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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
Why was complex numbers discovered?
They were discovered when Cardano solved the third degree equation.
In the formulas that arose to solve the third degree equation, Cardano needed to take the square root of negative numbers and add them up in a certain way. The strange thing that happened was that the formulas used these complex numbers, even if the solutions to the equation where all real. This baffled the mathematicians of the time, because how could these strange numbers turn out to be "real"?
Later this was considered totally correct, when the field of complex numbers was better undestood.
If z equals a plus ib then show that arg conjugate of z equals 2pi -arg z?
If z = a + ib
then
arg(z) = arctan(b/a)
Let z' denote the conjugate of z. Therefore, z' = a - ib
Then
arg(z') = arctan(-b/a) = 2*pi - arctan(b/a) = 2*pi - arg(z)
Which set of numbers does 9 belong to?
9 belongs in the sets:
-Natural number set, positive whole numbers
-Integer number set, whole numbers
-Rational number set, numbers that are not never ending
-Real number set, basic numbers without i and that can be expressed in say amounts of apples
-Complex number set, the set that contains both real and unreal numbers
What is the complex number of its distance for the origin in the complex plane?
This is called the magnitude. It can be found (for a complex number a + bi) as:
(where a & b are both real numbers and i is the imaginary unit)
sqrt(a^2 + b^2)
Find the fourth complex roots of w equals 81 and plot them?
if the question is w^4 = 81 {w raised to the power of 4},
Then the four roots are w = {3, -3, 3i, -3i}.
The plots on the real-imaginary plane would be the points:
- (3,0)
- (-3,0)
- (0,3)
- (0,-3)
Who first wrote about complex numbers?
The Italian mathematician, Gerolamo Cardano was the first to consider the concept of complex numbers. However, he did not develop a theory of complex numbers to any extent. That was left to Rafael Bombelli.
How do you use complex numbers?
Better get a textbook that explains this in more detail. You can also get a brief summary at Wikipedia, or other online sites.
In any case, here is a brief summary.
For addition and substraction, you add (or subtract) the real and imaginary parts separately. For example, (4 + 3i) + (7 - 2i) = 11 + 1.
For multiplication, multiply each part of one number with each part of the other number - and remember that i2 = -1. For example, (4 + 3i) x (7 - 2i) = 28 - 8i + 21i - 6i2 = 28 + 13i - 6(-1) = 34 + 13i.
Division is a bit more complicated. For example, to divide by (3 + 4i) you have to multiply numerator and denominator by the complex conjugate of this number, that is, change the sign of the imaginary part; in this case, (3 - 4i).
Multiplication and division are actually quite a lot easier if you convert the complex number to polar coordinates, that is, a distance and an angle. Here is a quick example: (4 angle 30 degrees) x (5 angle 20 degrees) = (4 x 5) angle (30 + 20 degrees) = 20 angle 50 degrees (a length of 20, at an angle of 50 degrees). Most scientific calculators have special functions to convert from rectangular to polar coordinates and back.
The answer is Yes, for the purposes of most Math students. There are, number systems that have been devised which are outside the set of Complex Numbers, though.
How could you use Descartes' rule to predict the number of complex roots to a polynomial?
Descartes' rule of signs will not necessarily tell exact number of complex roots, but will give an idea. The Wikipedia article explains it pretty well, but here is a brief explanation:
It is for single variable polynomials.
- Order the polynomial in descending powers [example: f(x) = ax³ + bx² + cx + d]
- Count number of sign changes between consecutive coefficients. This is the maximum possible real positive roots.
- Let function g(x) = f(-x), count number of sign changes, which is maximum number of real negative roots.
Note these are maximums, not the actual numbers. Let p = positive maximum and q = negative maximum. Let m be the order (maximum power of the variable), which is also the total number of roots.
So m - p - q = minimum number of complex roots. Note complex roots always occur in pairs, so number of complex roots will be {0, 2, 4, etc}.
Can you factorise numbers on a scientific calculator?
Up till now you can't but hopefully will be able to do so soon.
What is a nonzero real number?
It's any real number distinct from 0.
For instance, in the expression x/y, where x and y are real numbers, y needs to be a nonzero real number. This is because otherwise the expression x/y is undefined (viz. x/0).
Can you divide infinity by an imaginary number?
Yes, but the answer will be in infinities within the complex domain. Unless you know what you are doing there, stay away ;)
Can every point on the number line be named by a real number?
Yes, as long as the number line is entirely real.
I assume you mean googolplex (sounds similar).
Googolplex is real, its definition is based on Googol (which inspired the name Google, seriously).
A googol is 10^100.
A googolplex is 10^googol.
To try and imagine the size of this number, consider that
the number of atoms in the universe is about 10^80 (this is not yet a googol).
In other words, a googol is more than the number of atoms in the universe.
a googolplex is a number so big that you can not write it, because it has a googol+1 digits (all zero, except the leading 1). So, to write it you would need at least 1 atom to write a digit, but there aren't enough atoms in the universe to write it. Heavy!
Of course, there are bigger numbers than this!
Consider numbers with a name
Consider googolplexplex, googolplexplexplex, etc.
Consider googol(plex . . . . . . plex)
etc.
All these numbers are incredibly big natural numbers, but FINITE. They are all real.
How do you find caste certificate number on internet?
Depending on where the Caste certificate was applied for and registered, is where one might start to look. One may obtain it over the net by contacting the same place they registered with.
Assuming x is your number, x * 10^n = x moved n decimal places.
When n is positive, move the decimal point n places to the right.
When n is negative, move the decimal point n places to the left.
When n is 0, do nothing.
Is every complex nberan imaginary number?
No. A complex number consists of a real part and a imaginary part. If the real part equals zero, there is only the imaginary left and you could therefor argue that it is an imaginary number (or else it would still be a complex number -with a real part=0)
