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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 write a complex number in standard form?
The standard from for a complex number is a + bi, where a and b can have any real value including zero and i = √-1
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"?
What are the 3 complex roots of -1 using the DeMoivre's theorem?
Problem: find three solutions to z^3=-1.
DeMoivre's theorem is that (cos b + i sin b)^n = cos bn + i sin bn
So we can set
z= (cos b + i sin b),
n = 3
cos bn + i sin bn = -1.
From the last equation, we know that cos bn = -1, and sin bn = 0.
Three possible solutions are bn=pi, bn=3pi, bn=5pi. This gives three possible values of b:
b=pi/3
b=pi
b = 5pi/3.
Now using z= (cos b + i sin b), we can get three possible cube roots of -1:
z= (cos pi/3 + i sin pi/3),
z= (cos pi + i sin pi),
z= (cos 5pi/3 + i sin 5pi/3).
Working these out gives
-1/2+i*sqrt(3)/2
-1
-1/2-i*sqrt(3)/2
How do you calculate root of any complex number using casio Fx 991MS calculator?
Not sure about the Casio, but most calculators which have capability to handle complex numbers should be similar. Input the complex number according to however you normally do that, then raise to a power. In the case of roots, you want to raise to a reciprocal power: Square root is 0.5 power, cube root is 1/3 power, fourth root is 0.25 power, etc
Is two thirds and four ninths equivalent?
Umm, no. Two thirds and six ninths are equivalent. They look like this 2/3 and 6/9, to get to 9 you times the 3 by 3 so to get the 6 you have to times the 2 by 3. I can see that you've seen that you have squared 3 so you had to square 2, but unfortunately that isn't what happens.
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
How do you simplify complex numbers?
Hi,
See the related answers and the Related Link. My friends are working hard to supply
more answers.
What is the square root of -45 using imaginary numbers?
Square root of -45 = (-45)1/2 = (-3x3x5)1/2 = 3(-5)1/2 = 351/2i
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.
[a(0), ... , a(n) should be written with suffices but this browser has decided not to be cooperative!]
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).
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
What is the name for 1000000000000000000000000000000000000000000000000000000000000000000?
For such large numbers, just write the number in scientific notation, like this: 10x, replacing "x" with the number of zeroes.
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.
What is seven eighths times two fifths?
When multiplying fractions we multiply all the numerators to produce a new numerator and we multiply all the denominators to produce a new denominator, then we reduce the new fraction to its simplest form.
In this case - 7/8 x 2/5 gives 7 x 2 = 14 as the new numerator and 8 x 5 = 40 as the new denominator
The "new" fraction is now 14/40 or 7/20
This forms an equation which solves nicely:
2x-7 = x-1
x-7 = -1
x = 6
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!
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.
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.
