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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
When was Gee Whiz-z-z-z-z-z-z created?
1971
The Gee Whiz Story
So when Gilda Trozzolo, Pasquale's mother, decided to open a candy store, picking the name was easy. Gee Whiz Candy was established in 1971 in Elmwood Park, Illinois.
To convert a number to scientific notation, move the decimal point right or left to make the number greater than or equal to one but less than ten, and record the number of positions moved as a power of 10 - the exponent. That is, if the decimal point moves to the left by n positions, then the exponent is 10n. If the decimal point moved to the right by npositions, the exponent is 10-n (note the minus symbol).
For instance, the number 123,456,000,000 is larger than 10, so we move the decimal point 11 positions to the left to get 1.23456, which is greater than or equal to one, but less than ten. Since we moved the decimal point to the left by 11 positions, the exponent is 1011 (ten raised to the eleventh power, which is 100,000,000,000) so the scientific notation for 123,456,000,000 becomes 1.23456x1011.
If the original number were 0.000000123456, we need to move the decimal point to the right by seven positions to get 1.23456 (greater than or equal to one but less than ten). The exponent is therefore 10-7, thus the scientific notation for 0.000000123456 is 1.23456x10-7.
To convert from scientific notation to standard notation, we simply reverse the process. If the exponent is a positive power of 10, we multiply by the exponent. Thus 1.23456x1011 is 1.23456 x 100,000,000,000 which is 123,456,000,00. If the exponent is a negative power of 10, we divide by the exponent. Thus 1.23456x10-7 is 1.23456 / 10,000,000 which is 0.000000123456.
Note that scientific notation is only useful when you are not interested in the least significant portion of a number. For instance, a value such as 123,456,789,123,456,789,123,456,789 is better notated in full if you want the highest degree of accuracy. Scientific notation is generally only used to make the notation of an extremely large (or extremely small) number more concise. So 123,456,789,123,456,789,123,456,789 might be reduced to a more concise form such as 1.23456789x1026. This then equates to 123,456,790,000,000,000,000,000,000 in standard notation, which is clearly not the same value we started out with. In other words, the degree of accuracy is determined by the number of decimal places you retain in the scientific notation.
How many horsepower equals 1 mph?
Ha Ha. two different measurements. Horsepower is a measurement of the torque your engine can produce, MPH is a measurement of speed.
It is true you need horsepower to generate speed (MPH), and you can calculate the torque required if you have all the factors such as weight, air resistance, friction with road surface, etc.
How do you express a complex number like 5 plus 5i in polar form?
Polar form = re^i(angle)
r=(4+4)^.5
angle=atan(2/2)
PF = (8)^.5*e^atan(1)
pf = 2.83e^.7854i
Which of these are complex numbers 5 3i 1 2i?
All of them. Real numbers are a subset of complex numbers.
6x7=42 42x8=336 That is your answer, I hope I helped! ;)
Which set of real numbers is bigger set of integers or set of rational numbers explain?
The question is not well-posed, in that the term "bigger" can be understood in different ways.
If A is a subset of B, we can call B bigger than A.
However, in set theory, the cardinality of a set is defined as the class of sets with the "same number" of elements:
Two sets A and B have the same cardinality if there exists a bijection f:A->B.
Theorem: If there is an injection i:A->B and an injection i:B->A, then there is a bijection f:A->B. Not proved here.
The set of integers and the set of rational numbers can be mapped as follows.
Since the natural numbers are a subset of the rational numbers by i:N->R: n-> n/1, we have half of the proof.
Now, order the rational numbers as follows:
- assign to each rational number p/q (p,q > 0) the point (p,q) in the plane.
Next, consider that you can assign a natural number to each rational number by walking through them in diagonals:
(1,1) -> 1; (2,1) -> 2; (1,2) -> 3; (3,1) ->4 ; (2,2) ->5; (1,3) -> 6; (4,1) -> 7; (3,2) -> 8, (2,3) -> 9; (1,4) -> 10, etc. (make a drawing).
It is clear that in this way you can assign a unique natural number to EACH rational number. This means that you have an injection from the rational numbers to the natural numbers.
Now you have two injections, from the natural numbers to the rational numbers and from the rational numbers to the natural numbers.
By the theorem, there is a bijection, which means that the natural numbers and the rational numbers have the same cardinality. Neither of them is "bigger" than the other in this sense. The cardinality of these two sets is called Aleph-zero, and the sets are also called countable (because the elements can be counted with the natural numbers).
Why is the square root of a negative number not a real number?
Because there is no real number which you can square, which will result in a negative real number. So they came up with imaginary numbers, and denoted the letter i to represent the square root of negative one. At first, they were thought to be just that - imaginary - nonexistent, whose only purpose was to fill in and make equations solvable. But now these numbers are useful in solving equations which govern electrical waves and other types of wave motion.
What is the conjugate of a real number?
Since the imaginary portion of a real number is zero, the complex conjugate of a real number is the same number.
What is the answer to this -5-12i?
This is a complex number, not an algebraic expression. The letter i represents the imaginary unit (which is equal to sqrt(-1)). Graphiclly, with real numbers on a horizontal axis, and imaginary numbers on a vertical axis, this means starting at the origin, go to the left 5 units, and then go down 12 units.
No. i = square root of -1, and therefore imaginary. Only by squaring i can you have a real number.
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.
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.
What is the correct notation for the complex number square -81-14?
-81-14 is not a complex number. And its square is 9025.
Who is the father of complex number?
The answer to this question is more like an opinion than a solid fact. Several different mathematicians have been attributed to contributions in imaginary and complex numbers, but the work of Leonhard Euler gave new meaning to how imaginary and complex numbers behave, and how they can be used to simplify the analysis of something very real: waves (especially electromagnetic waves).
- Euler's Formula: e^(i*Θ) = cos(Θ) + *sin (Θ)
I assume you mean googolplex (sounds similar).
Googolplex is real, its definition is based on Googol (which inspired the name Google, seriously).
A googol is 10^100.
A googolplex is 10^googol.
To try and imagine the size of this number, consider that
the number of atoms in the universe is about 10^80 (this is not yet a googol).
In other words, a googol is more than the number of atoms in the universe.
a googolplex is a number so big that you can not write it, because it has a googol+1 digits (all zero, except the leading 1). So, to write it you would need at least 1 atom to write a digit, but there aren't enough atoms in the universe to write it. Heavy!
Of course, there are bigger numbers than this!
Consider numbers with a name
Consider googolplexplex, googolplexplexplex, etc.
Consider googol(plex . . . . . . plex)
etc.
All these numbers are incredibly big natural numbers, but FINITE. They are all real.
How could you use Descartes' rule to predict the number of complex roots to a polynomial?
Descartes' rule of signs will not necessarily tell exact number of complex roots, but will give an idea. The Wikipedia article explains it pretty well, but here is a brief explanation:
It is for single variable polynomials.
- Order the polynomial in descending powers [example: f(x) = ax³ + bx² + cx + d]
- Count number of sign changes between consecutive coefficients. This is the maximum possible real positive roots.
- Let function g(x) = f(-x), count number of sign changes, which is maximum number of real negative roots.
Note these are maximums, not the actual numbers. Let p = positive maximum and q = negative maximum. Let m be the order (maximum power of the variable), which is also the total number of roots.
So m - p - q = minimum number of complex roots. Note complex roots always occur in pairs, so number of complex roots will be {0, 2, 4, etc}.
Can you factorise numbers on a scientific calculator?
Up till now you can't but hopefully will be able to do so soon.
