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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 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.
The densest subset of real numbers is the set of fractions?
Your question is ill-posed. I have not come across a comparison dense-denser-densest.
The term "dense" is a topological property of a set:
A set A is dense in a set B, if for all y in B, there is an open set O of B, such that O and A have nonempty intersection.
The rational numbers are indeed dense in the set of real numbers with the standard topology. An open set containing a real number contains always a rational number.
Another way of saying it is that every real number can be approximated to any precision by rational numbers.
There are denser sets, if you are willing to consider more elements.
Suppose you construct a set consisting of the rational numbers plus all algebraic numbers. The set of algebraic numbers is also countable, but adding them, makes it obviously easier to approximate real numbers.
Can you perhaps construct a set less dense than the set of rational numbers?
Suppose we take the set of rational numbers without the element 0. Is this set still dense in the real numbers? Yes, because 0 can be approximated by 1/n, n>1.
In fact, you can remove finite number of rational numbers from the set of rational numbers and the resulting set will still be dense in the set of the real numbers.
What are two ways to get 280 in multiplication?
140 multiplied by 2
28 multiplied by 10
There are many more ways to multiply to get 280.
How do you read two complex numbers and perform addition and subtraction?
To add complex numbers, the real parts and the imaginary parts are added separately. For example, to add (3 + 3i) + (5 - 2i), the result is (3+5) + (3-2)i = 8+i.
Subtraction is quite similar - you subtract the real and the imaginary parts separately. For example, (3 + 3i) - (5 - 2i) = (3-5) + (3 - -2)i = -2+5i.
Can a logarithmic function have a complex number?
Yes and as an example:
1. the natural log (ln) of 2+i3~1.28+.98i (base e or 2.71828)
2. The Log of 2+i3~.57+.43i (base 10)
Complex roots occur in conjugate pairs?
Not for all types of equations. But always in second degree equations they do.
Consider a third degree equation with 3 different roots. Obviously, one of the roots can not be in a pair.
What is a number of the form a plus bi where a is the real part and b is the imaginary part?
a complex number
How do you find all real and complex roots?
In general this question is unanswerable.
However, you can consider Newton's method to make very good estimates.
Equations can be very complex in that their curves have poles and zeros where you do not expect them. Consider Riemann's Zeta function Z(z) = Sum(1/n^z, n>0). It has complex zeros on the line z=1/2, but up to this date, the distribution of the zeros is not entirely known!
Yes. -10.39 is real because it does not have an imaginary component.
What is the worded form of 100000000000000000000000000000000000000000000000?
1 with 47 zeros. Ten to the forty seventh power.
Depending on what country you are in:
- one hundred quattuordecillion in the United States and some other English speaking countries.
- one hundred septilliard in many European countries.
See related link.
What set of numbers is the acute and which is the obtuse numbers on a protractor?
A right angle sits at 90o, which is right in the middle. Anything less than 90o, or to the left on the protractor are acute angles. Anything to the right, or bigger than 90o, would be considered an obtuse angle. A good way to remember this is that an acute angle is "a cute little angle."
In metamorphoses who transforms into a laurel?
Daphne, a patroness of Diana (Goddess of the Hunt), asked to be turned into a laurel tree after Apollo fell in love with her. She did not love him back and wished to remain an unmarried virgin, true only to Diana, for the rest of her life. Apollo pursued her and she finally begged the gods to be turned into the tree. Thus, to show his devotion, Apollo adopted the laurel tree as his sacred tree and wore the laurel leaves on his head as a crown.
How would you define the zero vector by using the idea of components?
All components of the zero vector equal to zero.
The complex roots of an equation are the complex numbers that are solutions to the equation.
