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FAQs
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What makes a field?
A field is a set F, containing at least two elements, on which two operations. + and · (called addition and multiplication, respectively) are defined so that for each pair. of elements x, y in F there are unique elements x + y and x · y (often written xy) in F for. -
What is an ordered field in math?
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. ... Every ordered field contains an ordered subfield that is isomorphic to the rational numbers. -
Are rational numbers an ordered field?
By Rational Numbers form Field, (Q,+,×) is a field. By Total Ordering on Quotient Field is Unique, it follows that (Q,+,Ã) has a unique total ordering on it that is compatible with its ring structure. Thus (Q,+,Ã,\u2264) is a totally ordered field. -
Are integers an ordered field?
Examples. The rational numbers Q, the real numbers R and the complex numbers C (discussed below) are examples of fields. The set Z of integers is not a field. In Z, axioms (i)-(viii) all hold, but axiom (ix) does not: the only nonzero integers that have multiplicative inverses that are integers are 1 and \u22121. -
Are the rationals an ordered field?
By Rational Numbers form Field, (Q,+,×) is a field. By Total Ordering on Quotient Field is Unique, it follows that (Q,+,Ã) has a unique total ordering on it that is compatible with its ring structure. Thus (Q,+,Ã,\u2264) is a totally ordered field. -
Is Q an ordered field?
Every subfield of an ordered field is an ordered field with the same ordering as the original one. Since Q\u2264R, it is an ordered field. The same holds true, for example, for the field Q[\u221a2]\u2264R as well. -
What is the field Q?
A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted Q, together with its usual operations of addition and multiplication. -
What does Z * mean in complex numbers?
Representing Complex Numbers All complex numbers z = a + bi are a "complex" of just two parts: The real part: Re(z) = a. The imaginary part: Im(z) = b. -
Are natural numbers a field?
The Natural numbers, , do not even possess additive inverses so they are neither a field nor a ring. The Integers, , are a ring but are not a field (because they do not have multiplicative inverses). -
What is a field in linear algebra?
I LINEAR ALGEBRA. A. Fields. A field is a set of elements in which a pair of operations called multiplication and addition is defined analogous to the operations of multiplication and addition in the real number system (which is itself an example of a field).
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Draft ordered field
let's talk about ordered fields now we've talked about fields so I'm taking a field F and we've actually talked about ordered sets so we'll say F is a field in fact an ordered field if and Here I am assuming that simply the set F is an ordered set so f is just an ordered set but moreover that F is an ordered field if these two things these are the real two conditions if we have three elements here XY and z in F take any three elements well then X less than Y and remember here the less than sign this is our relation this is our ordered relation and it's one that you're probably very familiar with X less than Y well this would imply that X plus Z is less than y plus Z we can add to both sides of an inequality very convenient definitely something you'd like to do and then number two here second second part of the definition of an ordered field if we have two elements x and y in F which are both positive so X is greater than zero and Y is greater than zero well then this implies x times y is greater than zero in other words if I have two positive numbers here or they're not technically numbers and I'm talking about a generic field but you can think of this as how positive times a positive is is a positive right this makes sense and you can really just think about the fact that we're going to be talking about real numbers if I multiply two positive real numbers the result should be positive as well let's let F be an order field and let's take any four elements X Y Z and W in F then we're gonna get these following results and these are things that you would really expect but we're gonna state them just to be perfectly clear number one here if X is positive well then minus X is negative 4 of X is greater than 0 then minus X is less than 0 and we'll say and vice-versa here you know if X is negative then negative x is positive just like you would expect number 2 X positive and y less than Z well then x times y is less than x times Z or so X is positive and Y is less than Z well then multiplying by something less than Z should be less than multiplying by something which is Z number 3 very similar x- + y less than Z this would imply that X Y is greater then X is a very very similar result number 4 X not equal to 0 well that would imply x squared is positive or it's square square a real number it should be positive here this is the way you should be thinking but this is true of an arbitrary ordered field number 5 X positive but less than Y well this implies that 1 over Y is positive but also less than 1 over X think about how you would you know change things around with this inequality think about dividing both sides by x + y not really talking about what that means but this is a true fact number 6 if X is positive but less than Y well then their squares have this relation x squared is less than Y square exactly what you would expect think 1 is less than 2 well 1 squared is less than 2 squared of course and then finally we have a bit of transitivity here notice the less than or equal to sine X less than or equal to Y and Z less than or equal to W well this would imply that X plus Z is less than or equal to y plus W in a sense I can sort of add these inequalities together so we have all of these results these are all true facts if we have an ordered field I'm not going to prove all of them but maybe we'll just prove number one here for you so for number one we assume we assume that X X is in our order field and that X is positive okay so this is my only assumption along with the definition we had of an ordered field now if you remember what was the first definition of the ordered field that for any XY and Z and F with X being less than Y this implies that X plus Z is less than y plus Z this was our first condition of being an ordered field now we also have the fact that well we're working with a field and what were some of the things that happened with the field well we knew that there was a zero in the field so zero is in the field and we also have a minus X in the field these were two conditions of the field field axioms if you want to look back at a previous video on fields you can check those out but let's use all of these facts let's use this that we have we have this and we have this so we assumed X is greater than zero well let's just say that X plus our minus X has to be greater than 0 plus minus X all right these are all field elements and then we'll just use the properties of the field remember X plus minus X one of the field axioms this is in fact zero on the left and any element in the field here minus X plus zero is simply that element so this left hand side I'm using properties of inverses essentially and on the right hand side I'm using properties of the additive identity and hey there we go this is exactly what we wanted to prove draw a little square there if you like some people like that and and that's exactly what we want that if X is greater than 0 it implies that 0 is greater than minus X
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