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Complex Numbers
The square root of negative one, which we now call the imaginary unit, was once thought to be an absurd notion. However, through the diligent studies of open-minded mathematicians, it was shown that the real numbers were actually just one part of a larger set of numbers known as the complex numbers, the other part being the imaginary numbers. Please direct all questions about these surprisingly useful and applicable numbers into this category.
887 Questions
How do you do arithmetic complex math?
Addition and subtraction: add (or subtract) the real parts, then add (or subtract) the imaginary parts. Multiplication: treat just like multiplying binomials (like with a variable x).
After multiplying, convert any i² terms to -1.
Division: multiply both numerator and denominator by its conjugate, which will make the denominator a real, then divide real part by denominator, and then divide imaginary part by denominator.
Why does a negative discriminant yield imaginary numbers?
When solving equations (of e.g. 2nd grade) the discriminant is part of a squareroot.
A square root of 'x' is the number that multiplied y itself equals 'x'.
Any number we can think of - when multiplied by itself (squared) gives a positive number. (even negative number - e.g. -2*-2 = 4)
So we cannot think of a number which square is negative.
Then sq.root(-1) has no answer - no number squared gives -1.
Therefore the number that equals sq.root(-1) is imaginary.
Multiplication two complex numbers in c?
(a +bi)(c + di) : Use the distributive property and remember i*i = -1. In polar form:|ab| = |ab| and thetaab = thetaa + thetab.
Yes. All Real numbers are a proper subset of the Complex numbers.
What is the equation of circle in complex number?
z=e^(2 times pi times i times t)
If t goes from 0 to 1, then you get the unit circle.
What are examples of complex numbers?
z=x+iy, where x and y are real numbers. Complex numbers can produce interesting graphs. If you graphed the above, you would get vertical and horizontal lines. But what happens when you graph 1/z ? When you work it out, you get two equations which are at right angles to each other. You get u=x/(xx+yy) and v=-y/(xx+yy) which are families of concentric circles at right angles to each other.
3i. Combining the real number 3 with the imaginary number i creates a complex number.
The SMITH CHART is based on complex numbers and is used in electronic to find real and imanginary impedences. It is loosely based on the Mobius transform and is the formula w=(z-1)/(z+1)
Where does the negative sign go in imaginary numbers?
The basic theory of imaginary numbers is that because (-) numbers squared are the same as (+) numbers squared there is not a correct continueos line on a graph.
Can a null set be in union with a set of numbers in math?
Yes. Union is an operation in which all the members of any two sets are placed in a common set. The union operation can be applied to the null set and any set but since it has no members, it does not change the set the union is taken with.
It is rather like adding 0 to a number.
How do you change a number from scientific notation to standard form?
You are a complete idiot. Multiply your number by ten to whatever its to. such as 1.24x10^2 would be 1.24(100) which would be 124.
What is the relation of complex numbers to real numbers?
Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.
How to solve synthetic division with complex numbers?
I'm not sure about how to use complex numbers to do this, but I've posted a link to a pretty neat website about Synthetic Division.
Why complex numbers used in impedance?
Because there is an angle involved. If - for example - the resistance (the real part) is 10 ohms, and the reactance (the imaginary part) is also 10, then there is an angle of 45 degrees; which actually means that this will be the displacement angle between the voltage and the current.
Impedance may just be specified with an angle; but it turns out that the calculations between voltage, current, and impedance correspond precisely to the calculations with complex numbers.
Because there is an angle involved. If - for example - the resistance (the real part) is 10 ohms, and the reactance (the imaginary part) is also 10, then there is an angle of 45 degrees; which actually means that this will be the displacement angle between the voltage and the current.
Impedance may just be specified with an angle; but it turns out that the calculations between voltage, current, and impedance correspond precisely to the calculations with complex numbers.
Because there is an angle involved. If - for example - the resistance (the real part) is 10 ohms, and the reactance (the imaginary part) is also 10, then there is an angle of 45 degrees; which actually means that this will be the displacement angle between the voltage and the current.
Impedance may just be specified with an angle; but it turns out that the calculations between voltage, current, and impedance correspond precisely to the calculations with complex numbers.
Because there is an angle involved. If - for example - the resistance (the real part) is 10 ohms, and the reactance (the imaginary part) is also 10, then there is an angle of 45 degrees; which actually means that this will be the displacement angle between the voltage and the current.
Impedance may just be specified with an angle; but it turns out that the calculations between voltage, current, and impedance correspond precisely to the calculations with complex numbers.
What is the sum of two complex conjugate number?
Since the imaginary parts cancel, and the real parts are the same, the sum is twice the real part of any of the numbers. For example, (5 + 4i) + (5 - 4i) = 5 + 5 + 4i - 4i = 10.
What set of numbers does -50 belong?
Rational because it can be expressed as a fraction in the form of -50/1
What set of numbers does 5 belong in?
5 belongs in the sets:
-Natural number set, positive whole numbers
-Integer number set, whole numbers
-Rational number set, numbers that are not never ending
-Real number set, basic numbers without i and that can be expressed in say amounts of apples
-Complex number set, the set that contains both real and unreal numbers
The square root of a value is called an imaginary or complex number?
Not necessarily. The square root of 4 are +/- 2 which are Real numbers, NOT imaginary. Although, since the Reals are a subset of Complex numbers, the above roots would belong to the Complex numbers.
What is the word i in the math problem 6i plus i?
i is the constant for the imaginary unit.
i ^2 = -1
and therefore i = squareroot of -1
What is the meaning of the term complex numbers?
Complex numbers are numbers of the a + bi where a and b are real number and i is the imaginary square root of -1.
Is there any one-to-one correspondence between complex numbers and real numbers?
== == Yes there is, but it's a little tricky to prove. Here's a sketch. First, some notation:
P ~ Q means there is a 1-1 correspondence between all members of set P and all of Q
R is the set of real numbers, C is the complex numbers
R x R is the cross-product of R with itself, the set of ordered pairs of reals (ditto for any set)
(0,1] is the half-open interval of reals from 0 to 1, that is all real x with 0 < x <= 1
Now the sketch. It's pretty obvious that R x R ~ C, since a complex number is just an ordered pair of reals.
R ~ (0,1] by the function f(x) = 1/(1-x) (you can prove yourself pretty easily that this is a 1-1 function and covers all of R vs all of (0,1]
R x R ~ (0,1] x (0,1] by applying the previous rule to each element of the pair
[0,1) ~ [0,1) x [0, 1] is a bit tricky, but one way is to map a real number x in (0,1] into two real numbers y and z by taking every other digit in the decimal expansion of x. For example take x = pi/10...
x = .314159265358979323846... <-> .1196387334... and .34525599286...
You have to watch out for technicalities like .09999... = .100000,... but it does work and is 1-1 and covers all of (0,1].
So, stringing all this together...
C ~ R x R ~ (0,1] x (0,1] ~ (0,1] ~ R
I'm not sure you need the R ~ (0,1] part, but it's an important fact (all of R is ~ to a subset of R), a cute trick and worth showing.
QED
