Scalene Triangle

A scalene triangle has no congruent sides and no congruent angles. That is to say a scalene triangle has no sides or angles which are the same.

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Multiplying decimals

When multiplying decimals:

1. Ignore the decimal point
2. Add the number of digits to the right of the point of the numbers you are multiplying
3. Starting from the right of the answer, place that number of digits before the decimal point (add zeros if you have to)

Example:

.2 * .4

1. Ignore digits so 2*4 = 8
2. Count numbers 1 and 1, so 2 total
3. Count over two from right of answer so 0.08

Now

0.03 * 0.004

1. Ignore digits: 3*4 = 12
2. Count digits: 2 and 3 so 5 total
3. Count over: 0.00012

also 1.33 * 3.44

1. Ignore digits so 133*344 = 45752
2. Count: 2 and 2 so 4
3. Count over: 4.5752

Nice!

Topology

Topology (Greek Τοπολογία, from τόπος, “place”, and λόγος, “study”) is a major area of mathematics that has emerged through the development of concepts from geometry and set theory, such as those of space, dimension, shape, transformation and others.

Ideas that are now classified as topological were expressed as early as 1736, and toward the end of the 19th century a distinct discipline developed, called in Latin the geometria situs (“geometry of place”) or analysis situs (Greek-Latin for “picking apart of place”), and later gaining the modern name of topology. In the middle of the 20th century, this was an important growth area within mathematics.

The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms, which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line.

Topology includes many subfields. The most basic and traditional division within topology is point-set topology, which establishes the foundational aspects of topology and investigates concepts as compactness and connectedness; algebraic topology, which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology, which primarily studies manifolds and their embeddings (placements) in other manifolds. Some of the most active areas, such as low dimensional topology and graph theory, do not fit neatly in this division.

http://en.wikipedia.org/wiki/Topology

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.

http://en.wikipedia.org/wiki/Poisson_distribution

Radix and Base

In arithmetic, the radix or base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b. It is more commonly expressed as "the nth power of b", "b to the nth power" or "b to the power n". The term power strictly refers to the entire expression, but is sometimes used to refer to the exponent.

http://en.wikipedia.org/wiki/Base_(mathematics)

Sexagesimal

Sexagesimal (base-sixty) is a numeral system with sixty as the base. It originated with the ancient Sumerians in the 2000s BC, was transmitted to the Babylonians, and is still used—in modified form—for measuring time, angles, and geographic coordinates.

The number 60 has twelve factors, 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60, of which 2, 3, and 5 are prime. With so many factors, many fractions of sexagesimal numbers are simple. For example, an hour can be divided evenly into segments of 30 minutes, 20 minutes, 15 minutes, etc. 60 is the smallest number divisible by every number from 1 to 6.

http://en.wikipedia.org/wiki/Sexagesimal

Decimals

Decimal notation is the writing of numbers in a base-10 numeral system. Examples are Roman numerals, Brahmi numerals, and Chinese numerals, as well as the Arabic numerals used by speakers of English. Roman numerals have symbols for the decimal powers (1, 10, 100, 1000) and secondary symbols for half these values (5, 50, 500). Brahmi numerals had symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100 and another for 1000. Chinese has symbols for 1–9, and fourteen additional symbols for higher powers of 10, which in modern usage reach 1044.

http://en.wikipedia.org/wiki/Decimal

Proper and Improper Fractions

I had almost forgot about this distinction.

Proper fractions always have a smaller number in the numerator than in the denominator. (On top than under)

Examples of proper fractions

1/2 , 2/5 , 6/17

Improper fractions have a larger numerator than denominator like

5/2 , 7/4 , 20/17

These fractions can be rewritten with whole numbers included like

5/2 = 2(1/2)
7/4 = 1(3/4)
20/17=1(3/17)

The Golden Ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

http://en.wikipedia.org/wiki/Golden_ratio

The Quadratic Formula

The quadratic formula can help you solve any quadratic equation of the form

ax² + bx + c

To find the solutions to this equation we can use the quadratic formula which is written as follows

(-b (+or-) sqrt(b²-4ac)) / 2a

Let us consider an example of

x² + 6x + 7

a=1
b=6
c=7

so

(-6 (+or-) sqrt(6²-4*1*7)) / 2*1

=

-6 (+or-) sqrt(36 - 28) / 2

=

-6 (+or-) sqrt(8) / 2

We can simplify the square root so we get

(-6 (+or-) 2sqrt(2)) / 2

=

-3 +or- sqrt(2)

and that is our final answer.

Solving Equations with a Square Root

When we solve an equation by taking a square root, we have to consider both a positive and negative outcome.

For example consider -3 and 3

now 3² = 9 and -3²=9

Thus when we have an equation such that

x² = 9 to find x we need to take the square root of 9, however, then we have to say x= + or - 3 that is to say positive or negative 3 since it could be either and we don't know.

Multiplying and dividing square roots (radicals)

Square roots act just like other numbers when you multiply and divide them, consider the example

3sqrt(5) * 5sqrt(6) = 15sqrt(30)

Similarly

15sqrt(30) / 5sqrt(6) = 3sqrt(5)

Adding and Subtracting Radicals

When adding and subtracting radicals you treat the radicands as variables.

Example

3sqrt(5) + 4sqrt(5) = 7sqrt(5)

However, we cannot add together a radicand that is different as in

3sqrt(5) + 4sqrt(2)

The square root of a fraction

By definition we cannot derive the square root of a fraction. Thus we must find a way to get a fraction out of the radical.

Consider the square root of 1/2

sqrt(1/2)

how can we get the fraction out? We have to multiply and make the denominator a perfect square. But what we multiply to the bottom we must also multiply to the top

sqrt((1/2)*(2/2))

=

1/2*sqrt(2)

Now we have got the fraction out of the radical and created a square root we can rationalize.

Another example

sqrt(3/5)

To get the denominator out we multiply by (5/5) (which is equal to 1)

sqrt((3/5)*(5/5))

=

(1/5)*sqrt(3*5)

=

(1/5)*sqrt(15)

Simplifying Radicals

Like so much of algebra, it is good to know how to simplify radicals for purposes of canceling out or combining like terms.

Consider the square root of 27 or sqrt(27) there is no whole number that equals sqrt(27) but we can write the radical as sqrt(9*3) and that is equal to 3*sqrt(3)

Thus we have simplified the radical for purposes of mathematical calculation or canceling.

Square roots are radical

It is true, taking the square root of an expression can also be called "The radical"

We will write square root as sqrt on this website, thus

sqrt(4) = 2

and

sqrt(9x²y¹⁰) ?

To solve this, it is good to factor under the radical ( or factor the square root)

So we get

sqrt(9x²y¹⁰)

=

(3xy⁵)

Solving quadractics by factoring

One way to solve a quadratic equation like:

(4x² - 4)=0 is by factoring and setting both factors equal to zero. Because a quadratic contains a x² they often have two solutions.

(4x² - 4)=0 can be factored to

(2x - 2)(2x + 2)=0

Now set both factors equal to zero

2x-2=0
2x+2=0

we get
x=1 and x=-1
substituting 1 or -1 for x will solve the quadratic equation (4x² + 4)=0

Quadratic Equation

A quadratic equation is described as an equation where the highest exponent is 2.
The graph of a quadratic is a smooth curve known as a parabola.

All of the following are quadratic equations.
x² + 4 = 0

x² + 4x + 3 = 0

3x² + 34x + 7 = 50

Factoring a trinomial

As I said yesterday, I really think factoring comes down to trial and error till the process is internalized. As an example today we will factor trinomials.

Consider

5x² - 8x - 21

This is quite a complicated trinomial to factor, lets start with a guess

First off, we know that to get 5x² we need to multiply 5x and x, so that gives us our first two terms:

(5x + ) (x - )

As a further guess I also alternated the signs.

Now we can try guess what two numbers can multiply to give us -21. How about 7 and -3?

(5x + 7) (x - 3)

Checking with the foil method we get
5x² - 8x - 21

Factoring with the difference of squares

I think factoring is something which becomes internal, you see a problem, make a guess, and then check. The best method is trial and error till it becomes intuitive.

Still the difference of squares method is often taught, and so I will show it here.

Basically the difference of squares is always factored in the following form:

(x+y)(x-y)

Which equals (x² - y²)

Example

x² - 9

9 is a perfect square so we can use the memorized formula

(x-3)(x+3)

Again, I prefer gaining an intuitive understanding of factoring, but memorizing a rule like this can help till you gain an intuitive understanding.