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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
What is the name of 3.1415926535?
That's an approximation of "pi", truncated after ten decimal places.
Can every point on the number line be named by a real number?
Yes, as long as the number line is entirely real.
Can you divide infinity by an imaginary number?
Yes, but the answer will be in infinities within the complex domain. Unless you know what you are doing there, stay away ;)
How do you use complex numbers?
Better get a textbook that explains this in more detail. You can also get a brief summary at Wikipedia, or other online sites.
In any case, here is a brief summary.
For addition and substraction, you add (or subtract) the real and imaginary parts separately. For example, (4 + 3i) + (7 - 2i) = 11 + 1.
For multiplication, multiply each part of one number with each part of the other number - and remember that i2 = -1. For example, (4 + 3i) x (7 - 2i) = 28 - 8i + 21i - 6i2 = 28 + 13i - 6(-1) = 34 + 13i.
Division is a bit more complicated. For example, to divide by (3 + 4i) you have to multiply numerator and denominator by the complex conjugate of this number, that is, change the sign of the imaginary part; in this case, (3 - 4i).
Multiplication and division are actually quite a lot easier if you convert the complex number to polar coordinates, that is, a distance and an angle. Here is a quick example: (4 angle 30 degrees) x (5 angle 20 degrees) = (4 x 5) angle (30 + 20 degrees) = 20 angle 50 degrees (a length of 20, at an angle of 50 degrees). Most scientific calculators have special functions to convert from rectangular to polar coordinates and back.
The answer is Yes, for the purposes of most Math students. There are, number systems that have been devised which are outside the set of Complex Numbers, though.
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.
What is the correct notation for the complex number 113-square root-68?
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
What is the complex conjugate of -3-9i?
For any number (a + bi), its conjugate is (a - bi), so the real part stays the same, and the imaginary part is negated.
For this one, conjugate of [-3 - 9i] is: -3 + 9i
What is the conjugate of the complex number 7-4i?
To get the conjugate simply reverse the sign of the complex part.
Thus conj of 7-4i is 7+4i
How do you plot 6-i on a complex plane?
You go 6 in positive x-direction ("right") and one in negative y-direction ("down"), there is your complex number, drwa an arrow reaching from the center to this point.
When were complex numbers developed?
The concept of a complex number appears to be first conceived by Gerolamo Cardano (about 1545). See Related Link. The work of Euler in the 1700's helped to make them more useful, though.
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.
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).
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.
Who first wrote about complex numbers?
The Italian mathematician, Gerolamo Cardano was the first to consider the concept of complex numbers. However, he did not develop a theory of complex numbers to any extent. That was left to Rafael Bombelli.
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.
How did imaginary numbers get their name?
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.
