Understanding Definition of Complex Number

Question

In the book "Linear Algebra Done Right", a complex number is defined as: "an ordered pair $(a, b)$ , where $a, b \in \Bbb{R}$, but we will write this as $a + bi$."

"The set of all complex numbers is denoted by $\Bbb{C}$: $$ \Bbb{C}=\{a+bi\ : a, b \in\ \Bbb{R}\} $$"

Where did this imaginary number come from? What is wrong just an ordered pair?

I am not understanding what the imaginary number has to do with representing an ordered pair. Please note that I am revisiting math years after taking any type of formal math course. Currently I am interesting in linear algebra, and interested in understanding all math.

Answer 1

There is a confluence of notation here. The pattern "$a + b i$" could be interpreted in two different ways:

• As defined by the passage you quoted: this is just a funny way to write $(a,b)$.
• An arithmetic expression denoting the result of adding $a$ and the product of $b$ and $i$

As it turns out, both of these interpretations mean the same thing if we define $i$ to mean the ordered pair $(0,1)$ and make the complex numbers a real algebra by embedding the real numbers in the complexes by $r \mapsto (r,0)$.

Answer 2

This is just a definition, and not a proof. In this context, things do not need to "come from somewhere". You can think of it as a simple matter of notation, that is, how you represent things.

The text is just saying that you can equivalently represent any complex number as an ordered pair of real numbers. In fact, the same procedure is applied to other mathematical objects as well (e.g polynomials and matrices).

Finally, I would suggest how to look at the first 10 or 11 Chapters of 'Naive Set Theory' from Halmos if you want to get started with understanding mathematical formalism.

Answer 3

You can interpret the equivalente between $(a,b)$ and $a+ib$ in the following sense:

$(a,b)+(c,d) = (a+c,b+d)$

$(a+ib)+(c+id)=(a+c) + i(b+d)$ (as adition of polinomials!)

Do you see whats going on?

As we have a product in $\mathbb{C}$, we have by definition of the product that

$(a,b)(c,d) = (ac-bd, ad+bc)$

but we don't have to remember that formula if we remember that $i^2=-1$ and we write

$(a+ib)(c+id)=ac+aid+ibc+i^2bd = (ac-bd) +i(ad+bc) $ which is the same as $(ac-bd,ad+bc)$ above.

So we prefer to write $a+ib$ because it allow us to extrapolate usual notions of algebra and remember less stuff.

Hope it helps.