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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
What are the engineering applications of complex numbers and matrices?
I suggest asking separate questions for complex numbers, and for matrices.
Complex numbers are used in a variety of fields, one of them is electrical engineering. As soon as AC circuits are analyzed, it turns out that complex numbers are the natural way to do this.
Find the complex fourth root of 256i Express your answer in a plus bi form?
The four roots of 4√256 are {4, -4, 4i, and -4i}. Note that two of them are real numbers and the other two are pure imaginary, therefore 0 + 4i is the same as just 4i
Does trichotomy axiom hold true for complex numbers?
I think so. In x+iy, x and y are real numbers and have to be <0,0 or >0.
Why must you rationalize complex numbers and not divide them?
Since a Complex number has a real component and an imaginary component, it would be like trying to divide z / (2x + 3y)
Can a no real number be a pure imaginary number?
If a number is pure imaginary then it has no real component. If it is a real number, then there is no imaginary component. If it has both real and imaginary components, then it is a complex number.
What are the applications of complex numbers in real life?
- As with many other topics of advanced math, it depends whether your "real life" includes working in some engineering area, or not. If you work in electrical engineering or electronics, you will use complex numbers on a daily basis; similarly if you do a lot of math for some reason.
- Complex Numbers (ones involving the imaginary unit 'i' which is defined to handle the square root of -1) are a convenient way to describe the behavior of some electric circuits and waves.
- One thing you may encounter in your 'real life' are uninterruptible power supplies (UPS - what keeps your computer running when the power goes out). You will see ratings such as 500 VA / 300 W. This has to do with Complex Power. Electric circuits, which have inductances (such as transformers and electric motors) or capacitors, behave in a way that is described by complex numbers. While Volts X Amps equals Watts, the 300 W is the 'real power' (the amount available to do work), and the 500 VA is the apparent power (the total power that the power supply is capable of producing). The reactive power (due to capacitance & inductance) is represented by the imaginary component and it is at right angles to the real power (represented by the real component).
- Another use of complex numbers is in the Fast Fourier Transform (FFT) - one of the most ubiquitous algorithms - used heavily for signal processing. If you have a digital camera, a cell phone, an LCD - FFT is there, bringing complex numbers along.
- The complex equation w=z+1/z is a basic formula used for designing air foils-airplane wings and Figuring out flow forces around a circular object in water for instance.
Please see the related links for details.
How do you subtract imaginary numbers?
When adding and subtracting complex numbers, you can treat the "i" as any variable. For example, 5i + 3i = 8i, 5i -3i = 2i, etc.; (2 + 5i) - (3 - 3i) = (2 - 3) + (5 + 3)i = -1 + 8i.
Is Every complex number is a pure imaginary number?
No. All Complex Numbers are of the form a + bi where a and b are Real Numbers and i is the square root of -1. So only ones where a = 0 are pure Imaginary Numbers.
Applications and use of complex numbers in engineering?
Any physical motion which is periodic, such as an oscillating beam, string, wire, pendulum, electronic signal, or electromagnetic wave can be represented by a complex number function. This can make calculations with the various components simpler than with real numbers and sines and cosines.
Find the absolute value of the complex number z equals 3 plus 4i?
('|x|' = Absolute value of x)
|3+4i| = √(32 + 42)
= √(9+16)
= √25
= 5
Thus |3+4i| = 5.
How do we apply complex number in our life?
If you've ever flown then you used complex numbers. The basic equation w=z+1/z is used to design air foils (airplane wings). While you don't actually concern yourself with these equations anymore than the thermodynamic equations that govern the running of your car's engine.
Check out the related link for some interesting application of imaginary and complex numbers, though.
What is a leprecon is he real or imaginary?
Leprecon's only exist after generous portions of Irish stout. How do you know if it's a "he"?
How do you simplify 9x plus 6-4x-2x plus 1-15?
Assuming the equation is exactly as you described in words, it would be expressed algebraically as 9X+(6-4X-2X)+(1-15)
9X+(6-4X-2X)+(1-15) First, Combine all like terms inside the parenthesis
9X+(6-6X)+(-14) Next, clear the parenthesis. This is done by distributing the "1" coefficient implied by the parenthesis over the entire parenthesized term.
9X+1(6-6X)+1(-14)
9X+6-6X-14 Combine like terms again to get your answer!
=3X-8
What is the significance of complex roots?
One significant feature of complex numbers is that all polynomial equations of order n, in the complex field, have n solutions. When multiple roots are
Given any set of complex numbers {a(0),  … , a(n)}, such that at least one of a(1) to a(n) is non-zero, the equation
a(n)*z^n + a(n-1)*z^(n-1) + ... + a(0) has at least one solution in the complex field.
This is the Fundamental Theorem of Algebra and establishes the set of Complex numbers as a closed field.
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The above solution is the complex root of the equation.
In fact, if the equation is of order n, that is, if the coefficient a(n) is non-zero then, taking account of the multiplicity, the equation has exactly n roots (some of which may be real).
How the set of real no is uncountable?
There is no one to one correspondence between the real numbers and the set of integers. In fact, the cardinality of the real numbers is the same as the cardinality of the power set of the set of integers, that is, the set of all subsets of the set of integers.
An equilateral triangle is always inscribed in a circle.
This means that if you can prove that z1, z2 and z3 are the vertices of an equilateral triangle, they automatically lie on a circle subscribing it.
Compute |z1-z2|, |z1-z3| and |z2-z3|. These need to be equal for z1, z2 and z3 to lie on an equilateral triangle. If not, they aren't lying on an equilateral triangle.
for z=a+ib, |z| = (a^2+b^2)^(1/2).
To find the center c of the circle, note that (z1-c)+(z2-c)+(z3-c) = 0, hence,
c = (z1+z2+z3)/3.
You cannot show it in general since it need not be true!
Who first used the term imaginary to describe numbers?
It looks like it was most likely Rafael Bombelli in 1572, from what I can find. I've posted a related link to a Wikipedia article. Wikipedia should not be your sole source of information, but it's a starting point.
To get the complex conjugate, change the sign in front of the imaginary part. Thus, the complex conjugate of -4 + 5i is -4 - 5i.
