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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 plot complex numbers?
It's easy to do. Take the complex number (2+i3). On an x, y plane the 2 would be on the real number line or x=2. 3 would be on the y=3 line where x=2. So you would have a point at (2,3) in the plane.
You can get some interesting graphics from complex plotting programs.
The LINK below shows what the equation w=1/z can do to a photo.
If I understand the question correctly then a is a proper subset of u.
How do you pronounce this number 6580000000000000000000?
Six sextillion, five hundred and eighty quintillion.
How do you know if a set of numbers are arithmetic or not?
If they are all numbers - no letters other than e or pi, then they are arithmetic. Otherwise they are probably algebraic.
But beware, phi is also arithmetic.
What is the strategy for finding the cube root of complex numbers?
Here's how you can find any power (fractions would be a root of a number) of any number (complex or real). A real number is a subset of the complex number set, with the imaginary part = 0. I'll refer you to a related link on Euler's formula for information about how this is derived. A complex number can be graphed on the Real-Imaginary plane, with reals on the horizontal axis, and imaginary on the vertical. Convert the complex number from x-y style coordinates in this plane to polar coordinates.
For a complex number a + bi, here's how you do that. We will end up with a magnitude and an angle. The magnitude is sqrt(a² + b²). The angle is found by tan-1(b/a). Now to find a power, apply the power to the magnitude (for cube root this is exponent of 1/3). Then multiply the angle by the power (in this case you divide by 3). Really for a cube root there will be 3 distinct roots. Since a the angle of a circle is 360° or 2pi radians, you can add 2pi radians to the angle of the original complex number, then divide by 3 to determine the second root. Add 4pi radians to the original angle and then divide by 3 to determine the 3rd root. Then convert back to x-y coordinates if you want to:
Magnitude*(cos(angle) + i*sin(angle)), for each of the 3 angles that you determined.
See the question: 'Strategy for finding the cube root of complex numbers'
What is the abbreviation of a imaginary number?
The letter i represents the imaginary unit [square root of -1]. It's added after the number. So 5i means 5 units in the positive imaginary direction, or just 5 times sqrt(-1).
In electrical engineering, often the letter j is used, because i represents current in electrical notation.
CommentIn electrical engineering, the operator 'j' is usually placed in front of a number, not behind it: i.e. (a+jb) not (a +bj).
Is the universal set the set of all real numbers?
A set is a collection of objects such that you can tell whether or not it is in a set. This decision may be based on a list: {red, yellow, blue}, or description {primary pigments}. These elements need not share any characteristic: for example, {red, 3.6, Charles, water} is a set.
The universal set is simply the collection of objects from any of the sets under consideration.
Besides, all real numbers are not the largest set. The real numbers are a proper subset of the complex numbers, which are a subset of ... and so on.
What is the usefulness of the conjugate and its effect on other complex numbers?
The conjugate of a complex number is the same number (but the imaginary part has opposite sign).
e.g.: A=[5i - 2] --> A*=[-5i - 2]
Graphically, as you change the sign, you also change the direction of that vector.
The conjugate it's used to solve operations with complex numbers. When a complex number is multiplied by its conjugate, the product is a real number.
e.g.: 5/(2-i) --> then you multiply and divide by the complex conjugate (2+i) and get the following: 5(2+i)/(2-i)(2+i)=(10+5i)/5=2+i
Negative square roots are just the opposite of positive square roots. Since square roots (of positive numbers) are real, the negative square roots are also real.
Square roots of negative numbers are not real.
Note that -1 = exp(Pi*i), so (-1)^(1/2) = exp((1/2)*Pi*i) = i.
Note that exp(i*x) = cos(x) + i*sin(x), for instance by taking derivatives:
(d/dx)(exp(i*x)) = i*exp(i*x), and
(d/dx)^2(exp(i*x)) =(-1)*exp(i*x).
This means that the second derivative of exp(i*x) equals -exp(i*x).
The same property holds for cos(x) + i*sin(x):
(d/dx)(cos(x) + i*sin(x)) = -sin(x) + i*cos(x)
(d/dx)^2(cos(x) + i*sin(x)) = -cos(x) - i*sin(x) = -(cos(x) + i*sin(x)))
Hence cos(x) + i*sin(x)) = C + Dx + exp(i*x), for some C and D.
Comparing the values on both sides for x = 0, we find:
1 = C+1, so C = 0 and for the first derivative:
i = D + i, so D = 0.
So cos(x) + i*sin(x)) = exp(i*x) for all x.
by comparing x=0 for both functions and their first derivative. Since they coincide,
Why is log i 0.682i Where i is the imaginary number sqr rt -1?
By Euler's formula, e^ix = cosx + i*sinx
Taking natural logarithms, ix = ln(cosx + i*sinx)
When x = pi/2, i*pi/2 = ln(i)
But ln(i) = log(i)/log(e) where log represents logarithms to base 10.
That is, i*pi/2 = log(i)/log(e)
And therefore log(i) = i*pi/2*log(e) = i*0.682188 or 0.682*i to three decimal places.
How to solve imaginary complex numbers?
That depends a lot on what you want to solve. In general, you can do quite a lot by simply considering the complex number like any polynomial, and remembering that i2 = -1. For example, to add two complex numbers, you simply add the real and the imaginary part of both numbers.
What are some underlying causes for creating imaginary friends?
There are many causes to creating and having imaginary friends. Don't worry though, millions of kids have imaginary friends (and adults too, although adults it should be slightly worrying unless it's not affecting their health).
A huge, possibly the greatest contributor is loneliness. If you have nobody real to talk to, you'll probably make an imaginary friend to talk to.
But take heed! Don't let your imagination run away from you! Facing reality can sometimes be hard, but you must face your fears. Don't live in a fantasy world.
It's hard to heal from an addiction to imaginary friends, but if you are addicted, just listen to your heart. It will show you the right way.
Some younger children tend to make imaginary friends to create jealousy in their peers. For example, say Suzie and Ally got in a fight. Suzie made an imaginary friend so she could say to Ally, "My friend is better than you!" even though her new friend is imaginary.
And also, imaginary friends can help you with reality. They are someone you can boss around, and you can make them do anything; whatever the cause.
The square root of a negative value is called an or complex number?
The square root of a negative value is called an imaginary number.
What is the trigonometric form of a complex number?
The standard form of a complex number is the cartesian one; a plane with orthogonal axes for real parts and imaginary parts. A complex number has a pair of co-ordinates defining its position on the plane.
A trigonometric form is a plane with an origin, and one line from the origin to infinity. A complex number is defined by its distance from the origin and the angle between the datum line and the line joining the number to the origin.
It is just like co-ordinate geometry with co-ords r, theta instead of x,y.
678-471-xxxx whats that number for?
It's someone's personal cell phone number in Atlanta Georgia. I deleted everything except the area code and prefix so they won't be getting a million calls.
Set of real numbers and set of complex numbers are equivalent?
Real numbers are a proper subset of Complex numbers.
Does every real number have a sign that is plus or minus?
With the exception of zero, which is neither positive nor negative, every real number is either plus or minus. With positive number, the + sign is often implied and therefore is not always written [example: 3 and +3 both mean positive three]
Why do some quadratic equations have solutions which are imaginary or complex numbers?
The solutions of a quadratic equation in the form of ax^2+bx+c=0 are the points at which the parabola of ax^2+bx+c=y touches the x axis. An imaginary or complex solution to such a question implies that the parabola touches the x axis at a point not within the real x-y plane; to represent complex or imaginary answers, one must introduce a third dimension, and then the location at which the parabola crosses the y-axis will be apparent.
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