# What Type of Mathematical Knowledge Is Needed to understand these?

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## What type of mathematical knowledge is needed to understand these mathematical notations?https://arxiv.org/pdf/1603.01121

Each bit of mathematical notation is an abbreviation for English words. There are a few reasons for using a notation rather than words. Save space and be able to express complicated expressions in simple formulas. Imagine if you had to do algebra the way Al-Khwarismi did, all in words, even the numbers were written out. We can solve the equation x^2-3x+1=0 with the quadratic formula x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a} to find x=\frac12(3\pm\sqrt5). Imagine solving it all in English without the formula. Find a number such that the square of that number minus three times it plus one is equal to zero. Then when you square three halves less than that number and subtract nine fourths and add one, you'll get zero. Therefore, the square of three halves less than that number is equal to five fourths. Taking square roots, it follows that three halves less than that number is equal to plus or minus half the square root of five. Thus, the number is three halves plus or minus half the square root of five. Make conceiving more abstract concepts easier. Before functional notation mathematicians used variables and connected them by equations. For example, y=x^2-3x+1. You can call the right side of that equation a polynomial and denote it f(x). That f is a function, a second-order concept whereas the variables x and y were first-order.. A function f that takes real numbers as arguments and returns real numbers is often denoted f.\mathbf R\to\mathbf R where \mathbf R denotes the real numbers. (Collecting all real numbers into one thing is also an abstraction.) Sometimes you want to consider not just one function f.\mathbf R\to\mathbf R but all of them. The notation \mathbf R^{\mathbf R} is useful for that, and \mathbf R^{\mathbf R} is actually a third-order concept. When you come across a new mathematical notation, understand what it means in English. As you use it more and more, you won't have to translate it into English, but prefer to use it as is.

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