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Is there any one-to-one correspondence between complex numbers and real numbers?
Wiki User
∙ 16y ago
== == Yes there is, but it's a little tricky to prove. Here's a sketch. First, some notation:
P ~ Q means there is a 1-1 correspondence between all members of set P and all of Q
R is the set of real numbers, C is the complex numbers
R x R is the cross-product of R with itself, the set of ordered pairs of reals (ditto for any set)
(0,1] is the half-open interval of reals from 0 to 1, that is all real x with 0 < x <= 1
Now the sketch. It's pretty obvious that R x R ~ C, since a complex number is just an ordered pair of reals.
R ~ (0,1] by the function f(x) = 1/(1-x) (you can prove yourself pretty easily that this is a 1-1 function and covers all of R vs all of (0,1]
R x R ~ (0,1] x (0,1] by applying the previous rule to each element of the pair
[0,1) ~ [0,1) x [0, 1] is a bit tricky, but one way is to map a real number x in (0,1] into two real numbers y and z by taking every other digit in the decimal expansion of x. For example take x = pi/10...
x = .314159265358979323846... <-> .1196387334... and .34525599286...
You have to watch out for technicalities like .09999... = .100000,... but it does work and is 1-1 and covers all of (0,1].
So, stringing all this together...
C ~ R x R ~ (0,1] x (0,1] ~ (0,1] ~ R
I'm not sure you need the R ~ (0,1] part, but it's an important fact (all of R is ~ to a subset of R), a cute trick and worth showing.
QED
Wiki User
∙ 16y ago
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Are most numbers rational or irrational?
The set of irrational numbers is larger than the set of rational numbers, as proved by Cantor: The set of rational numbers is "countable", meaning there is a one-to-one correspondence between the natural numbers and the rational numbers. You can put them in a sequence, in such a way that every rational number will eventually appear in the sequence. The set of irrational numbers is uncountable, this means that no such sequence is possible. All rational and irrationals (ie real numbers) are a subset of complex numbers. Complex numbers, in turn, are part of a larger group, and so on.
What is the difference between imaginary numbers and complex numbers?
No difference. The set of complex numbers includes the set of imaginary numbers.
What is the relation of complex numbers to real numbers?
Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.
Could anyone list numbers not in the set of complex numbers?
No. Complex numbers is the highest set of numbers you can go, and there are no sets outside of complex numbers.
What is the difference between real numbers and complex numbers?
Real numbers are a proper subset of complex numbers. In fact each complex number, z, can be represented as z = x +iy where x and y are real numbers and i is the imaginary square root of -1.Thus the set of complex numbers is the Cartesian product of two sets of real numbers. That is, C = R x R where C is the set of complex numbers and R is the set of real numbers. Limitations of this browser prevent me from writing that in a mathematically precise and more helpful fashion.
Are most numbers rational or irrational?
The set of irrational numbers is larger than the set of rational numbers, as proved by Cantor: The set of rational numbers is "countable", meaning there is a one-to-one correspondence between the natural numbers and the rational numbers. You can put them in a sequence, in such a way that every rational number will eventually appear in the sequence. The set of irrational numbers is uncountable, this means that no such sequence is possible. All rational and irrationals (ie real numbers) are a subset of complex numbers. Complex numbers, in turn, are part of a larger group, and so on.
What is the difference between imaginary numbers and complex numbers?
No difference. The set of complex numbers includes the set of imaginary numbers.
What are the similarities between vectors and complex numbers?
If you add two complex numbers, the resulting complex number is equivalent to the vector resulting from adding the two vectors. If you multiply two complex numbers, the resulting complex number is equivalent to the vector resulting from the cross product of the two vectors.
What is the difference between a set of real numbers and a set of complex numbers?
The set of real numbers is a subset of the set of complex numbers. For the set of complex numbers, given in the form (a + bi), where a and b can be any real number, the number is only a real number, if b = 0.
What is complex math?
Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.
HOW to do addition between two complex numbers?
Addition between complex numbers is very simple if the complex numbers are in standard form (real part and imaginary part separated); just add the real part and the imaginary part separately. For example: (3 + 2i) + (-5 + 3i) = (-2 + 5i)
What is the relation of complex numbers to real numbers?
Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.
Could anyone list numbers not in the set of complex numbers?
No. Complex numbers is the highest set of numbers you can go, and there are no sets outside of complex numbers.
What real number corresponds to 1200 dollars?
We're not sure what the correspondence is between numbers and sums of money, but 1,200 is a real number.
What is the definition of number line?
number line- a straight line on which there is indicated a one-to-one correspondence between points on the line and the set of real numbers. basically a line with numbers on it.
What is the difference between real numbers and complex numbers?
Real numbers are a proper subset of complex numbers. In fact each complex number, z, can be represented as z = x +iy where x and y are real numbers and i is the imaginary square root of -1.Thus the set of complex numbers is the Cartesian product of two sets of real numbers. That is, C = R x R where C is the set of complex numbers and R is the set of real numbers. Limitations of this browser prevent me from writing that in a mathematically precise and more helpful fashion.
What is the difference between a complex number and a non real complex number?
Think of the complex numbers as points on a coordinate system. Instead of the usual x-axis you have the real numbers, instead of the y-axis, you have the imaginary numbers.The real numbers are on the horizontal axis.The imaginary numbers are on the vertical axis.The complex numbers are any number on the plane.The non-real complex are, of course, any complex numbers that are not on the real number axis - not on the horizontal axis.Think of the complex numbers as points on a coordinate system. Instead of the usual x-axis you have the real numbers, instead of the y-axis, you have the imaginary numbers.The real numbers are on the horizontal axis.The imaginary numbers are on the vertical axis.The complex numbers are any number on the plane.The non-real complex are, of course, any complex numbers that are not on the real number axis - not on the horizontal axis.Think of the complex numbers as points on a coordinate system. Instead of the usual x-axis you have the real numbers, instead of the y-axis, you have the imaginary numbers.The real numbers are on the horizontal axis.The imaginary numbers are on the vertical axis.The complex numbers are any number on the plane.The non-real complex are, of course, any complex numbers that are not on the real number axis - not on the horizontal axis.Think of the complex numbers as points on a coordinate system. Instead of the usual x-axis you have the real numbers, instead of the y-axis, you have the imaginary numbers.The real numbers are on the horizontal axis.The imaginary numbers are on the vertical axis.The complex numbers are any number on the plane.The non-real complex are, of course, any complex numbers that are not on the real number axis - not on the horizontal axis.
