At least those languages are precise, spoken language is not. It all started with adding but in the last 400 years or so some concepts have been added. To start let's see what adding does. It leads to the natural numbers ℕ and multiplikation. Since we want to invert addition, i.e. substract, we come to the integers ℤ. The inversion of multiplikation is a division which leads us to the rational numbers ℚ. Now we found, that a diagonal of a square with length 1 can't be measured in ℚ, it's √2. So we have to complete ℚ with all limits of serieses of rational numbers: 1, 1.4, 1.41, 1.414, 1.4142, 1.41421 etc ... → √2. And of course π as well, the ratio between circumference and diameter of a circle and so on. So we got the reals ℝ. Almost done. We came from a half-group ℕ, to the additive group ℤ, which is also a ring, since we can multiply, to our smallest field ℚ, in which we can additionally divide. As the smallest field in which this can be done, it's a prime field. ℝ is its completion since a convergent serie has its limit in it. But there is still a problem. We cannot solve X*X + 1 = 0. So we introduced an imaginary unit i = √-1 to solve this equation and added it to ℝ receiving the complex numbers ℂ which turn out to be very useful in all computational processes. We therefore call ℂ the algebraic closure of ℝ, i.e. all algebraic equations can be solved in it. ℂ is a field extension of ℝ of degree 2, the power of X in X*X +1 = 0.
Now things become funny. Sir William Rowan Hamilton asked himself whether there is another field (+ - * /) containing ℂ. After years he found the Quaternions. But it came to a price. a*b is no longer equal to b*a. A quaternion can be written by four real numbers a + i*b + j*c + k*d where i,j,k fulfill certain equations. It is a 4-dimensional space over ℝ. Such spaces are vectorspaces which merely means arrows. Vectorspaces in which we can multiply are algebras. There are several forms of them: some with a 1, some without, some in which a*(b*c) = (a*b)*c is valid and some where it doesn't hold, e.g. Lie Algebras. Each of the described concepts has many generalizations, special cases, and unusual rules.
E.g. a light switch and this computer obey the rule 1+1=0, the clock 1+1+1+1+1+1+1+1+1+1+1+1=0. A total bunch of structures are out there.
Once you know what it's about it looses its secrets. I mean everbody uses a switch not thinking about the fact, that switching it twice is equal to not switching it. The problem is that meanwhile there are really many concepts, esp. added in the last century.
And to make it complicated, physicist write, e.g. their vectors different than mathematicians do.
