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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
Can a real number be a pure imaginary number?
even if you can't see it.... the number is real. but question to that, what's the number?
* * * * *
Clearly, the above answer was provided by someone who had no knowledge of the complex field.
A real number cannot be a pure imaginary number, nor can it contain any part that is imaginary.
In the context of the coordinate plane, the real numbers can be seen as the x-axis, the pure imaginary numbers form the y-axis excluding the origin. Complex numbers are all the point in the four quadrants.
Which operation involves complex numbers requires the use of a conjugate to be carried out?
One operation that is used a lot in quantum mechanics is taking the absolute value of the square of a complex number. This is equivalent to multiplying the complex number by its complex conjugate - and doing this is simpler in practice.
What are the differences between real numbers and imaginary numbers?
The set of real numbers is not closed under powers. That is to say, there are some equations of the form y = xa which do not have a solution within the set. Typical example: x is negative, a = 0.5
What is triangular inequality of complex number?
The absolute value of the sum of two complex numbers is less than or equal to the sum of their absolute values.
How do you simplify the complex fraction 1 over 4 plus 3 over 5 divided by 1 over 2 plus 1 over 5?
1/4 + 3/5//1/2 + 1/5
attack the numerator and denominator separately by adding the fractions. Need like denominators. numerator first, 20 looks good
5*1/5*4 + 4*3/4*5
5/20 + 12/20
= 17/20
-------------------your numerator
Now, the denominator. 10 looks good
5*1/5*2 + 2*1/2*5
5/10 + 2/10
= 7/10
--------------your denominator
now,
17/20//7/10
same as saying
17/20 * 10/7
= 170/140
reduces to, ( of course )
17/14
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Can you evaluate all integration in complex plane?
No. There are many functions that cannot be integrated.
How do you change a number from standard form to normal form?
If you were to draw your complex number on the complex plane, theta would be the angle between the number and the positive x axis, and r would be the length of the line. r is easiest to find; simply put x and y into the equation Sqrt(x2 + Y2)and the result will be r, the length of the complex number.
To find theta, you must picture a triangle, imagining the length y to be the opposite, and the length x to be the adjacent, and performing arctan(y/x) to find theta - however, be careful as depending on what quadrant the number is in, you may have to perform further operations in order to find the true angle.
Conjugate and a complex number equal?
The conjugate will have equal magnitude. The angle from the real axis will be the same angle measure (but opposite direction).
A complex number is denoted by Z=X+iY, where X is the real part and iY is the
imanginary part. So the number 4 would be 4+i0 and is the real part of a complex
number and so 4 by itself is just a real number, not complex.
This might be a complex number and its conjugate: (a + bi) times (a - bi). More generally, any two complex numbers such that the angle formed by one is the negative of the angle formed by the other. In other words, you can multiply the conjugate by any real constant and still get a real result: (a + bi) times (ca - cbi).
Specific examples:
Multiply (3 + 2i) times (3 - 2i).
Multiply (3 + 2i) times (6 - 4i).
Are 2 purely imaginary complex numbers multiplication complex?
No.
Suppose bi and di are two complex purely imaginary numbers such that b and d are real.
Then
bi * di = bdi2 = -bd which is real.
Why square root of negative 36 is not a real number?
When you multiply any number by itself, the result is always positive. You cannot take any real number and square it to get something negative. The square root of -36 is 6i.
What 2 parts does complex number have?
A complex number has a real part and imaginary part. In x+iy, the x is real and the iy is imaginary. In 3d, you can have x+iy+ik. Complex numbers produce amazing fractals.
A wise man said complex numbers are useless and to be avoided at all cost.
Netflyer
Does the i in ipod have anything to do with imaginary numbers?
Back when the iMac first came out, I believe the i stood for Internet. The iMac was the first computer without a floppy disk drive. The idea was that with the Internet and email becoming so popular, more and more people were emailing files to each other, rather than saving them to a floppy disk.
This part is a guess, but I think the letter i became popular enough, that they decided to use it with the ipod, iphone, and now the icloud.
Where do you find complex numbers in algebra?
Certain functions, when solving to find the zeros (value which makes the function equal zero), the only value which will work has an imaginary component. Note that a parabola (graph of a quadratic or 2nd order polynomial) can touch the x-axis at a single point, or 2 points or no points. If it does not touch or cross the x-axis, then the root (or zeros) of the function are complex with imaginary components.
Technically, all real numbers are a subset of complex numbers, so all numbers are complex - but this is not how we normally refer to them. We usually say that a number is real, or it is imaginary, or it is complex.
Applications of complex numbers and complex variables in engineering?
Complex equations and their graphs are used to visualize electrical and fluid flow in the real world. For instance, the equation z+1/z can show the flow of water around a circular piling. The equation (z-1)/(z+1) and the graph can show the electrical force fields around two parallel wires carrying current.
Is -4 a nonreal complex number?
No. Negative four is a real number. All real numbers are also complex numbers, so it is a complex number (but it's real, not nonreal)
How do you calculate complex frequency?
As far as know, frequency is not complex and is a positive number
denoted in cycles per seconed like 60 cps.
0 cps is sometimes noted as DC, direct current with frequency of zero.
What is the symbol for imaginary numbers?
It's nice and simple, just i
So for example (-4)1/2
is 2i or -2i
According to my calculations, (-2)^.5 eq. 1.414i or -1.414i
Electrical engineers often use j to represent the imaginary unit, rather than i. This is to avoid confusion with the symbol for electric current, which is I.
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What is difference of complex numbers?
If y = a + bi and z = c + di are two complex numbers then
z - y = (c - a) + (d - b)i
