

3² + 4² = 5²

9 + 16 = 25

Here is a list of the first few Pythagorean Triples:

(3,4,5)	(5,12,13)	(7,24,25)	(8,15,17)	(9,40,41)
(11,60,61)	(12,35,37)	(13,84,85)	(15,112,113)	(16,63,65)
(17,144,145)	(19,180,181)	(20,21,29)	(20,99,101)	(21,220,221)
(23,264,265)	(24,143,145)	(25,312,313)	(27,364,365)	(28,45,53)
(28,195,197)	(29,420,421)	(31,480,481)	(32,255,257)	(33,56,65)
(33,544,545)	(35,612,613)	(36,77,85)	(36,323,325)	(37,684,685)
... infinitely many more ...

Scale Them Up

The simplest way to create further Pythagorean Triples is to scale up a set of triples.

Example: scale 3,4,5 by 2 gives 6,8,10

However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 and c = √2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is irrational.

Animation demonstrating the simplest case 

of the Pythagorean Triple: 3² + 4² = 5².

Endless

The set of Pythagorean Triples is endless.

It is easy to prove this with the help of the first Pythagorean Triple, (3, 4, and 5):

Let n be any integer greater than 1, then 3n, 4n and 5n would also be a set of Pythagorean Triple. This is true because:

(3n)² + (4n)² = (5n)²

Examples:

n	(3n, 4n, 5n)
2	(6,8,10)
3	(9,12,15)
...	... etc ...

So, you can make infinite triples just using the (3,4,5) triple.

Euclid's Proof that there are Infinitely Many Pythagorean Triples

However, Euclid used a different reasoning to prove the set of Pythagorean Triples is unending.

The proof was based on the fact that the difference of the squares of any two consecutive numbers is always an odd number.

Examples:

2² - 1² = 4-1 = 3 (an odd number),

15² - 14² = 225-196 = 29 (an odd number)

And also every odd number can be expressed as a difference of the squares of two consecutive numbers. Have a look at this table as an example:

n	n2	Difference
1	1
2	4	4-1 = 3
3	9	9-4 = 5
4	16	16-9 = 7
5	25	25-16 = 9
...	...	...

And there are an infinite number of odd numbers.

There is an infinite number of odd numbers. Since the perfect squares form a subset of the odd numbers, and a fraction of infinity is also infinity, it follows that there must also be an infinite number of odd squares. Therefore, there are an infinite number of Pythagorean Triples.

Properties

It can be observed that a Pythagorean Triple always consists of:

• all even numbers, or
• two odd numbers and an even number.

A Pythagorean Triple can never be made up of all odd numbers or two even numbers and one odd number. This is true because:

• (i) The square of an odd number is an odd number and the square of an even number is an even number.
• (ii) The sum of two even numbers is an even number and the sum of an odd number and an even number is in odd number.

Constructing Pythagorean Triples

It is easy to construct sets of Pythagorean Triples.

When m and n are any two positive integers (m < n):

• a = n² - m²
• b = 2nm
• c = n² + m²

Then, a, b, and c form a Pythagorean Triple.

Example: m=1 and n=2

• a = 2² - 1² = 4 - 1 = 3
• b = 2 × 2 × 1 = 4
• c = 2² + 1² = 5

Thus, we obtain the first Pythagorean Triple (3,4,5).

Similarly, when m=2 and n=3 we get the next Pythagorean Triple (5,12,13).