Top Ten Mathematical Tools Ever Invented

I'm not including algebra here, because it is elementary and almost all the mathematical tools listed here use algebra in some way or the other.

The Top Ten

1 Calculus

If newton did not discover calculus (and also one more mathematician whose name I'm currently having difficulty to remember), then Physics wouldn't have much scope at all and we all would be living in 17th century (even now). Although calculus is difficult to explain in common language terms, it will be very interesting field for you, if you get to know about it. - Kiteretsunu

I adore calculus. It is a language in itself. And look where it took us! Cannot imagine physics without it.
Kiteretsunu, you're thinking of Leibniz. - PositronWildhawk

2 Coordinate Geometry

Well, who can beat this beast! Descartes gave us this wonderful mathematical tool that could solve any geometry problem with ease. Physics would be very incomplete without this. It helped us use algebra in solving geometry problems.
This mathematical tool is simple in its making, but brilliant and ingenious in application. - Kiteretsunu

Oh totally agree. This is definitely the best... Or at least that's what I WOULD say if I knew what it was... - Britgirl

3 Trigonometry

Who thought that ratios of the length of sides of a right angled triangle would create miracle in the field of science! Although this field of mathematics is very easy to learn, it has such applications in physical sciences, that all the mathematical theorems of advanced mathematics and physics would collapse if trigonometry were not applied on them. - Kiteretsunu

4 Vector Algebra

Vector Algebra is very necessary for the fields of both Physics and Three Dimensional Geometry. It is an algebra concerning directed line segments. All Newtonian Mechanics will collapse if the human race somehow, forgets this one. - Kiteretsunu

5 Determinants

Determinants is the most beautiful mathematical device to look at! It just looks so pretty! You just have to draw two vertical lines with sufficient space between them and write numbers in imaginary rows and columns to create one.
However there are special rules to solve a determinant, and its algebra is somewhat complicated too.
So why was Determinant ever invented?
It was invented to make "solving of system of linear equations" a little easier job to do. It saves time. And as I have said, because the determinants are beautiful to look at, equations and mathematical formulas are easier to remember which involve them. - Kiteretsunu

6 Matrices

It just looks very similar to determinant but is very different from it, both in terms of its applications and its way of solving. - Kiteretsunu

7 Set Theory

Set theory gave birth to both relation and function fields of mathematics. Although set theory has very limited applications, it is very useful in solving Permutations and Combinations problems, Probability problems and others. - Kiteretsunu

8 Binomial Theorem

This is yet another masterpiece. - Kiteretsunu

9 Progressions and Series

Like arithmetic progression, Geometric progressions and Harmonic progressions. - Kiteretsunu

10 Probability

This is the only field of mathematics which was not invented because science needed it, but because the game of gambling needed it. Probability has many applications, like in weather forecasting, and of course in quantum mechanics. Probability is what gives us reality picture for different wave functions in quantum mechanics - Kiteretsunu

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