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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
Find the complex conjugate of 14 plus 12i?
To find the complex conjugate of a number, change the sign in front of the imaginary part. Thus, the complex conjugate of 14 + 12i is simply 14 - 12i.
How do you calculate root of any complex number using casio Fx 991MS calculator?
Not sure about the Casio, but most calculators which have capability to handle complex numbers should be similar. Input the complex number according to however you normally do that, then raise to a power. In the case of roots, you want to raise to a reciprocal power: Square root is 0.5 power, cube root is 1/3 power, fourth root is 0.25 power, etc
What are the 3 complex roots of -1 using the DeMoivre's theorem?
Problem: find three solutions to z^3=-1.
DeMoivre's theorem is that (cos b + i sin b)^n = cos bn + i sin bn
So we can set
z= (cos b + i sin b),
n = 3
cos bn + i sin bn = -1.
From the last equation, we know that cos bn = -1, and sin bn = 0.
Three possible solutions are bn=pi, bn=3pi, bn=5pi. This gives three possible values of b:
b=pi/3
b=pi
b = 5pi/3.
Now using z= (cos b + i sin b), we can get three possible cube roots of -1:
z= (cos pi/3 + i sin pi/3),
z= (cos pi + i sin pi),
z= (cos 5pi/3 + i sin 5pi/3).
Working these out gives
-1/2+i*sqrt(3)/2
-1
-1/2-i*sqrt(3)/2
Is two thirds and four ninths equivalent?
Umm, no. Two thirds and six ninths are equivalent. They look like this 2/3 and 6/9, to get to 9 you times the 3 by 3 so to get the 6 you have to times the 2 by 3. I can see that you've seen that you have squared 3 so you had to square 2, but unfortunately that isn't what happens.
How do you enter an imaginary number in a scientific calculator?
Some scientific calculators can't handle complex or imaginary numbers. If you happen to have a special calculator that does, probably the manual will tell you how to enter them.
The HP 48 and up series does. It depends on if your calculator is in Polar Coordinate mode or X-Y coordinate mode, but a quick way to get the imaginary number i (regardless of which mode the calculator is currently in), is to press -1, then 'square root' button.
Are imaginary numbers rational or irrational?
Imaginary numbers are not intrinsically rational or irrational.
Of course, all real numbers are either rational or irrational numbers.
Imaginary numbers are not real numbers.
Imaginary numbers have a real part and an imaginary part, sometimes written like z=x+i y.
The two parts, i.e. the x and the y, are real numbers. As real numbers, they are either rational or irrational. Its just that the two parts of a complex number may both be either rational or irrational or one may be rational and the other irrational. One could always make up a new name for these cases, but right now there is no such classification.
No. i = square root of -1, and therefore imaginary. Only by squaring i can you have a real number.
Is this number real 100000000000000000000000000000?
Yes and it would normally be expressed in scientific notation.
What kind of numbers belongs to the family of real numbers?
Numbers that include real numbers are natural numbers, whole numbers, integers, rational numbers and irrational numbers.
Is the difference of a complex number and it's conjugate an imaginary number?
Yes. By definition, the complex conjugate of a+bi is a-bi and a+bi - (a - bi)= 2bi which is imaginary (or 0)
What is a non-zero complex number?
A non-zero complex number is like 5-i2.
A lot of complex numbers can make interesting designs. Please see the below
LINK for an example and an image.
What is the correct notation for the complex number square -81-14?
-81-14 is not a complex number. And its square is 9025.
Do any of these sets have element in each common of real number?
It kind of depends on what "these" sets are.
What are the 3 undefined terms in geometry and their definition?
Points, lines & planes.
Point - a dot on a page. A point has no dimensions (length, width, height), it is usually represent by a capital letter and a dot on a page. Think of it as an infinitely small place or position on a map.
Line - an unlimited number of points along the same path. The set of points may be straight or form a curve. Normally, the term 'line' means a straight line. The 'line' has no dimensions (length, width, height) and extends unlimited in both directions. (The part of a line defined by two points, called 'endpoints', is call a line segment or 'segment'.) A 'line' is represented by a drawn line with arrowheads on each end to represent that it doesn't have endpoints. A line can be named in two ways: (1) use the capital letters representing two points on the line and place a double-headed arrow above the two letters (2) use a lowercase letter beside the line to represent it.
Plane - a flat surface that extends indefinitely in all directions. It is usually represented by a parallelogram (four sided figure) with a capital letter in one corner. Remember the plane is not limited by the parallelogram, it extends infinitely. The plane can also be represented by using three points that lie on the plane surface but not on the same line (noncollinear).
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)
Are imaginary numbers real numbers?
When people started classifying numbers in different ways Some numbers were grouped together and called Real numbers. Solutions that would create Imaginary numbers were simply explained away as impossible, later the rules for working with these numbers, but, even though they are not considered Real numbers some math operations will create Real number answers.
Why do you say that pi is an imaginary number?
If I ask Answers™ "what is pi squared?" I find "It is approximately equal to 3.14 but in reality pi is an imaginary number that has no end." The answer also goes on to tell me that imaginary numbers cannot be multiplied by themselves. Now i must see what y'all have to say about imaginary numbers...
When were imaginary numbers developed?
Imaginary numbers were first recognised in the first century CE by Heron of Alexandria but development was slow because "the establishment" did not consider these to be proper numbers. Gerolamo Cardano, in his work on finding roots of cubic equations in early 16th century CE, set out some of the rules for manipulating complex numbers. Rafael Bombelli set down the rules for multiplication of complex numbers later in that century. However there was no serious work done on these numbers for a long time: their name did not help. It was not until two of the giants of mathematics, Leonhard Euler and Carl Friedrich Gauss in the 18th century worked on them that they were accepted as worthy of attention by serious mathematicians! And the rest, as they say, is history!
What is the rest of this number pattern 1 5 30 210 1680?
The next term after the 1680 is 15,120 and the one after that is 151,200 .
We can't give you the "rest" of it, because it has an infinite number of terms.
Yes. Both ziggurats and pyramids were huge tasks that required the labor of thousands to complete.
