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For an imaginary number R*i, where R is a real number. It helps to know about complex numbers, and how they can be rewritten using polar coordinates.
A complex number can be considered a vector in the real-imaginary plane, starting at the origin, and ending at the coordinates of the complex number.
For a complex number a + bi, the magnitude of this vector is sqrt(a2 + b2) and the angle that it makes in a counterclockwise direction from the real axis is tan-1(b/a). The number then can be written (using Euler's Identity) as Magnitude*e(i*angle), with i being the imaginary number: sqrt(-1).
So if you want to take the fourth root of a number, then this is the same as raising it to the (1/4) power, so you'd take the (1/4) power of the real magnitude, then multiply by e(i*angle/4). Note that angles are in radians in this relationship, but for simplicity, I'll use degrees.
So in this real-imaginary plane, pure imaginary numbers lie on the vertical axis (angle = 90° for positive imaginary or 270° for negative imaginary). So for 'positive' imaginaries, just divide 90° by 4 = 22.5°.
But we want 4 roots, so note that once we go 360°, then we are at the same place as 0°, so if we add or subtract 360° to an angle, we get an angle pointing in the same direction.
- 90° + 360° = 450°; 450° / 4 = 112.5°
- 450° + 360° = 810°; 810° / 4 = 202.5°
- 810° + 360° = 1170°; 1170° / 4 = 292.5°
The corresponding radians are: 22.5° = pi/8; 112.5° = 5*pi/8; 202.5° = 9*pi/8; and 292.5° = 13*pi/8.
If you want the complex number root back in the format a + b*i, a = magnitude* cos(angle), and b = magnitude* sin(angle)
You can find all of the roots for any order root (square, cube, etc) by using this same method. It works for real numbers, too. Just take angle = 0°, 360°, etc. for positive numbers, and angle = 180°, 540°, etc. for negative numbers. Example square root of 1: 0° / 2 = 0° (positive 1), 360° / 2 = 180° (negative 1)
See the PDF in the related links: Using the Shannon Sampling Theorem to Design Compact Discs - page 2 for a brief explanation on Euler's Identity and how it was derived. There are many other references available.
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Is the square root a real number?
The square roots of any positive real number are a positive and a negative real number. The square roots of any negative real number are a positive and a negative imaginary number. The square roots of any imaginary number or any complex number are two complex numbers.
What is the fourth root of 1?
The two real roots are -1 and +1.There are also two roots in the complex field and these are ±i where i is the imaginary square root of -1.
What are the fourth roots?
Fourth roots are the inverse operation of raising a number to the fourth power. For a given number, the fourth root is a number that, when raised to the fourth power, equals the original number. For example, the fourth root of 16 is 2, since 2^4 = 16. In mathematical notation, the fourth root of a number x is denoted as √√x or x^(1/4).
How many cubed roots are there in number 27?
There are 3 cube roots of 27. There are 2 square roots of 27 ( or any real number ). There are 4 fourth roots of 27 and so on:)
What is the square root of -92 to the nearest whole number?
Square roots of negative numbers are imaginary.
