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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
How do you see that 108200000?
It is: 108,200,000 and can be expressed as 1.082*108 in scientific notation
What are some imaginary sights?
Imaginary sights can include fantastical landscapes like floating islands adorned with vibrant, luminescent flora, or a bustling city in the clouds where buildings are made of glass and light. You might envision a serene forest where trees whisper secrets and streams shimmer with colors unseen in reality. Other examples could be a grand castle perched on the edge of a rainbow or mythical creatures like dragons soaring through a starry sky. These sights evoke wonder and creativity, allowing the mind to explore realms beyond our own.
By "the nth term" of a sequence we mean an expression that will allow us to calculate the term that is in the nth position of the sequence. For example consider the sequence 2, 4, 6, 8, 10,... The pattern is easy to see. # The first term is two. # The second term is two times two. # The third term is two times three. # The fourth term is two times four. # The tenth term is two times ten. # the nineteenth term is two times nineteen. # The nth term is two times n.
In this sequence the nth term is 2n.
What are the cube roots of zero plus 8i?
The 3 cube-roots of 8i are:
- √3 + i
- -√3 + i
- -2i
Think of 8i in polar form as 8∠90°. A number raised to a power (in this case 1/3) is the magnitude raised to the power and the angle is the angle times the power.
So 8 raised to 1/3 power is 2. And 90° * (1/3) = 30°. To find the angles of the other 2 cube-roots, find equivalent angles (add 360° & 720°). So you have 450°/3 = 150° and 810°/3 = 270°.
So the three roots: 2∠30°, 2∠150° & 2∠270°. Which are the three answers, above (in rectangular coordinates)
That's the average value of all the members of the set.
What is a counterexample to show that the repeating decimals are closed under addition false?
There cannot be a counterexample since the assertion is true.
This requires you to accept the true fact that the terminating decimal 1.25, for example, is equivalent to the repeating decimal 1.25000... (or even 1.24999.... ).
Rational and irrational numbers are complex numbers?
Rational and irrational numbers are real numbers.
A complex number is represented by a+bi where a and b are real numbers.
Zero is a real number therefore any real number is also complex whenever b=0
What are the Eulers formulas for the Fourier coefficients?
I think the following Wikipedia link on Fourier Series (see related links below), has the information that you're looking for.
If they told you to click on the lightest blue square which one would you click on?
I would go for the blue that is for example aqua blue
How can one understand the essence of the imaginary unit of a complex number?
Imaginary numbers are useful in describing how waves (such as electrical signals) relate and interact with one another. The picomonster website in the related link has one of the coolest explanations, along with animations, that I've seen.
How do you tell if a complex number isn't real?
If the coefficient of i is not zero then the number is not real.
Are there more even numbers or whole numbers?
Yes and no.
That is the correct answer because dealing with infinity sometimes is nonsense.
For every even number there are two whole numbers (an even and an odd).
But if we count on - to infinity - we have ∞ (infinity) even numbers.
That means we have 2 * ∞ whole numbers.
But 2 * ∞ = ∞ (because we are dealing with infinity)
So, there are as many whole numbers as even numbers, eventhough there are twice as many.
Why complex no is denoted by z?
There are no real reason why it is denoted by z, but that the real number axis is denoted by x, imaginary number is denoted by y, the real part of a complex number is denoted by a, the imaginary part of a complex number is denoted by b, so there is z left.
How do you convert the complex number to standard form 1 plus 2i over root2 plus i?
Multiply the numerator and denominator by the complex conjugate
of the denominator ... [ root(2) minus i ].
This process is called 'rationalizing the denominator'.
