- #1
Sobhan
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What are imaginary numbers?
Does anyone know a good book for it?
Does anyone know a good book for it?
Exploring Imaginary Numbers for Beginners
- Thread starter Sobhan
- Start date
In summary, imaginary numbers are a way to extend the real numbers so that we can do mathematical operations in a different way. They are an extension of the real numbers and can be written as (a, b) where a and b are real numbers.
- #2
jedishrfu
Mentor
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There's this popular book on the subject:
https://www.amazon.com/gp/product/0691146004/?tag=pfamazon01-20
https://www.amazon.com/gp/product/0691146004/?tag=pfamazon01-20
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- #3
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Sobhan said:
What level are you at? What level are you looking for?
- #4
Sobhan
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i need a book for basics of it.
- #5
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You're not answering my question. What level are you at?
- #6
Sobhan
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high school
- #7
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High school can mean a lot of things. It can mean you know calculus, or it can mean you don't even know algebra yet.
- #8
Sobhan
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i know calculus but not completely
- #9
BvU
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Well, my respect for your mathematical curiosity !
As you may know, we can't find a real number for which the square is -1. Imaginary numbers appear if we pretend we can: we call ##i## the number for which ##i^2 = -1## (so we simply imagine things
). From that moment on we can (pretend to ) do a whole lot of things: we can also solve for ##\sqrt {-1}##, although a small duality arises: because if ##i^2 = -1##, then also ##(-i)^2 = -1## !
Imaginary numbers are an extension of the real numbers so that we can invert a mathematical operation: the operation of "raising to a power".
In the same way we extended natural numbers with integer numbers so that we could invert addition:
with natural numbers we can't find the natural number to solve ? + 5 = 2
So we invented negative numbers
And with integer numbers we couldn't solve ? * 3 = 2
So we invented fractions and had a set of rational numbers
With rational numbers we still couldn't solve ? 2 = 2
so we invented real numbers
And with real numbers we couldn't solve ? 2 = -1
And that's where imaginary numbers appear on stage !
As you may know, we can't find a real number for which the square is -1. Imaginary numbers appear if we pretend we can: we call ##i## the number for which ##i^2 = -1## (so we simply imagine things
). From that moment on we can (pretend to ) do a whole lot of things: we can also solve for ##\sqrt {-1}##, although a small duality arises: because if ##i^2 = -1##, then also ##(-i)^2 = -1## !Imaginary numbers are an extension of the real numbers so that we can invert a mathematical operation: the operation of "raising to a power".
In the same way we extended natural numbers with integer numbers so that we could invert addition:
with natural numbers we can't find the natural number to solve ? + 5 = 2
So we invented negative numbers
And with integer numbers we couldn't solve ? * 3 = 2
So we invented fractions and had a set of rational numbers
With rational numbers we still couldn't solve ? 2 = 2
so we invented real numbers
And with real numbers we couldn't solve ? 2 = -1
And that's where imaginary numbers appear on stage !
- #10
jk22
- 731
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$$?^2=2$$ needs algebraic numbers.
There are transcendental numbers too for example sin(?)=1 even I am not sure it is transcendent.
Most real numbers are transcendent. Complex number is a extension of real numbers you can represent the imaginary unit by a real 2x2 matrix $$i=\left(\begin{array}{cc} 0&-1\\1&0\end{array}\right)$$
There are transcendental numbers too for example sin(?)=1 even I am not sure it is transcendent.
Most real numbers are transcendent. Complex number is a extension of real numbers you can represent the imaginary unit by a real 2x2 matrix $$i=\left(\begin{array}{cc} 0&-1\\1&0\end{array}\right)$$
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- #11
HallsofIvy
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Another way of defining complex numbers is this: the set of complex numbers is the set of ordered pairs of real numbers, (a, b), with addition defined by (a, b)+ (c, d)= (a+ c, b+ d) and multiplication defined by (a, b)*(c, d)= (ac- bd, ad+ bc).
One thing we can show immediately is that pairs with second number, 0, (a, 0), have addition (a, 0)+ (b, 0)= (a+ b, 0+ 0)= (a+ b, 0) and multiplication (a, 0)*(b, 0)= (a*b- 0*0, a*0+ b*0)= (ab, 0) so that we can "identify" the real number, a, with the pair (a, 0) and, in that sense, think of the real numbers as being a "subset" of the complex numbers.
But (0, 1)*(0, 1)= (0*0- 1*1, 0*1+ 1*0)= (-1, 0) so that, in this "number system", unlike the real number, there exist a "complex number" (pair) is -1.
We can write (a, b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, 1). We have already identified (1, 0) with the real number, 1. If we use "i" to represent the pair (0, 1), in the same way that we are using "1" to represent the pair (1, 0), we can write (a, b)= a+ bi in the more usual notation.
One thing we can show immediately is that pairs with second number, 0, (a, 0), have addition (a, 0)+ (b, 0)= (a+ b, 0+ 0)= (a+ b, 0) and multiplication (a, 0)*(b, 0)= (a*b- 0*0, a*0+ b*0)= (ab, 0) so that we can "identify" the real number, a, with the pair (a, 0) and, in that sense, think of the real numbers as being a "subset" of the complex numbers.
But (0, 1)*(0, 1)= (0*0- 1*1, 0*1+ 1*0)= (-1, 0) so that, in this "number system", unlike the real number, there exist a "complex number" (pair) is -1.
We can write (a, b)= (a, 0)+ (0, b)= a(1, 0)+ b(0, 1). We have already identified (1, 0) with the real number, 1. If we use "i" to represent the pair (0, 1), in the same way that we are using "1" to represent the pair (1, 0), we can write (a, b)= a+ bi in the more usual notation.
- #12
QuantumForumUser
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When a negative number is under a square root radical sign (√), you can use the multiplication property of square roots (√(ab)=√(a)×√(b)) to change how the answer "looks". Let's say √(-3). The number -3 can factor out as -1 times 3 so √(-3)=√(-1)×√(3). Since i is equal to √(-1), one can write √(-3) equals i×√(3).
FAQ: Exploring Imaginary Numbers for Beginners
What are imaginary numbers?
Imaginary numbers are numbers that when squared, result in a negative number. They are represented by the letter i and are expressed as a multiple of i, such as 3i or -2i.
Why are they called "imaginary" numbers?
They are called imaginary numbers because they do not exist on the real number line. They were initially thought to be "imaginary" or not real, but they are now recognized as important mathematical tools.
How do we use imaginary numbers in real life?
Imaginary numbers are used in many real-life applications, such as in engineering, physics, and electronics. They are also used in music and art to create complex patterns and harmonies.
What is the difference between imaginary and complex numbers?
Complex numbers are numbers that have both a real and imaginary component, while imaginary numbers only have an imaginary component. Complex numbers can be expressed as a + bi, where a is the real part and bi is the imaginary part.
What is the importance of imaginary numbers in mathematics?
Imaginary numbers are important in mathematics because they allow us to solve equations that would otherwise have no solution. They also play a crucial role in understanding and solving complex equations and systems in various fields of science and engineering.