Complex Numbers

Always. The set of imaginary numbers is a subset of complex numbers. Think of complex numbers as a plane (2 dimensional). The real numbers exist on the horizontal axis. The pure imaginary are the vertical axis. All other points on the plane are combinations of real and imaginary. All points on the plane (including imaginary axis and real axis) are complex numbers.

I got -1.

"a + bi" is a common way to write a complex number. Here, "a" and "b" are real numbers.Another common way to write a complex number is in polar coordinates - basically specifying the distance from zero, and an angle.

A diamond has two rotation symmetry. It is possible to have a diamond that does have four of rotation symmetry.

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...

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!

It is 6+2i. But -6-2i will also serve.

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.

The answer is relatively simple if you know hyperbolic functions.

Suppose x is real so that ix is an imaginary number.

Then tanh x = -i*tan(ix)

So tan(ix) = (tanh x )/-i = i*tanh x

= i * sinh x/csh x = i*(ex - e-x)/(ex + e-x) = i*(1 - e-2x)/(1 + e-2x)

You can, but the process is slightly complicated, because addition in the Complex field is like vector addition.

If z1 = (r1, a1), and If z2 = (r2, a2)

Then, if z = (r, a)

r = sqrt(r12 + r22)

and

a = arctan[(r1sina1 + r2sina2)/(r1cosa1 + r2cosa2)]

Yes. Both ziggurats and pyramids were huge tasks that required the labor of thousands to complete.

113-2i sqr 17

Gerolamo Cardano is an Italian mathematician who introduced complex numbers. Complex numbers are those that can be expressed in the form of a+bi where a and b represent real numbers.

Well, there is doubtlessly more than one way to get from 9999 to 9, but the most direct way would be, 9999 - 9990 = 9.

Not necessarily.

the mean or average of the set

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.

complicated numbers

Don't see any "following" and this I's guessing is what you want?

113-(-68)^.5 =

113-((-1)(68))^.5 =

113-(68)^.5 (-1)^.5 =

113-i(68)^.5

To get the conjugate simply reverse the sign of the complex part.

Thus conj of 7-4i is 7+4i

Counting numbers

Yes it is. All pure imaginary numbers (such as 5i) as well as all real numbers and any combination of real & imaginary (by adding, subtractin, multiplying, dividing) makes a complex number.

That's an approximation of "pi", truncated after ten decimal places.

five hundred quintillion