# Why do we call $\sqrt{-1}$ imaginary and $-1$ real?

Looking at the numbers I can see in nature only positive real numbers. Because many problems couldn't be solved using only positive real numbers a new number set, called "negative numbers" was introduced. Later on, for similar reasons, complex numbers were introduced and the term of "imaginary".

Can anyone please explain to me why $-1$ is "more real" than $i$? For me both seem to be a convention, a name given to the result of a function or a set of functions to the positive real numbers.

Disclaimer: I'm not a mathematician, I'm just trying to better understand some basic concepts and conventions in math

• This is a philosophical question. See also Does a negative number really exist? and Do complex numbers really exist? – Alex Silva Dec 22 '16 at 9:59
• "For me both seem to be a convention": exactly that. You are adding your own interpretation "more real", which is not intended. Had the qualifier "dense" be used f.i., you would be asking "is $-1$ more dense than $i$ ?" – Yves Daoust Dec 22 '16 at 10:10
• Note that negative integers are defined before reals are defined. – Yves Daoust Dec 22 '16 at 10:12
• It's just an unfortunate naming. And it's extremely narrow minded to say only natural numbers are found in nature. Most natural concepts reflect rationals & reals, a lot of oscillatory phenomena are naturally described with complex numbers, and quantum mechanics is complex on the fundamental physical level. I extremely recommend this entire series of videos: youtube.com/watch?v=T647CGsuOVU – orion Dec 22 '16 at 10:52
• This SMBC comes to mind. – Asaf Karagila Dec 22 '16 at 16:04

In mathematics, every modifier, such as "continuous, real, rational, ...", is reassigned meaning, lest there be unnecessary confusion such as "your 'continuous" is different than mine" that wastes time. So, if you come across a terminology in some math book, find its definition within the book; do not take a dictionary and check the meaning of the term...

I am not sure if you are asking about the origin of the reason why the word "imaginary" was chosen. If not, then leave the "secular" meaning of the word alone. If you do this, then it is crystal clear that "imaginary number" is no more than a word used to name the relatives of $\sqrt{-1}$. Note also that "positive" and "negative" in math have nothing to do with any sentiments.

Logically, you can call the real numbers "imaginary" and the imaginary numbers "real". Then in your book, the theorem "the square of every real number is $\geq 0$" becomes "the square of every imaginary number is $\geq 0$". Noticed? The name thing is rather superficial :), which in math serves as a mnemonic device and a shortcut of reference.

Natural numbers is a very intuitive mathematical construction, corresponding to counting, and this is why they are termed natural. Together with natural numbers, addition appears.

Then the need appears to solve problems like $3+x=7$, how much is $x$ ? This is solved by means of subtraction. But then you face frustrating problems such as $7+x=3$, how much is $x$ ? To cope with these, negative numbers are introduced, forming the integers. (A close friend of entire.)

The next step is to look at multiplication, i.e. repeated addition. All is fine until you want to solve problems like $4\times x=24$, how much is $x$ ?, then $7\times x=17$, how much is $x$ ? This is how division and the rational numbers are introduced. (From ratio.)

The ancient Greeks once discovered that rational numbers are not all, much to their resentment, when they asked the question $x\times x=2$, how much is $x$ ? Then came the real numbers. (Possibly evoking the continuous characteristic of our real world.)

Another step was reached in the Middle-Age when mathematicians started to deal with the equation $x\times x=-1$, also annoyingly unsolvable, and the imaginary and complex numbers were introduced.

You may attach a "romantic" meaning to these terms, but this is not the intent. On the opposite, they are conventional and universally adopted words with a well-defined understanding, and probably a more mnemonic intent.

Even though some of these number categories may look somewhat artificial, there are cases where they come handy just for intermediate results in solving a real-world problem. A famous example is the need for complex numbers to find the three real roots of some cubic equations.

We call them imaginary because people made fun of the person who first used them. They mocked his methods, saying he used "imaginary" numbers. Seems like it stuck.

Ultimately, they're no more fictitious than anything else in mathematics.

• Nor less fictious :) – Yves Daoust Dec 22 '16 at 10:15

For humans, certain abstract objects are more natural than others. For example, natural numbers like $1$, $2$, and $3$ are second nature to us. Rational numbers are useful in the context of "relative" size. Three nickels are $\frac{15}{100}$ as valuable as a dollar. Real numbers are intuitive as well; they can be used to model "continuous" quantities like length and area. Negative numbers can be used to model things like debt and (cold) temperature.

Conversely, complex numbers are not so intuitive. We can't actually "see" $\sqrt{-1}$. Of course, complex numbers are present in the real world; they are used, for example, to mathematically model quantum theory. However, for non-quantum physicists, $\sqrt{-1}$ doesn't mean anything concretely. It is detached from reality, in a sense.

When complex numbers were first introduced, the prevailing ethos in mathematics was that a number is something which lies on the number line. Complex numbers were used, but, for the mathematicians of this time, because these "numbers" did not correspond to anything we could actually imagine, they were regarded as imaginary. For these mathematicians, the complex numbers were only something to be used in formal calculations; they weren't actually "real". Evidently, the name "imaginary" stuck.

Nowadays, whether or not something is "real" in the sense that it corresponds to the real world is largely irrelevant to mathematicians. Mathematics is purely a study of the abstract nowadays. The commutative field $\mathbb{C}$ of complex numbers is the two-dimensional vector space $\mathbb{R}^2$ of real numbers with multiplication defined by $(a,b)(c,d) = (ac - bd, ad + bc)$. "$\sqrt{-1}$" may be identified as the pair $(0, 1)$. This is called the imaginary unit. The name is a relic.

-1is real no because it can represented on real number line but$\sqrt{-1}$ can't be represented on this line

• This meaning has no sentence. – Yves Daoust Dec 22 '16 at 10:17
• @YvesDaoust: In fact, it does: it says that if you define $\sqrt {-1}$ purely algebraically, as one of the roots of $x^2 = -1$, then when you try to picture it geometrically you'll discover that it cannot sit among the real numbers, therefore it is in a way of a different "nature". Notice that in the above answer "no" is a shorthand for "number". While terse and with poor punctuation, this answer is essentially correct. – Alex M. Dec 22 '16 at 10:41

Whenever you have got a question about why we call something so.... refer to the definition.

According to Wikipedia:

A real number is a value that represents a quantity along a line.

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit $i$, which is defined by its property $i^2 = −1$.

There is no problem in understanding that why $-1$ is a real number as you can easily construct a line and represent $-1$ on that line. The problem lies in the definition of imaginary numbers. It is a real number times $i$. Why $i$ ?? Because imaginary numbers are defined that way and changing the definition is not so good I think. :) :) :)

• "why i?" maybe because imaginary? – Ring Ø Dec 22 '16 at 10:15