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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 cube roots of an imaginary number in trigonometric form?
z = y i where y is a real number
1: z^(1/3) = y^(1/3) (cos(30 deg) + i sin(30 deg))
2: z^(1/3) = y^(1/3) (-sin(60 deg) + i cos(60 deg))
3: z^(1/3) = y^(1/3) (- i)
What is the twenty seventh power of an imaginary number?
See the Wikipedia article on Imaginary Numbers. i^n = i^(n mod 4). With n = 27, 27 mod 4 = 3, and i^3 = -i. This is easier to visualize when you consider the graphical representation of complex numbers, and use polar coordinates. Writing i as exp(i*pi/2), (from Euler's formula), then i^27 = {using exp() to mean the natural base e, raised to a power} exp(i*pi/2)^27 = exp(27*i*pi/2) = exp(13.5*i*pi) = exp((12 + 1.5)*(i*pi)) = exp(12*i*pi)*exp(3*i*pi/2).
Since the coefficient of i in the exponent is an angle (in radians), then even multiples of pi are the same angle as 0 {exp(0) = 1} so we are back to the same as exp(3*i*pi/2), which is pointing straight down [-i]. Note that 3*pi/2 radians is the same as 270°.
Since the question asked about 27th power of an imaginary number, that could mean a multiple of i, such as bi, where b is any real number. In this case, you would have (bi)^27 = (b^27)(i^27) = (b^27)(-i). So if b = 1.5 for example, then you would have (-i)(1.5^27) ≅ -56815i.
Natural numbers, integers, rational numbers are all part of the real numbers.
-1 is an integer (and a rational number), so it is also real.
How can you square a complex number in trigonometric form?
Multiply the angle by 2, and square the magnitude. The angle can be rewritten between (-180° & +180°) (or -pi and +pi radians), after multiplying.
To any set that contains it!
It belongs to {27},
or {45, sqrt(2), 27, pi, -3/7},
or all whole numbers between 23 and 53,
or multiples of 3,
or composite numbers,
or counting numbers,
or integers,
or perfect cubes,
or rational numbers,
or real numbers,
or complex numbers,
etc.
I will be glad to give you a complex number. (i^i)^i is i raised to the i power and
raised to the i power again. This is not the same as i^3 which is -i but computes to .947 + .321i
The canonical example is the square root of -1.
Mathematicians use the symbol i to represent it, electrical engineers use j because i is already busy.
What is used to divide complex numbers?
You can use another complex number, a real number or an imaginary number. Complex number equations make interesting images. The link shows the image
produced by (z-1)/(z+1) and inverses the checkerboard around two points.
What is the meaning of the number sets?
A number set is simply a collection of numbers. The numbers in a set need not share any property whatsoever - the only requirement is that they are all numbers.
How would you solve an imaginary number with -31 as its exponent?
The following discussion is for complex numbers; this includes (pure) imaginary numbers as a special case. This type of powers (a complex number to the power of a real number) are very simple if you write the complex number in polar coordinates, specifying an absolute value and an angle. Raise the absolute value to the specified power, and multiply the angle by the power.
Example (writing on a piece of paper is clearer; it is difficult to represent some of the symbols here):
(1 + i)6 = [(square root of 2) angle (45 degrees)]
Square root of 2 to the power 6 is 8.
45 degrees x 6 = 270 degrees, which is the same as minus 90 degrees.
The result is, then, 8 at an angle of -90 degrees. Converting this back to rectangular coordinates, this is equal to -8i.
What is icon dance complex number?
No but check it up in ICON dance complex and there it is and also that go to Atlantic city ad go to Texas ave school and ask for girls that can auction for them because some girls like they called them self THE JERSEY KIDS and they are girls who dance and sing and act too:)
Is any number divisible by zero is a multiple of a complex number?
no o is nothing if you try dividing it you will get a fraction
Are all imaginary numbers are real numbers?
No. None are because the opposite of a real number is an imaginary number. In real numbers there are rational, irrational, counting, whole numbers, and integers.
What are two complex numbers where neitjer a nor b are zero whose product is a real number?
Lets see............
(a+ib)(a-ib) = a^2+iab-iab+b^2
= a^2 + b^2
a and b can be any values and a^2+b^2 will be real.
What is the largest number of pieces you could cut a pie into with three straight cuts of a knife?
Eight. Use the first two cuts to cut the pie into four pieces, then use the third cut to slice the top from the bottom, doubling the number of pieces to 8.
