Field of Rational Functions On page 4 of the textbook, we read
... 12, but with one additional wrinkle. Page 12 defines f + g to be the function with domain dom f ∩ dom g whose value at x is f (x) + g(x), and defines multiplication similarly. The extra wrinkle is that we also need to remove any removable singularities. Algebraically, this amounts to canceling any c ...
... 12, but with one additional wrinkle. Page 12 defines f + g to be the function with domain dom f ∩ dom g whose value at x is f (x) + g(x), and defines multiplication similarly. The extra wrinkle is that we also need to remove any removable singularities. Algebraically, this amounts to canceling any c ...
alg6.1
... 6-1 Integer Exponents Check It Out! Example 3b Evaluate the expression for the given values of the variables. for a = –2 and b = 6 Substitute –2 for a and 6 for b. Simplify expressions with exponents. Write the power in the denominator as a product. Simplify the denominator. ...
... 6-1 Integer Exponents Check It Out! Example 3b Evaluate the expression for the given values of the variables. for a = –2 and b = 6 Substitute –2 for a and 6 for b. Simplify expressions with exponents. Write the power in the denominator as a product. Simplify the denominator. ...
Complex Numbers Notes 1. The Imaginary Unit We use the symbol i
... The modulus, or length, of a complex no. is the distance from the origin to the point representing the complex number on an Argand diagram. We use 2 lines either side of a complex no. to represent the modulus. E.g the modulus of 3 - 4i is written |3 - 4i|. This is then the distance from (0,0) to (3, ...
... The modulus, or length, of a complex no. is the distance from the origin to the point representing the complex number on an Argand diagram. We use 2 lines either side of a complex no. to represent the modulus. E.g the modulus of 3 - 4i is written |3 - 4i|. This is then the distance from (0,0) to (3, ...
THE ARITHMETIC OF COMPLEX NUMBERS Contents 1. Basic
... and imaginary parts, respectively, give the identity eit = cos t + i sin t. It is worth pointing out that this is not entirely rigorous, because we have not yet proven that power series make sense when we allow the variable to be a complex instead of real. We will make this rigorous over the next fe ...
... and imaginary parts, respectively, give the identity eit = cos t + i sin t. It is worth pointing out that this is not entirely rigorous, because we have not yet proven that power series make sense when we allow the variable to be a complex instead of real. We will make this rigorous over the next fe ...
x - Net Start Class
... You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invent ...
... You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invent ...
FP1 Complex Numbers
... Dividing complex numbers The second important technique shown in this section is demonstrated in Examples 2.3 and 2.4. It is difficult to work with complex numbers in the denominator of an expression, so it can be very useful to be able to make the denominator real. This is sometimes known as “reali ...
... Dividing complex numbers The second important technique shown in this section is demonstrated in Examples 2.3 and 2.4. It is difficult to work with complex numbers in the denominator of an expression, so it can be very useful to be able to make the denominator real. This is sometimes known as “reali ...
Infinite sets of positive integers whose sums are free of powers
... that an example of such a set S is provided by the set of all Fermat numbers n {Fn }n≥0 , where Fn := 22 + 1 for n ≥ 0. In this note, we show how to construct an infinite set of positive integers S such for every n ≥ 1 and any Pthat n n distinct elements x1 , . . . , xn of S the sum i=1 xi is not a ...
... that an example of such a set S is provided by the set of all Fermat numbers n {Fn }n≥0 , where Fn := 22 + 1 for n ≥ 0. In this note, we show how to construct an infinite set of positive integers S such for every n ≥ 1 and any Pthat n n distinct elements x1 , . . . , xn of S the sum i=1 xi is not a ...
Complex Numbers, Polynomials, and Symmetry
... relates a number and its complex conjugate? 2. We measure the “size” of a real number by using absolute value: |x| is the distance from the real number x to the origin. In the same way we define the absolute value of a complex number to be its distance from the origin. This distance will always be a ...
... relates a number and its complex conjugate? 2. We measure the “size” of a real number by using absolute value: |x| is the distance from the real number x to the origin. In the same way we define the absolute value of a complex number to be its distance from the origin. This distance will always be a ...
