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I've just started learning about the quaternions and it's raised some interesting questions for me about the complex numbers.

The $+$ sign in the expression $a+bi$ is confusing me. Usually when we use the $+$ sign it represents an operation combining two elements of the same space; $3+2=5$, $+$ is the operation of combining these two integers into a new integer. With the complex numbers the + sign in for example $1+i$ doesn't represent an operation. How could you combine an (strictly) imaginary number and a real number?

Apologies for the poorly expressed question, but any help would be greatly appreciated.

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  • $\begingroup$ It does represent addition of the real $a$ with the imaginary $bi$. $\endgroup$ – robjohn Dec 3 '13 at 19:19
  • $\begingroup$ But what could that possibly mean? Isn't that like adding an apple and an orange? $\endgroup$ – James Machin Dec 3 '13 at 19:22
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    $\begingroup$ It is like adding two vectors in $\mathbb{R}^2$. I've added an answer to this effect. $\endgroup$ – robjohn Dec 3 '13 at 19:25
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$a+bi$ represents the addition of a real number $a$ and an imaginary number $bi$. One can also represent a complex number as a point in $\mathbb{R}^2$: $(a,b)$, where $a$ is the real component and $b$ is the imaginary component. We can write these representations as $$ a+bi=a(1,0)+b(0,1)=(a,b) $$ This is why we generally talk about the real line and the complex plane.

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Addition of complex numbers is handled as addition of vectors in $\mathbb{R}^2$. Multiplication of complex numbers is performed using the identity $i^2=-1$, so that the product of two complex numbers yields another complex number.


Clarification:

The complex numbers are an extension field of the reals (of degree 2). $\mathbb{R}$ can be embedded in $\mathbb{C}$ as $\{(x,0):x\in\mathbb{R}\}$. $1+0i\in\mathbb{C}$ is not the same number as $1\in\mathbb{R}$ since they live in different sets.

$\mathbb{R}$ can be embedded in $\mathbb{C}$, but it is not a subset of $\mathbb{C}$; however, the embedding of $\mathbb{R}$ as $\{(x,0):x\in\mathbb{R}\}$ is an isomorphism, and so often people say that $\mathbb{R}\subset\mathbb{C}$ because of this.

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  • $\begingroup$ What does the addition of a real and an imaginary number mean though? Does it just mean that they are associated with each other? Could I replace the + sign with "and"? $\endgroup$ – James Machin Dec 3 '13 at 19:36
  • $\begingroup$ $i$ is a complex number just as $1$ is, just like $a$ and $bi$ are complex numbers. $a$ represents the point $(a,0)$ in the complex plane and $bi$ represents the point $(b,0)$. $a+bi$ is the sum of these two complex numbers, the point $(a,b)$. Just think of a complex number as a point in the plane $\mathbb{R}^2$. We usually write these points as a vector sum using the basis vectors $1=(1,0)$ and $i=(0,1)$. $a+bi$ is a different complex number than $a-bi$, so the $+$ and $-$ are important. $\endgroup$ – robjohn Dec 3 '13 at 19:44
  • $\begingroup$ But when you say i is just the complex number 0+1i e.t.c. and 1 is 1+0i you're implying that there's some other definition for the complex numbers than c=a+bi. What I mean is if 1=1+0i, what does this addition sign mean? I understand the addition of vectors in a plane, but I think of this as two seperate additions, in the x and y directions, where the x and y directions are distinct and to add displacements in two different directions together merely means to associate them together. By the way thanks for all the help so far, I have really appreciated it. $\endgroup$ – James Machin Dec 3 '13 at 20:10
  • $\begingroup$ @JamesMachin: I have added a clarification section to the answer that may cover most of what is bothersome. $\endgroup$ – robjohn Dec 3 '13 at 20:28
  • $\begingroup$ That clarification part really helped! Thanks again for putting in the time. $\endgroup$ – James Machin Dec 3 '13 at 22:09
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Some would view the complex number $1$ as $1+0i$ and the complex number $i$ as $0+1i$, so it's always viewed as a pair of real numbers, and then addition is addition of pairs, so that when you add $1+0i$ to $0+1i$ you get $1+1i$.

You can also view it geometrically: A complex number is an arrow pointing from the origin to a point in the plane, and then addition is completing a parallelogram.

Viewed in that way, multiplication is rotation and dilation. The simplest instance is that multiplying by $i$ means rotating $90^\circ$ counterclockwise and by $-i$ is $90^\circ$ clockwise.

Similar question troubled the brilliant physicist Edwin Jaynes concerning tensor algebras: How can you add a vector and a scalar? He thought it seemed as bad as adding meters to kilograms.

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  • $\begingroup$ I can understand the definition of addition for complex numbers, but it's the actual definition of the complex numbers in the first place that's giving me trouble. If a complex number c is defined as [latex]c=a+bi[/latex] then this + sign is adding a real number with a real number times an imaginary number. What I'm finding hard to understand is the interaction between real and imaginary numbers, as by the definition of complex numbers you have to have defined what an imaginary number is first. $\endgroup$ – James Machin Dec 3 '13 at 19:33
  • $\begingroup$ As to what complex are, think of them either as pairs or as arrows, as described above, and then single out the ones whose real part is $0$ and call them "imaginary" (a misnomer, of course). $\endgroup$ – Michael Hardy Dec 3 '13 at 19:40
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There is a difference between interpreting '+' as a sign or as a binary operation. The sign '+' although often omitted, serves to distinguish a number, say $+x$ from its additive inverse $-x$. So viewed in this way the sign '+' or '-' is used in the same way as one would use $x^{-1}$ to indicate a unique inverse relating to a particular binary operation. A complex number, as also indicated in the other answers here, really consist of two components, and might as well be written $(a,b)$ - so writing it as $a+bi$, the '+' is merely a sign, and not a binary operation between the two components. You might as well also write $ib+a$: although this is not the convention, this is not "wrong".

It should be noted that complex numbers are not the same as $\mathbb{R}^2$ specifically in terms of how $i$ is defined and multiplication is defined (you can multiply the real part with the imaginary part of another complex number, etc.), as described in other answers here - and that is why a complex number is not usually denoted as a pair in brackets - so as not to confuse with $\mathbb{R}^2$, I think.

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  • $\begingroup$ How is multiplication usually defined in $\mathbb{R}^2$? $\endgroup$ – robjohn Dec 3 '13 at 20:09
  • $\begingroup$ well in terms of vector products the inner product is perhaps the most common - the result is a scalar - read about it here: geomalgorithms.com/vector_products.html $\endgroup$ – Christiaan Hattingh Dec 3 '13 at 20:17
  • $\begingroup$ sorry i wanted to add there are other products such as cross product, tensor product, etc. not all defined in $\mathbb{R}^2$, but nothing like complex multiplication. Intuitively, and I know this is not mathematically precise at all, the two components of a complex number are related through $i$ in ways that the two components of $\mathbb{R}^2$ are not. You are really 'splitting' a 'number' into (square root of positive number) and (square root of negative number) - so you can treat the parts as components, as you do not "know" how to do addition on them, but it is still part of same number. $\endgroup$ – Christiaan Hattingh Dec 3 '13 at 20:29
  • $\begingroup$ My point is that there is no usual multiplication defined on $\mathbb{R}^2$. The complex numbers are $\mathbb{R}^2$ endowed with complex multiplication, which makes them a field. $\endgroup$ – robjohn Dec 3 '13 at 20:33
  • $\begingroup$ hi robjohn, yes exactly, agreed...$\mathbb{R}^2$ together with scalar multiplication and vector addition is a vector space and the complex numbers together with complex addition and multipication is a field... $\endgroup$ – Christiaan Hattingh Dec 3 '13 at 20:37

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