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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 numbers are used in Boolean algebra?
Boolean algebra uses the numbers 0 and 1 to represent statements which are False and True respectively.
Is Algebra A or Algebra B is more advanced and what is the differences?
I think you mean Algebra 1 & 2
If you did, Algebra 1 is the basic stuff. Algebra 2, you get more complex, but it's still a basic idea.
How are matrices used in cryptography?
Suppose you had a secret message you needed to give someone. You can use a matrix to make it unreadable to anyone except the recipient.
First, you must have an encoding matrix. Let's suppose your encoding matrix is...
-2 2 3
-1 3 2
2 1 3
And your message is... I like matrices
You must then write you message with numbers.
Suppose you use the code A=1, B=2, C=3 etc. Spaces are 27.
That would make your message this:
9 27 12 9 11 5 27 13 1 20 18 9 3 5 19
You translate this into a matrix, going down the columns. The matrix must have three rows to be able to multiply.
9 9 27 20 3
27 11 13 18 5
12 5 1 9 19
When you multiply the two, you get...
72 19 -25 23 61
96 34 14 52 50
81 44 70 85 68
To find the original message, you multiply by the inverse
of the encoding matrix--the decoding matrix. You will find the first pattern of numbers and be able to find the message.
(Sorry, I don't know how to make matrix brackets. You'll just have to deal with it. I tried to make it pretty clear.)
What is surjective in algebra mapping?
A mapping, f, from set S to set T is said to be surjective if for every element in set T, there is some element in S such that it maps on to the element in T.
Thus, if t is any element of T, there must be some element, s, in S such that f(s) = t.
What is pi written as an exact fraction?
Since pi is irrational, there is (by definition) no way to express it as an exact fraction. Even so, many people write it as 22/7, which is a fairly accurate representation. But it is not exact!
Which shape has at least a obtuse angle?
if you are talking about regular polygons, then nothing has an obtuse angle. otherwise, pretty much any shape can have an obtuse angle
How can you get help with Math work?
There are many ways to get help for your math work. You could call a homewowrk helpline, get a friend or another person you know to help you, you could see a tutor, or even get your math teacher to help you if you need it. There are also many tutorials online and you could also find many websites that could help you out. I hope this helped.
What is the benzisoxidil group?
composed of resperidone (Resperidal) and ziprasidone (Geodon). Resperidone has been found useful for controlling bipolar mood disorder, while ziprasidone is used primarily as second-line treatment for schizophrenia.
BIonic Matrix is tiny particles in the atmosphere that grow if they leave the outside of Earth. It is scientifically proven if you touch one you immediately die.
In mathematics, unary operations are functions having only one number for an input. These include functions such as finding squares, square roots, and reciprocals for a number.
What is the proof of the ''Fundamental Theorem of Algebra''?
The Fundamental Theorem of Algebra:
If P(z) = Σnk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0, then P(z0) = 0 for some z0 Є C. Descriptively, this says that any nonconstant polynomial over the complex number space, C, can be written as a product of linear factors.
Proof:
First off, we need to apply the Heine-Borel theorem to C. The Heine-Borel theorem states that if S is a closed and bounded set in an m-dimensional Euclidean space (written as Rm), then S is compact.
From above, P(z) = Σnk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0. Let m = inf{|P(z)| : z Є C} where inf is the infinum, or the greatest lower bound of the set.
From the triangle inequality, |P(reit)| ≥ rn(|an| - r-1|an-1| - … - r-n|a0|),
so limr --> ∞ |P(reit)| = ∞. Therefore there is a real number R that |P(reit)| > m + 1 whenever r > R.
If S = {reit : r ≤ R}, then S is compact in C, by the Heine-Borel Theorem; and let m = inf{|P(z)| : z Є S}. |P| is a continuous and real-valued function in S, so, using the result from another proof not done here, it has a minimum value on S; i.e., there is a value for z0 Є S that makes |P(z0)| = m. So, if m = 0 then the theorem is proved.
We're going to show that m = 0 by proving that m can't equal anything else, and since we know m exists, it has no choice but to be zero. So, suppose m ≠ 0 and let Q(z) = P(z + z0)/P(z0), z Є C.
Q is therefore a polynomial with degree n and |Q(z)| ≥ 1 for all z Є C.
Q(0) = 1 so Q(z) can be expressed via P's series as:
Q(z) = 1 + bkzk + … + bnzn where k is the smallest positive integer ≤ n such that bk ≠ 0.
Since |-|bk|/bk| = 1, there exists a t0 Є [0, 2π/k] such that eikt0 = -|bk|/bk.
Then Q(reit0) = 1 + bkrkeikt0 + bk+1rk+1ei(k+1)t0 + … + bnrneint0
= 1 - rk|bk| + bk+1rk+1ei(k+1)t0 + … + bnrneint0.
So, if rk|bk| < 1 then |Q(reit0)| ≤ 1 - rk(|bk| - r|bk+1| - … - rn-k|bn|).
That means that if we pick a small enough r, we can make |Q(reit0)| ≤ 1 which contradicts the statement above that |Q(z)| ≥ 1 for all z Є C. Therefore m ≠ 0 doesn't hold and P(z0) = 0
Q.E.D.
Another proofSuppose P has no zeroes. Then we can define the function f(z) = 1 / P(z), and f is analytic. By the proof above, P(z) tends to infinity as z tends to infinity; hence f(z) tends to 0 as z tends to infinity. So there is a disc S such that f, restricted to the outside of S, is bounded. Also by the proof above, f is bounded inside the disc as well; therefore f is bounded. Now we apply a theorem called Liouville's Theorem, which says that any analytic function which is defined on all of C and is bounded must be a constant. So f is a constant; therefore P is constant. But we were assuming that P is not constant, so this is a contradiction.(To prove Liouville's Theorem: Suppose M is a bound for the function f, i.e. |f(z)| < M for all z. Suppose a and b are complex numbers, and we want to show f(a) = f(b). Use the theorem that f(a) = integral of f(z)/(z-a) / (2 * pi * i) around the circle of radius R and centre 0. Then, if R is sufficiently large:
|f(b) - f(a)|
= | integral, around circle, of (f(z) * (1/(z-b) - 1/(z-a))) | / (2*pi)
= | integral around circle of (f(z) * (b-a) / ((z-a)(z-b)) ) | / (2*pi)
<= (M * |b-a| / ((R-|a|)(R-|b|)) ) * (2*pi*R) / (2*pi)
The last line uses the formula |integral| <= |pathlength| * |maximum value|. Then we get |f(b) - f(a)| <= M * |b-a| * R / ((R-|a|)(R-|b|)). Letting R tend to infinity, we can prove that |f(b)-f(a)| is as small as we like; therefore f(a) = f(b).
)
A line segment is a line with two points. When drawn, there are no arrows coming out of the points.
Why does temple struggle with algebra?
Let's assume that 'Temple' is a person. Is this true?
The ability to learn the later part of any subject may depend on things one did not understand earlier in that subject. If one did not understand how to 'multiply', then it is reasonable to assume one would then struggle with any later study that required one to understand 'multiplication' to learn it. Right?
Also, if one then clarified the earlier thing, and restudied the material up to the current part they were struggling with it is very likely they would have to struggle less with the subject. Right?
Here is an example from my tutoring experience: An eighth-grade girl is struggling with Pre-Algebra and has been failing at math for several years. I clariefied several things in her current assignment with her, but noticed she was struggling with 'multiplication'.
So I went back to second-grade 'multiplication' to clarify that, and found she was struggling with that and always had been. So I looked for the earliest thing I could find she didn't get, and found 'Number'. I defined 'Number' from a very good 1872 math book and found she was now very happy and good at math and had a renewed vigor towards math.
What the 1872 definition of 'Number' contained was the word or concept of 'Unit'. When she recognized this, the struggling with Math vanished. I had her define the word 'Unit' from that textbook. Unit is what you are talking about with numbers. If we say '3 eggs', then one 'egg' is the unit. If we say 3 feet, then a 'foot' is the unit. That textbook then defined a 'Number' as a quantity and a unit. When this was clarified with her she could then understand 'Number', it made sense now, and move forward in the subject of math.
We then clarified 'multiply' and 'multiplication' and she could now do multiplication without a calculator and told me she could only do 'Calculator Math' before, but didn't understand what they were talking about.
We then went back to her current study of 'Pre-Algebra and she was flying and had no trouble learning and clarifying what she needed to in the current homework. No more struggling. At school she moved from failing to straight A's in math.
The moral is 'Do not go past a word in a subject that you do not understand.' I guess that includes the one's the teacher left out, because when I checked with a retired teacher group on-line. The concept of 'Unit' has been deleted from Elementary Education in America and they said they first encountered the concept in 'Earth Sciences' in the 7th-grade.
Good luck. I hope 'Temple' can use this information to end their struggle with algebra and all other subjects for that matter. - 411 Leon
I researched this a little and now believe 'Temple' in this question refers to 'Dr. Temple Grandin' a spokesperson for Autism that could not pass Algebra and then was not allowed to take Geometry and Trionometry which she feels she can learn. I would be glad to tutor the first 10 autistic people that apply to me for free, to see if the above technique works as well for them in Algebra as it does for my students. This includes Dr. Grandin and her Algebra teacher.
What is the product of two binomials having dissimilar terms?
Let a, b, c, d Є C, where C is the field of complex numbers.
Let m, n, p, q Є N, where N is the field of natural numbers, including 0.
If w, x, y, z Є C are unknown, the product of the two binomials (awm + bxn) and (cyp + dzq) is equal to the following:
acwmyp + adwmzq + bcxnyp + bdxnzq.
How many seconds is 10 minutes?
There are 60 seconds in 1 minute.
Times 60 by 10 and 1 by 10 to get 600 seconds in 10 minutes.
60 x 10 = 600
What is the value of a Model 12 Remington pump 22 value?
The value of a model 12 is greatly determined by condition and variation, for a general 12A (standard grade).
Mint (100%) $800-1,000
Excellent (98%) 700-800
Very good (95%) 600-700
? (90%) 500-600
Below it's a shooter from 100-500 depending on overall condition of blue, wood, and rifling.
What is the value of a model 100?
Winchester model 100 (rifle) in 100% shape is worth $550.00 in the 308 caliber. Winchester model 100 (carbine) in 100% shape is worth $795.00 in 308 caliber. If you find a model 100 in the 284 caliber consider yourself lucky and buy it, dinosaur bones are easier to find.
What is the value of a 12 gauge Diawa Model 500 shotgun?
The DAIWA Model 500 was made in Japan and is a copy of the Browning Auto-5. They sell in the $250-$350 range.
What is the value of a 1995 Suzuki DR 650?
$1500-$2500 Depending on miles and condition Kelley Blue Book has value at $1640 as of 10/31/08
What are the basic laboratory operation?
The basic laboratory operations are:
Chemistry, Physics, Science Experiments
Cleaning the Laboratory
Measuring and Estimating Liquid Volume
Transferring Liquids
Heating Liquids
Precipitation
Filtration
Decantation
Evaporation
Waste Disposal
...logically related concepts or statemtnts that seek to describe and explain development and to predict what kinds of behavior might occur under certain conditions.
