complex origin as a point?

Question

Why $a+\sqrt{-1}\times b$ is called as a point $(a,b)$ it is actually $2$ different numbers in addition, what can be the original significance of $\sqrt{-1}$ as a position and also those only in $\text{2D}$ plane why not $\text{3D}$ or more dimensionally, how and why it is treated as $(a,b)$

Answer 1

Technically I don't think $a + b\sqrt{-1}$ is a point in a two-dimensional space; it's just that when we map $a + b\sqrt{-1} \mapsto (a,b),$ that is, when we plot each complex number on a Cartesian plane using the real part of the number for one coordinate and the imaginary part for the other, we find a lot of good intuition in the shapes that these points make.

For example, plotting complex numbers in this way, the $n$th roots of $1$ are always on the unit circle, and the argument of a complex number actually is an angle measured from the positive real axis.

This mapping of complex numbers to the plane is so useful that we tend to identify the numbers with the points they map to and forget that they were two different things.

As for a three-dimensional mapping, it doesn't make much sense to try that when you can only extract two independent real numbers, but in the 19th century a mathematician named Hamilton tried really hard to find a new kind of "complex number" that would have a three-dimensional representation. He eventually found a four-dimensional representation instead, and developed the quaternions, which have one real part and three independent "imaginary" parts.

I forget whether Hamilton or later mathematicians finally answered the question regarding three-dimensional complex numbers, but I believe it was eventually found that we cannot make such numbers with the kinds of properties we find useful in complex numbers.

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Addendum: I also recall from the history of mathematics that the complex plane was not generally understood and accepted until complex numbers had been in use for many years. At least one early attempt to plot complex numbers in two dimensions was quite different from what we use today.

Answer 2

(1) A point can be a point on a line (one-dimensional), which can be represented by one real number; or a point on a plane, for example the complex plane (two-dimensional), which can be represented by a pair of real numbers, or equivalently by a single complex number; or a point in three-dimensional space, which can be represented by a triplet of real numbers; and so on.

(2) The algebra of complex numbers, originally created as solutions to algebraic equations (e.g. quadratic equations) that lacked real solutions, naturally corresponds to a planar geometric interpretation, where multiplication by $\mathrm i$ corresponds to rotation through $90^\circ$, and addition is vector addition.

(3) The extension of real numbers by solutions to the equation $x^2=-1$ doesn't have to be two-dimensional; in fact, there is a very nice four-dimensional extension called the quaternions, in which the above quadratic has an infinite three-dimensional set of solutions (which can be written as linear combinations of three basic solutions). However, in quaternion space, multiplication is not generally commutative (i.e. $xy\neq yx$ for most choices of $x$ and $y$). The usual complex numbers are the only extension that retains commutativity.

Answer 3

Anything can be called a point. A point is just a member of a set.

We can define $\mathbb C$ as the set of ordered pairs $(a,b)$ of real numbers, with $(a,b)+_c (a',b')=(a+a',b+b')$ and $(a,b)\times_c (a',b')=(aa'-bb',ab'+a'b).$

Then the $+_c$ (additive) identity of $\mathbb C$ is $(0,0)$ and the $\times_c$ (multiplicative) identity is $(1,0)$. The $+_c$ (additive) negation of $(1,0)$ is $(-1,0)$ and the square roots of $(-1,0)$ are $(0,\pm 1).$

With this def'n, $\mathbb C$ has a substructure, namely $\mathbb R \times \{0\},$ that behaves algebraically just like $\mathbb R.$

It does not mean anything to ask which of two structures is the "real" real numbers if they are algebraically isomorphic, so we speak of $\mathbb R$ as a subset of $\mathbb C,$ and write $+$ and $\times$ rather than $+_c$ and $\times_c$. And write $x+iy$ rather than $(x,0)+_c (0,y).$

Answer 4

$a+ib$ (or $a+\sqrt{-1}b$) is an alternative notation to $(a,b)$, which represents a 2D point in the frame of analytical geometry. The notation as a sum with a factor $i$ ensures that the two numbers remain separated.

The following arithmetic laws are associated to the notation:

$$(a,b)\pm(c,d):=(a\pm c,b\pm d),$$

which is coherent with

$$(a+ib)\pm(c+id)=(a+c)\pm i(b\pm d);$$

Then

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

which is coherent with

$$(a+ib)\cdot(c+id)=(ac+i^2bd)+i(ad+bc).$$ There is a perfect match if you state $i^2=-1$.

It turns out that these conventions are quite coherent and lead to very powerful tools. In addition, there is a direct connection to geometry via the rules

• the complex number $(a,b)$ can be associated to the vector $(a,b)$;

• the sum of two complex numbers corresponds to the sum of the related vectors;

• the product of a complex numbers by $i$, $(0,1)\cdot(a,b)=(-b,a)$ corresponds to the vector rotated by $90°$ counterclockwise.

The general multiplication rule is a little more difficult. It states that the product of two complex numbers is the vector with a length equal to the product of the lengths of the arguments, and the direction angle is the sum of the direction angles.

In the geometric interpretation, the term without the $i$ factor (the "real part") is the horizontal component of the vector, while the term with the $i$ (the "imaginary part") is the vertical component. You don't need to assign $i$ another meaning than that. [Whether $-1$ "really" has a square root is an irrelevant philosophical question.]

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For the 3D case, if you try on your own to generalize by writing

$$(a,b,c):=a+ib+jc,$$

you will soon realize that this "doesn't work".

Only in 4D will you find a viable system called quaternions,

$$(a,b,c,d):=a+ib+jc+kd$$ with which you can define coherent arithmetic laws. This system has the peculiarity that multiplication isn't communtative anymore,

$$ij\ne ji.$$

Answer 5

The expression $a+bi$ is not really "two different numbers in addition", as you state. Similarly, the expression $3x+2$ is a single value, but we are unable to combine the terms, since they are not the same.

Read the other answers; they explain really well why we represent complex numbers on a plane (and the other parts of your question, too). But do not think that $a+bi$ is the sum of two numbers; like a vector of two components, it is a single value.