Complex Numbers

Yes, it is.

Moreover, it is also a rational number.

1.25 * 4 = 5, so 1.25 = 5/4.

All rational numbers are real numbers, so 1.25 is real.

Any number you can think of, using decimal notation is real.

Real numbers are allowed to have an infinity of digits (behind the decimal point).

Not necessarily.

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)

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.

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

113-2i sqr 17

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.

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!

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

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.

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)]

there on a whole different plane. your x axis is labeled real not x and your y axis is labeled imaginary not y. then just plot the points like you normally would.

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

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

Rene Descartes came up with the word imaginary in 1637 to describe them. It was a derogatory term. He (and many other mathematicians of that age) did not like imaginary numbers. Many people didn't believe in them, because they were not real.

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.

Counting 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

Complex numbers are basically "numbers in two dimensions". You can extend them to more dimensions; one superset that is sometimes used is the quaternions, which are numbers in four dimensions.

Any physical motion which is periodic, such as an oscillating beam, string, wire, pendulum, electronic signal, or electromagnetic wave can be represented by a complex number function. This can make calculations with the various components simpler than with real numbers and sines and cosines.

('|x|' = Absolute value of x)

|3+4i| = √(32 + 42)

= √(9+16)

= √25

= 5

Thus |3+4i| = 5.

-125 is NOT an imaginary number.

No. All Complex Numbers are of the form a + bi where a and b are Real Numbers and i is the square root of -1. So only ones where a = 0 are pure Imaginary Numbers.

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