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| The word algebra
comes from the title of a book by Muhammad ibn
Musa al-Khwarizmi. |
Algebra
Algebra (from Arabic: الجبر, transliterated "al-jabr",
meaning "reunion of broken parts") is a part of
mathematics (often called math in the United States and
maths or numeracy in the United Kingdom ). 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 that "x equals 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 because it
always turns one input into one output.
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,
engineering and 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 use the rules of numbers and
mathematic operations used on numbers. The simplest are
adding, subtracting, multiplying, and dividing. More
advanced operations involve exponents, starting with
squares and square roots.
Algebra was first used to solve equations and
inequalities. Two examples are linear equations (the
equation of a straight line, y=mx+b or y=mx+c) and
quadratic equations, which has variables that are
squared (multiplied by itself, for example: 2*2, 3*3, or
x*x).
History
Early forms of algebra were developed by the Babylonians
and the Greek geometers such as Hero of Alexandria.
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 Persian mathematician, 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). |
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Examples
Here is a simple example of an algebra problem: |
- Sue has 12 candies, and Ann has 24
candies. They decide to share so that they have the same
number of candies. How many candies will each have?
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These are the steps you can use to solve the problem: |
- To have the same number of candies,
Ann has to give some to Sue. Let x represent the number
of candies Ann gives to Sue.
- Sue's candies, plus x, must be the
same as Ann's candies 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 candies, they will have the same number of candies.
- 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 each now have 18 candies.
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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. |
|
Writing algebra
As in most parts of mathematics, adding z to y (or y
plus z) is written as y + z.
Subtracting z from y (or y minus z) is written as y − z.
Dividing y by z (or y over z: y/z) is written as y ÷ z
or y/z. y/z is more commonly used.
In algebra, multiplying y by z (or y times z) can be
written in 4 ways: y × z, y * z, y·z, or just yz. The
multiplication symbol "×" is usually not used, because
it looks too much like the letter x, which is often used
as a variable. Also, when multiplying a larger
expression, parentheses can be used: y (z+1).
When we multiply a number and a letter in algebra, we
write the number in front of the letter: 5 × y = 5y.
When the number is 1, then the 1 is not written because
1 times any number is that number (1 × y = y) and so it
is not needed.
As a side note, you do not have to use the letters x or
y in algebra. Variables are just symbols that mean some
unknown number or value, so you can use any variable. x
and y are the most common, though.
Functions and Graphs
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 machine you can
put a number (or numbers) into and get a certain number
(or numbers) 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 the slope 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. |
|
Rules
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 1/a.
|
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!
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!
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.
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
* (b * c) = (a * b) * c. Note that this does not hold
for division.
Distributive property
The distributive property states that the multiplication
of a number by another term can be distributed. For
instance: a * (b + c) = ab + ac. Do not confuse this
with the associative properties!
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 * 1 = a.
This also holds for division: 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. Additive inverse of 1 is (-1).
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: a/a =
1. Multiplicative inverse of 2 is 1/2.
Advanced Algebra
In addition to "elementary algebra", or basic algebra,
there are advanced forms of algebra, taught in colleges
and universities, such as abstract algebra, linear
algebra, and universal algebra. This includes how to use
a matrix to solve many linear equations at once.
Abstract algebra is the study of things that are found
in equations, going beyond numbers to the more abstract
with groups of numbers.
Many math problems are about physics and engineering. In
many of these physics problems time is a variable. Time
uses the letter t. Using the basic ideas in algebra can
help reduce a math problem to its simplest form making
it easier to solve difficult problems. Energy is e,
force is f, mass is m, acceleration is a and speed of
light is sometimes c. This is used in some famous
equations, like f = ma and e=mc^2 (although more complex
math beyond algebra was needed to come up with that last
equation). |
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|
 Kiddle: Algebra
Wikipedia: Algebra |
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