4. + and x are identity elements 0 and 1 in F respectively
5. every element has an additive inverse in F
6. every non zero element has a multiplicative inverse in F
7. x distributes over +
1. r is closed under + and x
2. + and x are associative
3. + is commutative (x is not)
4. + and x are identity elements 0 and 1 in r respectively
5. every element has an additive inverse in r
6. (no multiplicative inverse)
7. x distributes over +
Most Common Rings
integers, set of squares matricies, set of polynomials
Why are natural numbers not a ring?
Because they do not contain an additive identity. Also, the additive inverse of natural numbers are not natural numbers. For instance, the additive inverse of 3 is -3, which is not a natural number.
are the ordinary whole numbers used for counting ("there are 6 coins on the table") and ordering ("this is the 3rd largest city in the country").
is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number.
Expressed as the sum of a real number and an imaginary number. So, any complex number can be written as a+bi, where a and be are real numbers I^2 = -1
What does it mean to be ordered?
A set is "totally ordered" (or just ordered) if, give any two elements a and b in the set either a < (=) b or
b < (=) a. Notice that beacuase of the trichotomy axiom fo real numbers. we know that if x and y are real numbers, eitehr i. x < y ii. x>y or iii. x= y. Thus the real numbers and any subset of the real numbers is ordered.
examples: ordered field: rational numbers, integers, and natural numbers because they are ordered via < , because they are subsets of trhe (ordered) real numbers.
Fundamental Theorem of Algebra
if f(x) is a polynomial with real coefficients, then f(x) can be factored into linear and quadratic factions, each of whcih with real coefficients.
Rational Root Theorem
Conjugate Roots Theorem
If f(x) is a polynomail with real coefficients anf if f(a+bi) = 0, then the Conjugate Roots Theorem says that f(a-bi) = 0
How do you use the Fundamental Theorem of Algebra?
The main way to use the Fundamental Theorem of Algebra is when determining the number of roots a polynomial has. For example. a polynomial of degree n has at most n roots. combined with previous theorems, we can often say more.
A relation from a set A to a seat B is a set of ordered pairs (x,y). all real numbers x all real numbers = all real numbers squared.
domain and range
Two go to one
x= 1 y= 3
x=-2 y= 5
Polynomials, radicals, and absolute value functions have no asymptotes. Radical functions have horizontal asymptotes exactly when the degree of the number is less than or equal to the degree of the denominator. Rational functions can have vertical asymptotes or holes
The basic exponential functions are continuous on the entire domain of real numbers.
The basic logarithmic functions are continous on their domains.
vector is a mathematical object that has a magnitude and a direction. People often think of two dimensional (2-D) vectors as arrows drawn on teh plane, and three dimension (3-D) vectors as arrows in space. The starting point of a vector is not important to the defintion. Consequently, vectors often depicted in standard position. The magnitude (or length) of
The dot product is the easiest way to determine the angle between two vectors. so, it can be used to tell when two vectors are perpendicular. Physicists use the dot product to decompose a vector into its various components.
A matrix is a rectangular array of numbers. Matricies can be very useful in solving systems of linear equations.
What is divisiblity in the natural numbers?
We say a and b be natural numbers with
Polynomials, Radicals, and Absolute Values
Rational Functions can have vertical asymptotes or holes at the points where the denominator is zero.