0
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.
Wiki User
∙ 13y ago
Add your answer:
Earn +20 pts
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?
Square roots of square roots. The fourth root of a number, x, is another number, y such that y*y*y*y = x ie y multiplied by itself 4 times is x.
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:)
Is the square root of 2 imaginary?
No, but it is irrational, because there is no rational number whose square is two. Imaginary numbers are the square roots of negative numbers.
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?
Square roots of square roots. The fourth root of a number, x, is another number, y such that y*y*y*y = x ie y multiplied by itself 4 times is x.
How does the sign of a determine the number of real fourth roots of a and the number of real fifth roots of a?
If "a" is positive, it will have two fourth roots, one will be positive and one will be negative it will have one fifth root, which will be positive. If "a" is negative, it will have one fourth root, which will be negative. it will have one fifth root, which will be negative.
What is a real root?
In the context of algebra, the term real root refers to the solution to an equation which consists of a real number rather than an imaginary or complex number (a complex number being a combination of real and imaginary numbers). You may recall that any given equation will have the same number of roots (or solutions) as the highest exponent in the equation, so that if you are dealing with x squared, you have two roots. Often there would be one real root and one imaginary root. In general, the real roots are more useful, although there are some circumstances in which imaginary or complex roots are also relevant to what you are doingl.
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:)
Is the square root of 2 imaginary?
No, but it is irrational, because there is no rational number whose square is two. Imaginary numbers are the square roots of negative numbers.
What is the square root of -92 to the nearest whole number?
Square roots of negative numbers are imaginary.
Are square roots real numbers?
Yes, if the number whose square root we are taking is greater than 0. Only if you try to take the square root of a negative number will you get back an imaginary number. Square roots are often irrational, but that's different from real versus imaginary.
How do you find the imaginary roots of p of x equals x to the fourth plus 2x cubed plus x squared plus 8x minus 12?
p(x) = 0 => x4 + 2x3 + x2 + 8x - 12 = 0=> (x + 3)*(x - 1)*(x2 + 4) = 0So the imaginary roots of p(x) are the imaginary roots of x2 + 4 = 0that is x = ±2i
What is the fourth roots of 324?
It does not come out as a whole number but it's 4.242640678
The polynomial 4x2 plus 5x plus 4 has how many roots?
None, it involves the square root of a negative number so the roots are imaginary.
