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Abstract Algebra
Have you ever wondered what would happen if you redefined some of the basic rules of algebra so that concepts you took for granted, like AB = BA, no longer apply? Abstract algebra does just that through the study of the properties that define algebraic structures. Post all questions about fields, rings, group theory, vector spaces, and the axioms that define them into this category.
1,849 Questions
What is the use of eigenvalues in daily life?
In N dimensions i.e. 3 space dimensions and 1 time dimension (our perceived universe) there is a way of collapsing all of the contain information (any nontrivial function i.e. x^4 + y^4 +Z^4 = any real number other than zero (Eigen function) at time being discrete towards 0 gives an informational representation of 3 dimensions in 4 dimensions after the first derivation and a 2 dimensional information representation of 4 dimensional space. After deriving any Eigen function the Eigen Values in a domain and range can be represented by a singularity. (i.e. the singularity that expanded "in the beginning) which is commonly known as the Big Bang theory).
In real life this can be useful in reducing large amounts of information into a "simplest" symbol. i.e. a 2 dimensional string (string theory) can be described by an Eigen Function. All the information of said string can be described in binary code i.e. 00111010101010110011101010 can be reduced by defining a string of binary code by a symbol say: q (where there can be an infinite amount of symbols)
Therefor a string of two dimensional information represented by symbols can then be reduced again by another symbol.
This can be done until all of the information in any space time of n dimensions is represented by a symbol defined by all of the information in said space time dimensions up to time t with complete regression assuming that information CAN be regressed with no error.
A real life example would be huge amounts of binary code is reduced and then transported to a television in which the simplified information is then expanded back to the original information which is then transformed by the television into a moving picture.
A more controversial assessment of a real life example in the field of Mathematics and Physics is that all of the information in our current space time (4-D) up to t=present can be represented by binary code at a snapshot t=plank time reduced to a symbol. The amount of plank times from the Big Bang to now can be reduced back down to the singularity from whence The Big Bang occurred. (This assumes that there is no error in the possible irreducible holonomies. The decomposition and classification of information can be represented by a singularity)
What is the collection of all output values called?
All the output values of a function are collectively called the "range" of that function. For example, consider the function x2. Any number squared will give a positive. Thus, the "range" of the function is positive numbers.
Algebra is important because it teaches how to manipulate equations. This is necessary in any type of math that follows.
When did algebra get its name?
Algebra comes from the Arabic word al-jabr, which appeared in the title of a book Hidab al-jabr wal muqubala (roughly translated as The book of restoration and cancellation".
This was written in Baghdad in the early 9th Century by the mathematician Mohammed ibn-Musa al-Khowarizmi.
What are the basic logic operators in Boolean algebra?
AND, OR, and NOT are the basic operators in Boolean Algebra.
How do you do boolean algebra?
It is used in science and proofs of proofs in maths, taking an example:
Hypothesis: Bananas are red.
(There exists) x x (is an element of) Red (and) Banana
Banana (implies) Red
(not) Red (implies) (not) Banana
If a yellow banana is found,
(There exists) y y(is an element of)Banana (therefore) y(is an element of) Red
There is a contradiction, as the banana must be red, but it is yellow.
The above is not the only use, but Boolean algebra exclusively contains symbols of (not), (and), and (or).
What does simplify mean in math?
Solve, or to work the problem until there are no other or smaller possible answers.
A matrix in mathematics is a rectangular array of quantities or expressions set out by rows and columns.
What is another way to write the expression 44n?
44n is the same thing as 44 * n. By the way "*" means "times/multiply".
It is a vector space with a quasi norm instead of a norm. A quasi norm is a variation of a norm which follows all the norm axioms except for the triangle inequality where we have x+y< or = K(x+y)for some K>1
Describe a collision ring theory?
The Collision Ring Theory is the hypothesis concerning the moons origin. Scientists theorize that a massive, planet-sized object struck Earth 63 million years ago, when the space around the Earth was full of rocky debris, resulting in ejected material to form a ring around the still young Earth. The material around the Earth fused into one orbital satellite known as the moon due to the laws of gravity. Parts of the outside layer of the Earth broke off and later were pulled together to form the moon
Boolean algebra is a division of mathematics that deals with operations on logical values and incorporates binary variables.
Is an invertible idempotent matrix the identity matrix?
The assertion is true.
Let A be an idempotent matrix. Then we have A.A=A. Since A is invertible, multiplying A-1 to both sides of the equality, we get A = I.
Q. E. D
If two rows of a matrix are interchanged what is true of the resulting determinant?
The determinant will change sign.
What is a self inverse function?
They are called involution functions.
All inverses are expressed as:
f-1(f(x)) = x for every x in f(x)'s domain.
Because of their unique property, involution functions are expressed simply as:
f(f(x)) = x for every x in f(x)'s domain.
An easy example is the reciprocal function f(x) = 1/x. If you do that twice, you get x.
Who is credited with the invention of Boolean Algebra?
George Boole, not to be confused with George Boolos
Ø Algebra is used in companies to figure out their annual budget which involves their annual expenditure. It, along with statistics is also used to predict their annual turnover. Ø Even various stores use algebra to predict the demand of a particular product and place their orders likewise. Algebra is also used in preparing the bills of the customers in the case of the purchase multiple number of the same object. Ø Algebra is used in schools by the teachers to prepare the annual reports of the children. Also, it can be used during examination to predict the usage of supplementary sheets, therefore helping in finding out the required number. It can be used to figure out the pass percentage for a test. Ø It also has individual applications in the form of calculation of annual taxable income, payable income tax and bank interest and instalment loans. Ø While reading the temperature in a thermometer, we can use this simple formula for converting °Celsius temperature into °Fahrenheit. Degrees in Celsius (9/5) + 32 = Degrees in Fahrenheit Ø During the construction of buildings algebra can be used to formulate the various measurements of the building in a proper ratio. Ø In chemistry, algebra can be used to calculate the quantity of a substance required in an experiment. Also, it is used to calculate the quantity of various substances formed in an experiment. Ø Unconsciously we are using algebra in every day life like when we go shopping and compare the pricing of the various products according to their quantity. Ø It may be used by a person while hosting a party to calculate the total consumption of food according to the average one person consumption. Ø It may be used when you estimate the distance from point A to point B by the time and speed of your travel or when you estimate the average speed at which you need to move to reach your destination in a particular time frame. Ø Algebraic expressions and equations serve as models for interpreting and making inferences about data. Algebraic reasoning and symbolic notations also serve as the basis for the design and use of computer-spreadsheet models. Ø Algebra is also used in various disciplines of physics like mechanics, electronics and acoustics. For instance, in SONARs algebra is used to calculate the distance of the sea bed from the surface. It is also used in astronomy in calculating the distances of planets, stars etc. from us. Most people don't believe algebra is used very often, but we use it every day in different applications, sometimes without even knowing it!
What is the associative property of addition?
the associative property of addition means that changing the grouping of the addends doesn't affect the sum
Boolean algebra deals with what?
Boolean algebra deals with logic and truth as it pertains to sets and possibilities. It uses the and, or and not operators to set up truth tables to define if a statement is true or not.
