Sequences and Series: Terminology and Notation
Zoom Passcode: CSUSB_Math. Abstract: Mathematics can be more than just a subject in school; it can be a career. The government is the number one single employer of mathematicians in the country. Many of those mathematicians end up at the National Security Agency, where they find careers in research, information assurance, and cryptanalysis. Jun 04, · In this section we take a quick look at some of the terminology we will be using in the rest of this chapter. In particular we will define a linear operator, a linear partial differential equation and a homogeneous partial differential equation. We also give a quick .
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share rv plans how to build a rv within a single location that is structured and easy to search.
Rings, groups, and fields all feel similar. What are the differences between them, both in definition and in how they are used? They should feel similar! In fact, every ring is a group, and every field is a ring. A ring is a ij with an additional operation, where the second operation is associative and the distributive properties make the two operations "compatible".
A field is a ring such that the second operation also satisfies all the group properties after throwing out the additive identity ; i. You're right to think that the definitions are very similar. The main difference between groups and rings is that rings have two binary operations usually called addition and multiplication instead of just one binary operation.
If you forget about multiplication, then a ring becomes a group with respect to addition the identity is 0 and inverses are negatives. This group is always commutative! If you forget about addition, then a ring does not become a group with respect to multiplication. The binary operation of multiplication is associative and it does have an identity 1, but some elements like 0 do not have inverses. This structure is called a monoid.
A commutative ring is a field when all nonzero elements have multiplicative inverses. In this case, if what is terminology in math forget about addition and remove 0, the remaining elements do form a group under multiplication.
This group is again commutative. A division ring is a not necessarily commutative ring in which all nonzero elements have multiplicative inverses. Again, if you forget about addition and remove 0, the remaining elements do form a group under multiplication. This group is not necessarily commutative.
An example of a division how to design a sports logo which is not a field are the quaternions. A group is an abstraction of addition and subtraction—except that the group operation what is terminology in math not be commutative. But the important part is what to do in namba osaka there is an operation, which is something like addition, and the operation si be reversed, so there is also something like subtraction.
I won't explain what a ring or a group is, because that's already been done, but I'll add something else. One reason groups and rings feel similar is that they are both "algebraic structures" in the sense of universal algebra.
So for instance, the operation of quotienting via a normal subgroup for a group and a two-sided ideal for a ring are basically instances of quotienting via an invariant equivalence relation in universal algebra. Every field is a ring, and every ring hwat a group. A group has one operation which satisfies closure, associative property, commutive property, identity, and inverse property.
A ring satisfies all properties of a group; it also has a second operation which has closure, associative, and distributive property between these two operations. A field has two operations and both satisfy all group properties.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Terimnology for Teams — Collaborate and share knowledge with a private group.
Create a free Team Kn is Teams? Learn more. What are the differences between rings, groups, and fields? Ask Question. Asked 10 years, 9 terminoloy ago.
Active 1 year ago. Viewed 86k times. Improve this question. CogitoErgoCogitoSum 3, 1 1 gold badge 15 15 silver badges 32 32 bronze badges. Add a comment. Active Oldest Votes. Improve this answer. BBischof BBischof 5, 1 1 gold badge 33 33 silver badges 43 43 bronze badges.
What is a timesheet used for is actually an additional property to the one requiring nonzero elements to form a multiplicative group; without it one has a division ring.
Thank you! Show 4 more comments. Ittay Weiss Dorais 4, 29 29 silver badges 36 36 bronze badges. Dorais Jan 7 '13 at In the case of a field, it's actually one shy of a group in the sense that 0 is the only element without an inverse.
Dorais Aug 6 '18 at To this, a ring adds multiplication, but not necessarily division. To this, a field adds division. Going the other way from a group, we have the monoid, which has addition, but not subtraction. Akhil Mathew Akhil Mathew Fountain Lee Fountain Lee Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
Accept all cookies Customize settings.
11 thoughts on “Some Important Reasons for Studying Terminology”
Feb 02, · Terminology is a necessity for all professionals involved in the representation, expression, communication and teaching of specialized knowledge. Scientists, technicians or professionals in any field require terms to represent and express their knowledge to inform, transfer or buy and sell their products. Sequences and series are most useful when there is a formula for their terms. For instance, if the formula for the terms a n of a sequence is defined as "a n = 2n + 3", then you can find the value of any term by plugging the value of n into the formula. For instance, a 8 = 2(8) + 3 = 16 + 3 = In words, "a n = 2n + 3" can be read as "the n-th term is given by two-enn plus three". Oct 19, · Mathwords: Terms and Formulas from Beginning Algebra to Calculus. An interactive math dictionary with enoughmath words, math terms, math formulas, pictures, diagrams, tables, and examples to satisfy your inner math geek.
Intro Examples Arith. Series Geo. A "sequence" called a "progression" in British English is an ordered list of numbers; the numbers in this ordered list are called the "elements" or the "terms" of the sequence. A "series" is what you get when you add up all the terms of a sequence; the addition, and also the resulting value, are called the "sum" or the "summation". A sequence may be named or referred to by an upper-case letter such as " A " or " S ".
The terms of a sequence are usually named something like " a i " or " a n ", with the subscripted letter " i " or " n " being the "index" or the counter. So the second term of a sequnce might be named " a 2 " pronounced "ay-sub-two" , and " a 12 " would designate the twelfth term. The sequence can also be written in terms of its terms. The sequence of terms starting with index 3 and going on forever could be written as:.
Some books use the parenthesis notation; others use the curly-brace notation. Either way, they're talking about lists of terms. The beginning value of the counter is called the "lower index"; the ending value is called the "upper index". The formatting follows the English: the lower index is written below the upper index, as shown above.
Just try always to make sure, whatever resource you're using, that you are clear on the definitions of that resource's terms and symbols. In a set, there is no particular order to the elements, and repeated elements are usually discarded as pointless duplicates. Thus, the following set:. The terms of a sequence can be simply listed out, as shown above, or else they can be defined by a rule.
Often this rule is related to the index. Sometimes the rule for a sequence is such that the next term in the sequence is defined in terms of the previous terms. This type of sequence is called a "recursive" sequence, and the rule is called a "recursion". The most famous recursive sequence is the Fibonacci fibb-oh-NAH-chee sequence. Its recursion rule is as follows:. What this rule says is that the first two terms of the sequence are both equal to 1 ; then every term after the first two is found by adding the previous two terms.
The first few terms of the Fibonacci sequence are:. To indicate a series, we use either the Latin capital letter "S" or else the Greek letter corresponding to the capital "S", which is called "sigma" SIGG-muh :. The whole thing is pronounced as "the sum, from n equals one to ten, of a-sub-n".
The summation symbol above means the following:. The written-out form above is called the "expanded" form of the series, in contrast with the more compact "sigma" notation. Any letter can be used for the index, but i , j , k , m , and n are probably used more than any other letters.
There are some rules that can help simplify or evaluate series. If every term in a series is multiplied by the same value, you can factor this value out of the series. This means the following:. This means that, if you've been told that the sum of some particular series has a value of, say, 15 , and that every term in the series is multiplied by, say, 2 , you can find the value as:. The other rule for series is that, if the terms of the series are sums, then you can split the series of sums into a sum of series.
In other words:. If you add up just the first few terms of a series, rather than all possibly infinitely-many of them, this is called "taking or finding the partial sum". If, say, you were told to find the sum of just the first eight terms of a series, you would be "finding the eighth partial sum".
Sequences and series are most useful when there is a formula for their terms. The word " n -th" is pronounced "ENN-eth", and just means "the generic term a n , where I haven't yet specified the value of n. Of course, there doesn't have to be a formula for the n -th term of a sequence.
The values of the terms can be utterly random, having no relationship between n and the value of a n. But sequences with random terms are hard to work with and are less useful in general, so you're not likely to see many of them in your classes. Page 1 Page 2 Page 3 Page 4 Page 5. All right reserved. Web Design by. Skip to main content.
Purplemath A "sequence" called a "progression" in British English is an ordered list of numbers; the numbers in this ordered list are called the "elements" or the "terms" of the sequence.
<- How to eat cordyceps sinensis - How much sugar is too much per serving->