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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 is the domain of -1 to the x if you restrict the range to the real numbers?
(-1)^-2 = 1
(-1)^-1 = -1
(-1)^0 = 1
(-1)^1 = -1
(-1)^2 = 1
(-1)^3 = -1
For every whole number n (-1)^n equals either positive or negative 1.
How do you multiply a number containing a complex number?
One way, use binomial multiplication. Example: (w + x)*(y + z) = {using the FOIL method} w*y + w*z + x*y + x*z, so if we have two complex numbers multiplied:
(a + b*i)*(c + d*i) = a*c + a*d*i + b*c*i + b*d*i*i, but i*i = -1, so this becomes:
(a*c - b*d) + (a*d + b*c)*i
Another way to express complex numbers is as a magnitude and an angle. If this is the case, then you multiply the two magnitudes, and add the angles, then reduce the resultant angle to within -180° and +180°.
If you have a real X complex, then just use {b=0} in (a + b*i), so then you have:
(a*c) + (a*d)*i *Or if using the polar coordinates, take the angle as 0° for a positive real number and 180° for a negative real number, then add the angles.
How do you write 0.033579584 in words?
Thirty-three million, five hundred seventy-nine thousand, five hundred eighty-four billionths.
What is the result of the addition of two pure imaginary numbers?
An imaginary number.
Think of imaginary numbers as being on a vertical line while real numbers are on a horizontal line. (the lines cross at zero).
Adding and subtracting won't change the axis.
integers
What do you think is true of the square roots of a complex number?
I posted an answer about cube roots of complex numbers. The same info can be applied to square roots. (see related links)
Why is 3459 not an even number?
Because 3459 is not completely divisible by 2. Or 3459 is not multiple of 2.
What is the number for someone not wanting to give you there real number?
The answer depends on where real number!
Is a negative sign has a square root?
- Yes, a negative sign has a square root.
- This is done through using the imaginary unit defined as: i = .
- for example the square root of -36 is square root of (-1) multiplied by square root of 36. Accordingly it equals 6i
How the differences of the two squares relates to the product of a complex number and its conjugate?
In real number, when multiplying the difference of two squares, the middle term disappears:(a + b)*(a - b) = a^2 - ab + ab + b^2 = a^2 - b^2
Similarly, with the complex number and its conjugate, the imaginary term cancels out.
(a + ib)*(a - ib) = a^2 - iab + iab + (ib)^2 = a^2 - i^2b^2 = a^2 + b^2.
If you put it into matrix form, A*x = p. With A a square matrix, and x & p are column matrices, then you can separate matrix A becoming B + Ci where B contains the real parts of the coefficients, and C contains the imaginary components, then p = m + ni, and x = y + zi, then the real parts must be equal, and the imaginary parts must be equal, so:
B*y = m, and C*z = n, solve each set of equations for y & z, then your solution set will be x = y + zi.
A complex number, equations and graphs can show electromagnetic forces-for instance two wires carrying current. A formula like (z-1)/(z+1) can show the fields around two parallel wires.
What are Real and Complex numbers?
Real numbers are numbers that you are already familiar with: integers, fractions, and irrational numbers: 1, 2, 0, -5, ¾, sqrt(2), pi, etc. Next, you need to know about imaginary numbers. These are numbers that, when squared, will give a negative real number. No real number can do that. Now imagine a graphical plane, with the real axis on the horizontal, and the imaginary axis on the vertical. This is called the complex plane, and any combination of real and imaginary numbers can be plotted on the complex plane.
The set of complex numbers includes all real numbers as well as all imaginary numbers, and the combination of the two.
