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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 are complex numbers used in electricity?
1. One way is to show the relationship between different components such as a
resistor and a coil. That would be R+iXL. This shows that the coil impedence is
90 degrees out of phase with R resistance.
2. Another way we can use complex numbers is as an aid to visualize the complex
fields around two conductors carrying current.
Please refer to the RELATED LINK for a picture.
How far is it between 3-i and 3-i to the -1 power?
| 3-i - 1/(3-i)| = | 3-i - (3+i)/10|= | (27-11i)/10| = sqrt(17/2)
What is 0.0005681818 as a fraction?
0.0005681818 as a fraction is 2,840,909/50,000,000
If you want to know what the repeating decimal (repeating 18's) is, the answer is: 556875/990000000.
Find this by multiplying the number by 100 (10^2, since there are two digits that repeat), then subtract the original number. The decimal point shifts by 2 places, so all of the 18's 'cancel'.
So you have the original number x, and then subtracted: y = 100x - x = (100 - 1)x = 99x, so x = y/99.
Divide the result by 99 and the equivalent number is: 0.0556875/99, then multiply both numerator and denominator by the appropriate power of ten to get an integer in both numerator and denominator. In this case, 10^7.
Why only complex numbers are used in unitary matrix?
It looks like that's part of the definition of a unitary matrix. See related link, below.
Why are complex numbers imaginary?
An imaginary number has the form b*i, where b is any real number, and i is the imaginary unit number sqrt(-1). A complex number is in the form a + b*i, where a and b are any real number. Complex numbers are called complex, because they combine numbers from the real domain and the imaginary domain.
I thought the Wikipedia article was very good. Also, Answers.com has a video about complex and imaginary numbers.
Where are complex numbers used?
Complex numbers are necessary in order for the fundamental theorem of algebra to be complete.
The fundamental theorem of algebra states that for every polynomial function of degree n (the highest exponent on the variable) we will get n number of solutions.
Take this quadratic for example:
f(x) = x2 - 4
The fundamental theorem of algebra states that we will get two values of x for every value of f(x) we pick. In most cases we are concerned with the "roots" (typically x-intercepts) where the function crosses the line f(x) = 0.
So for the quadratic above when f(x) is set to 0 we get x = 2 or x = -2.
Now take a look at this quadratic:
f(x) = x2 + 1
A quick look at the graph of this function will show that it never crosses the line f(x) = 0. But according to the fundamental theorem of algebra we should get two values for x at f(x) = 0.
This is where the complex numbers come in.
When we solve this equation for f(x) = 0 we get the following conundrum:
-1 = x2
Now we know that we cannot take the square root of a negative number and get a real solution. There is no real number asuch that a2 will equal a negative number. So we move into the complex numbers.
The following solutions work for the above problem, x = i, x = -i.
The complex numbers make it to where the fundamental theorem of algebra holds for any polynomial function.
Complex numbers are also used in physics and electrical engineering. I do not know the specifics of those applications though so feel free to add to this.
Through the use of Euler's formula, derived by in the 1700's Leonhard Euler, we have: ei*ÆŸ = cos(ÆŸ) + i*sin(ÆŸ), where 'e' is the base of natural logarithms (approx 2.718281828), i is the imaginary unit number sqrt(-1), and ÆŸ is an angle in radians.
Some examples of where this is useful is: analysis of certain electric circuits, signal processing, control systems, electric power transmission lines. The angle ÆŸ usually represents a phase shift between two signals. Graphing in the complex plane can reveal behavior of the system, that is not readily evident with other methods.
Write a program to find sum of two complex numbers?
#include<iostream.h> #include<conio.h> class complex { int a,b; public: void read() { cout<<"\n\nEnter the REAL PART : "; cin>>a; cout<<"\n\nEnter the IMAGINARY PART : "; cin>>b; } complex operator +(complex c2) { complex c3; c3.a=a+c2.a; c3.b=b+c2.b; return c3; } complex operator -(complex c2) { complex c3; c3.a=a-c2.a; c3.b=b-c2.b; return c3; } complex operator *(complex c2) { complex c3; c3.a=(a*c2.a)-(b*c2.b); c3.b=(b*c2.a)+(a*c2.b); return c3; } complex operator /(complex c2) { complex c3; c3.a=((a*c2.a)+(b*c2.b))/((c2.a*c2.a)+(c2.b*c2.b)); c3.b=((b*c2.a)-(a*c2.b))/((c2.a*c2.a)+(c2.b*c2.b)); return c3; } void display() { cout<<a<<"+"<<b<<"i"; } }; void main() { complex c1,c2,c3; int choice,cont; do { clrscr(); cout<<"\t\tCOMPLEX NUMBERS\n\n1.ADDITION\n\n2.SUBTRACTION\n\n3.MULTIPLICATION\n\n4.DIVISION"; cout<<"\n\nEnter your choice : "; cin>>choice; if(choice==1choice==2choice==3choice==4) { cout<<"\n\nEnter the First Complex Number"; c1.read(); cout<<"\n\nEnter the Second Complex Number"; c2.read(); } switch(choice) { case 1 : c3=c1+c2; cout<<"\n\nSUM = "; c3.display(); break; case 2 : c3=c1-c2; cout<<"\n\nResult = "; c3.display(); break; case 3 : c3=c1*c2; cout<<"\n\nPRODUCT = "; c3.display(); break; case 4 : c3=c1/c2; cout<<"\n\nQOUTIENT = "; c3.display(); break; default : cout<<"\n\nUndefined Choice"; } cout<<"\n\nDo You Want to Continue?(1-Y,0-N)"; cin>>cont; }while(cont==1); getch(); }
The additive inverse.
Is zero real or imaginary number?
Zero is in the set of the whole numbers and also integers, which are subsets of the real numbers. So you could say it's real, but it lies on the imaginary axis (it is the origin). It's kind of like asking: Is zero positive or negative. For most purposes, letting zero be real would be the logical choice, though.
First answer:
It is real. Imaginary numbers only involve square roots of negative numbers.
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