domingo, 15 de fevereiro de 2015

Number theory

From Wikipedia, the free encyclopedia

Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (Diophantine approximation) .Number theory (or arithmetic) is a branch of pure mathematics devoted primarily to the study of the integers, sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).

The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by "number theory".^{[note 2]} (The word "arithmetic" is used by the general public to mean "elementary calculations"; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating point arithmetic.) The use of the term arithmetic for number theoryregained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is preferred as an adjective to number-theoretic.

Rules of algebra

In algebra, there are a few rules that can be used for further understanding of equations. These are called the rules of algebra. While these rules may seem senseless or obvious, it is wise to understand that these properties do not hold throughout all branches of mathematics. Therefore, it will be useful to know how these axiomatic rules are declared, before taking them for granted. Before going on to the rules, reflect on two definitions that will be given.

Opposite - the opposite of $a$ is $-a$ .
Reciprocal - the reciprocal of $a$ is $\frac{1}{a}$ .

Rules

Commutative property of addition

'Commutative' means that a function has the same result if the numbers are swapped around. In other words, the order of the terms in an equation do not matter. When the operator of two terms is an addition, the 'commutative property of addition' is applicable. In algebraic terms, this gives

a + b = b + a

Note that this does not apply for subtraction! (i.e.

a - b \ne b - a

)

Commutative property of multiplication

When the operator of two terms is an multiplication, the 'commutative property of multiplication' is applicable. In algebraic terms, this gives

a \cdot b = b \cdot a

Note that this does not apply for division! (i.e.

\frac{a}{b} \ne \frac{b}{a}

, when

a \neq b

)

Associative property of addition

'Associative' refers to the grouping of numbers. The associative property of addition implies that, when adding three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives

a + (b + c) = (a + b) + c

. Note that this does not hold for subtraction, e.g.

1 = 0 - (0 - 1) \neq (0 - 0) - 1 = -1

(see the distributive property).

Associative property of multiplication

The associative property of multiplication implies that, when multiplying three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives

a \cdot (b \cdot c) = (a \cdot b) \cdot c

. Note that this does not hold for division, e.g.

2 = 1/(1/2) \neq (1/1)/2 = 1/2

Distributive property

The distributive property states that the multiplication of a number by another term can be distributed. For instance:

a \cdot (b + c) = ab + ac

. (Do not confuse this with the associative properties! For instance,

a \cdot (b + c) \ne (a \cdot b) + c

Additive identity property

'Identity' refers to the property of a number that it is equal to itself. In other words, there exists an operation of two numbers so that it equals the variable of the sum. The additive identity property states that the sum of any number and 0 is that number:

a + 0 = a

. This also holds for subtraction:

a - 0 = a

Multiplicative identity property

The multiplicative identity property states that the product of any number and 1 is that number:

a \cdot 1 = a

. This also holds for division:

\frac{a}{1} = a

Additive inverse property

The additive inverse property is somewhat like the opposite of the additive identity property. When an operation is the sum of a number and its opposite, and it equals 0, that operation is a valid algebraic operation. Algebraically, it states the following:

a - a = 0

Multiplicative inverse property

The multiplicative inverse property entails that when an operation is the product of a number and its reciprocal, and it equals 1, that operation is a valid algebraic operation. Algebraically, it states the following:

\frac{a}{a} = 1

By Wikipedia

sexta-feira, 13 de fevereiro de 2015

Algebra and Funcions

An important part of algebra is the study of functions, since functions often appear in equations that we are trying to solve. A function is like a box you can put a number or numbers into and get a certain number out. When using functions, graphs can be powerful tools in helping us to study the solutions to equations.

A graph is a picture that shows all the values of the variables that make the equation or inequality true. Usually this is easy to make when there are only one or two variables. The graph is often a line, and if the line does not bend or go straight up-and-down it can be described by the basic formula y = mx + b. The variable b is the y-intercept of the graph (where the line crosses the vertical axis) and m is theslope or steepness of the line. This formula applies to the coordinates of a graph, where each point on the line is written (x, y).

In some math problems like the equation for a line, there can be more than one variable (x and y in this case). To find points on the line, one variable is changed. The variable that is changed is called the "independent" variable. Then the math is done to make a number. The number that is made is called the "dependent" variable. Most of the time the independent variable is written as x and the dependent variable is written as y, for example, in y = 3x + 1. This is often put on a graph, using an x axis (going left and right) and a y axis (going up and down). It can also be written in function form: f(x) = 3x + 1. So in this example, we could put in 5 for x and get y = 16. Put in 2 for x would get y=7. And 0 for x would get y=1. So there would be a line going thru the points (5,16), (2,7), and (0,1) as seen in the graph to the right.

If x has a power of 1, it is a straight line. If it is squared or some other power, it will be curved. If it uses an inequality (< or >), then usually part of the graph is shaded, either above or below the line.

quinta-feira, 12 de fevereiro de 2015

Algebra II

Early forms of algebra were developed by the Babylonians and the Greeks. However the word "algebra" is a Latin form of the Arabic word Al-Jabr ("casting") and comes from a mathematics book Al-Maqala fi Hisab-al Jabr wa-al-Muqabilah, ("Essay on the Computation of Casting and Equation") written in the 9th century by a famous Persianmathematician, Muhammad ibn Mūsā al-Khwārizmī, who was a Muslim born in Khwarizm in Uzbekistan. He flourished under Al-Ma'moun in Baghdad, Iraq through 813-833 AD, and died around 840 AD. The book was brought into Europe and translated into Latin in the 12th century. The book was then given the name 'Algebra'. (The ending of the mathematician's name, al-Khwarizmi, was changed into a word easier to say in Latin, and became the English word algorithm.)^[3]

Examples

Here is a simple example of an algebra problem:

Sue has 12 jellybeans, Ann has 24 jellybeans. They decide to share so that they have the same number of jellybeans.

These are the steps you can use to solve the problem:

To have the same number of jellybeans, Ann has to give some to Sue. Let x represent the number of jellybeans Ann gives to Sue.
Sue's jellybeans, plus x, must be the same as Ann's jellybeans minus x. This is written as: 12 + x = 24 - x
Subtract 12 from both sides of the equation. This gives: x = 12 - x. (What happens on one side of the equals sign must happen on the other side too, for the equation to still be true. So in this case when 12 was subtracted from both sides, there was a middle step of 12 + x - 12 = 24 - x - 12. After a person is comfortable with this, the middle step is not written down.)
Add x to both sides of the equation. This gives: 2x = 12
Divide both sides of the equation by 2. This gives x = 6. The answer is six. If Ann gives Sue 6 jellybeans, they will have the same number of jellybeans.
To check this, put 6 back into the original equation wherever x was: 12 + 6 = 24 - 6
This gives 18=18, which is true. They both now have 18 jellybeans.

With practice, algebra can be used when faced with a problem that is too hard to solve any other way. Problems such as building a freeway, designing a cell phone, or finding the cure for a disease all require algebra.

quarta-feira, 11 de fevereiro de 2015

Algebra

Algebra is a part of mathematics (often called math in the United States and maths in the United Kingdom^[1] ). It uses variables to represent a value that is not yet known. When an equals sign (=) is used, this is called an equation. A very simple equation using a variable is: 2 + 3 = x In this example, x = 5, or it could also be said, "x is five". This is called solving for x.

Besides equations, there are inequalities (less than and greater than). A special type of equation is called the function. This is often used in making graphs.

Algebra can be used to solve real problems because the rules of algebra work in real life and numbers can be used to represent the values of real things. Physics, engineeringand computer programming are areas that use algebra all the time. It is also useful to know in surveying, construction and business, especially accounting.

People who do algebra need to know the rules of numbers and mathematic operations used on numbers, starting with adding, subtracting, multiplying, and dividing. More advanced operations involve exponents, starting with squares and square roots. Many of these rules can also be used on the variables, and this is where it starts to get interesting.

Algebra was first used to solve equations and inequalities. Two examples are linear equations (the equation of a line, y=mx+b) and quadratic equations, which has variables that are squared (power of two, a number that is multiplied by itself, for example: 2*2, 3*3, x*x). How to factor polynomials is needed for quadratic equations.

By Wikipedia.