0
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 standard form of complex numbers?
a + bi. a & b are both real numbers. the number i is the imaginary unit equal to the positive square root of -1.
What are the kinds of complex numbers?
Complex numbers include real numbers, pure imaginary numbers, and the combination of those two.
no because it has length
In complex geometry, an imaginary line is a straight line that only contains one real point.
What are the order of operations when dealing with polynomials and imaginary numbers?
The order or operation remains the same as that for ordinary numbers.
- Brackets or Parentheses
- Index or Exponent
- Division and multiplication (left to right)
- Addition and subtraction (left to right).
Can the product of two non real complex numbers be a real number?
Yes. Consider as the simplest example: i * i = -1. But there are others: (a + bi)(a - bi) = a² + b². When you multiply conjugates, the result is always real. This is useful when dividing to get a pure real number in the denominator.
2.5 is not a natural number.
as rational numbers are divided into Integers and Fractions
2.5=5/2 which is a fraction
What kind of jobs use logarithms and complex numbers?
Various engineering fields will use them. Electricity and electronics.
Is every complex number a pure imaginary number?
No. For example the number 1+i.
Pure imaginary complex numbers are of the form 0 + a*i, where a is a non-zero real number.
It is not always better.
Although quadratic equations always have solutions in the complex system, complex solutions might not always make any sense. In such circumstances, sticking to the real number system makes more sense that trying to evaluate an impossible solution in the complex field.
What are the fourth roots of an imaginary number?
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.
What is the numerical value of one mole of a substance?
1 mol = 6.02x1023 atoms or molecules or individual units
What are some imaginary things?
Imagination includes whatever u think should have or must have happened,Imagination differs from every person i cant guess thing u want as answer
What is the meaning of Complex conjugate reflection?
For a complex number (a + bi), its conjugate is (a - bi). If the number is graphically plotted on the Complex Plane as [a,b], where the Real number is the horizontal component and Imaginary is vertical component, the Complex Conjugate is the point which is reflected across the real (horizontal) axis.
Why complex numbers are denoted by the symbol z?
Possibly because x and y are used to denote the real and imaginary parts, respectively.
The two points would have to be plotted in 4-dimensional hyperspace. Conceptually straightforward, physically impossible.
Who discovered complex numbers in mathematics?
That's difficult to say. Rafael Bombelli defined an imaginary number in 1572, but Rene Descartes actually gave the term imaginary. Nobody seemed to have much use for them until the work of Euler and Gauss in the 1700's and 1800's. This information I got from the Wikipedia article on Imaginary Numbers.
What is the need for complex numbers?
The square of any real number cannot be negative. However, there are equations whose solutions require the square root of negative numbers. The real number system was extended to the set of complex number to allow such operations. In some ways, this is analogous to the set of integers being extended to the set of rational numbers to allow division (when the denominator was not a factor of the numerator), or the set of rational numbers being extended to real numbers to allow square (and other) roots.
No, but √-23 is. An imaginary number is the square root of a negative number.
Yes, that's right, because it converts the denominator into a purely real number. It makes the problem easier to deal with because you don't have to divide by a complex number any more.
So (a+jb) / (c+jd) = (a+jb)*(c-jd) / [(c+jd)*(c-jd)] = (a+jb)*(c-jd) / [c*c+d*d]
= (ac+bd) / [c*c+d*d] + j (bc-ad) / [c*c+d*d] . . . job done (note lack of proper font for squares).
The term inside the square root symbol is called the radicand. There isn't a specific term for it based on its sign; whether it's positive or negative, it's still the radicand.
I'm a little confused by your reference to the quadratic equation.
If the radicand is negative, the root is an imaginary number, though that doesn't specifically have anything to do with the quadratic equation in particular.
If the quantity b2 - 4ac is negative in the quadratic equation, the root of the quadratic equation is either complex or imaginary depending on whether or not b is zero.
---------------------------
Thank you to whoever answered this first; you saved me a bit of trouble explaining this to the asker :)
However, in the quadractic equation, the number under the radical is called the discriminant. This determines the number of solutions of the quadratic. If the radicand is negative, this means that there are no real solutions to the equation.
What is the differrent of imaginary numbers to complex numbers?
Imaginary number is a number that consist of only Imaginary part. Such as i, 40i, 1/2i, etc.
While the difference between the imaginary numbers and the complex numbers are that complex number also contains Real numbers, and can be written as a + bi. For example, 30+i, 1/2+1/2i, etc.
