How can you represent a point by "z = x + iy" as shown here?

In summary: So, yes, the representation of a point on a plane can be thought of in terms of (x,y), (x+p, y+q), or z=x+iy. But it's not really that important which one we use. Both (x,y) and x+y⋅i are simply a pair of numbers. What those numbers mean is determined by the problem you are solving. You can use either form to arrive at a solution.In summary, Dave has helped me understand that z=x+iy can be thought of as a complex number
  • #1
Slimy0233
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TL;DR Summary
Ways to represent a point which are not in the form (x,y)
Snapshot of Mary L. Boas' Mathematical Physics book
1685724522333.png


So, the marked lines say `If we think of P as the point z = x +iy in the complex plane, we could replace (2.3) by a single equation to describe the motion of P`

But, until now I have only learned of representing points in the form (x,y), now I wonder how I can represent a point on any plane (trying not to say "imaginary" here) using z = x + iy

PS: Please forgive me if I read the information wrong.
 
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  • #2
What do have in mind when you say "represent"? If you analyze your own words, perhaps you can answer your own question.

There are various things in mathematics called "repesentations" and they have technical definitions. I suspect you are not thinking about those concepts.

Perhaps you are asking how x + iy can be interpreted as coordinates in two dimensions. Isn't it obvious how to associate (x,y) and x + iy ? The test of whether that association is useful is whether the arithmetic of complex numbers imitates any operations in coordinates that are useful. In physics, if we regard (x,y) and (p,q) as the ends of vectors that begin at the origin, we can add the two vectors as (x+p, y+q). Adding the complex number x + iy to p + iq , we get (x+p) + (y+q)i. So the arithmetic of complex numbers produces an answer that can be associated with (x+p, y+q).
 
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  • #3
Stephen Tashi said:
Isn't it obvious how to associate (x,y) and x + iy ?
It is obvious as you say and I thought of that first when I read that, but how do you differentiate between when z = x + iy is a complex number and when it's co-ordinate in the imaginary plane.

Do we depend solely on the context of the text/given question?
 
  • #4
Slimy0233 said:
It is obvious as you say and I thought of that first when I read that, but how do you differentiate between when z = x + iy is a complex number and when it's co-ordinate in the imaginary plane.
A complex number has two coordinates when it is plotted in the complex plane. A point (x, y) in the real plane is at exactly the same position as the complex number x + iy. It's not clear to me what you're asking here.
 
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  • #5
Slimy0233 said:
It is obvious as you say and I thought of that first when I read that, but how do you differentiate between when z = x + iy is a complex number and when it's co-ordinate in the imaginary plane.

Do we depend solely on the context of the text/given question?
Both (x,y) and x+y⋅i are simply a pair of numbers. What those numbers mean is determined by the problem you are solving. You can use either form to arrive at a solution.

There are special properties of 2-dimensional vectors (x,y) that allow you to represent them as numbers z=x+y⋅i. So the complex number representation opens up some algebraic tools to manipulate them into the answer you want.

However, they aren't really different things. The representation on a plane is trivial, it's the same. That's not the point. The reason we do this is for the mathematical operations that may be (usually are) easier to perform to arrive at a solution.

PS: You can start a problem with a point representation (x,y), switch to a vector (from the origin to the point)
##v =
\begin{bmatrix}
x \\
y
\end{bmatrix} ##, then switch to a complex number z=x+y⋅i, or switch back whenever it suits you. You will confuse your reader, and maybe yourself, if you aren't clear about it. But they are all just a pair of numbers in the end.
 
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  • #6
Mark44 said:
A complex number has two coordinates when it is plotted in the complex plane. A point (x, y) in the real plane is at exactly the same position as the complex number x + iy. It's not clear to me what you're asking here.
hey mark, I was wondering if how I interpret z = x + i y depended on the context.

i.e., for example in the example I sited from Mary L Boas, I am supposed to interpret it as a co-ordinate rather than a complex number.

As you see Dave has already helped me. I thank you for your willingness to help :)
 
  • #7
Thank you very much @DaveE ! You have been very helpful today :smile:
 
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  • #8
Slimy0233 said:
It is obvious as you say and I thought of that first when I read that, but how do you differentiate between when z = x + iy is a complex number and when it's co-ordinate in the imaginary plane.

Do we depend solely on the context of the text/given question?
In general, it's useful to think about one thing in different ways. When doing completely rigorous pure mathematics we have to be careful about defining something one way (e.g. "an ordered pair of real numbers") and then writing about it as if it was something different (e.g. a point in 2D space). To establish that a mathematical object( e.g. linear transformation in a finite dimensional vector space) may be treated as a different mathematical object (e.g matrix of scalars) requires that we define what we mean by "may be treated as" and prove theorems showing that this is possible. In vague language, that is how pure mathematics defines and deals with "representations" .

We must distinguish between pure mathematics versus applications of mathematics and informal mathematics. In applications, people keep multiple mathematical representations of the same thing in mind - all at once! - and often without seeking any formal proof that this is a safe way of thinking. People who use the calculus of complex numbers regard (x,y) in the imaginary plane and x + iy as two aspects of the same thing. They don't think of these aspects as being mutually exclusive. As an analogy, a citizen in the US might have a social security number and have a different number as a driver's license number.
 
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  • #9
Whether you want to think of the point P as being in ##\mathbb{R}^2## or in the complex plane, ##\mathbb{C}##, depends on whether you want to take advantage of the additional mathematical operations of the complex numbers. The complex numbers have a defined multiplication that can represent rotations around the origin. And a rotation in the opposite direction can be represented by division by a non-zero complex number. When periodic functions are the subject, rotations are fundamental. Therefore, the complex plane is fundamental to periodic functions. As you will learn if you continue to study math and physics, even non-periodic functions are closely tied to the periodic components within them.
 
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  • #10
FactChecker said:
Whether you want to think of the point P as being in ##\mathbb{R}^2## or in the complex plane, ##\mathbb{C}##, depends on whether you want to take advantage of the additional mathematical operations of the complex numbers. The complex numbers have a defined multiplication that can represent rotations around the origin. And a rotation in the opposite direction can be represented by division by a non-zero complex number. When periodic functions are the subject, rotations are fundamental. Therefore, the complex plane is fundamental to periodic functions. As you will learn if you continue to study math and physics, even non-periodic functions are closely tied to the periodic components within them.
That's informative! Thank you!
 
  • #11
The two realms are the same thing. Complex numbers are a major convenience for describing cyclic phenomena.
 
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