# 1.3: How to Read and Write Mathematics

- Page ID
- 8381

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reading mathematics is difficult for beginners. It takes patience and practice to learn how to read mathematics. You may need to read a sentence or a paragraph several times before you understand it completely. There are writing styles and notational conventions that you acquire only by reading and paying attention to how mathematics is written. As we proceed with the course, we will discuss the details. As a starter, let us offer several suggestions.

- Make sure you know the definition of mathematical terms, the meaning and proper usage of mathematical symbols and notations. Although this may sound obvious, many beginners have difficulty understanding a mathematical argument because they fail to recall the exact meaning of certain mathematical concepts.
- Often, the reason behind a claim lies in the sentence before it. Sometimes it could be found in the preceding paragraph, and it is not unusual that you may need to check several sentences or paragraphs before it. You need to take an active role in reading mathematics, and you need to remember what you have read.
- Mathematicians prefer short and elegant proofs. To do this, they suppress the details of what they consider as “obvious” reasons. But what is obvious to one reader may not be that obvious to another. At any rate, for practical reasons, it is impossible to include every minute step in a mathematical argument. Consequently, keep your pencil and paper next to you, and be ready to check the calculation and fill in the missing details.
- It may help to try out some examples just to see how an argument works.
- After you finish reading a proof, go over it one more time, and try to summarize its key steps (in other words, try to draw an outline of the proof) in your own words.

Writing mathematics is even harder! It takes much longer to learn how to write mathematics. Of course, the most important thing about a mathematical argument is its correctness. When we say “good” mathematical writing, we are talking about precision, clarity, and sound logic.

- Be precise! For example, do not just say “it” when it is unclear which quantity you are referring to. This is particularly true in a lengthy argument. In this regard, it helps to identify and hence distinguish different quantities by their names such as \(x\), \(y\), \(z\), etc.
- Use mathematical terms correctly! A common mistake is confusing an expression with an equation. An equation has an equal sign, as in \[x+y = 5, \nonumber\] but an expression does not, as in \[x+y. \nonumber\]
- Likewise, the following is an inequality: \[x+y \geq 5. \nonumber\] Do not call it an equation!
- Do not abuse the word “solve.” For instance, many students would say “solve \(5^2+7^3\).” A more appropriate saying should be “compute the value of \(5^2+7^3,\)” or simply “evaluate \(5^2+7^3\).”

In the beginning, it helps to follow what others do. This again means you need to read a lot of mathematical writing, and pick up styles that you are comfortable with. We often follow some conventions (unwritten rules, if you prefer) that everyone follows.

Example \(\PageIndex{1}\)

Consider this argument for showing that \((x-y)(x+y) = x^2-y^2\):

We want to show that

\[ (x-y)(x+y) = x^2-y^2. \label{eg:readmath-01}\]

After expanding the product on the left-hand side, we find

\[ {} = x^2+xy-yx-y^2 = x^2-y^2, \nonumber\]

which is what we want to prove.

The logic and mathematics in the argument are correct, but not the notation. In formal writing, each equation should be a stand-alone equation. The last equation is incomplete, because it does not have anything on the left-hand side of the equal sign. Here is a proper way to write the argument:

**Solution**

We want to show that

\[ (x-y)(x+y) = x^2-y^2. \nonumber\]

After expanding the product on the left-hand side, we find

\[ (x-y)(x+y) = x^2+xy-yx-y^2 = x^2-y^2, \nonumber\]

which is what we want to prove.

The fix is simple: just repeat the left-hand side.

Example \(\PageIndex{2}\)

Short and simple mathematical expressions or equations such as \(a^2+b^2=c^2\) can be written within a paragraph. Longer ones and expressions or equations that are important should be displayed separately, and centered, on their own lines, as in \[x^3-y^3 = (x-y) (x^2+xy+y^2). \nonumber\]

If we intend to refer to the equation later, assign a number to it, and enclose the number within parentheses:

\[x^2-y^2 = (x-y) (x+y). \label{eqn:example}\]

Now, for example, we can say, because of \(\ref{eqn:example}\), we find

\[135 = 144-9 = 12^2-3^2 = (12-3) (12+3) = 9\cdot 15. \nonumber\]

For a longer equation such as

\[(x+y)^2 = (x+y)(x+y) = x^2+xy+xy+y^2 = x^2+2xy+y^2, \nonumber\]

it may look better and easier to follow if we break it up into several lines, and line them up along the equal signs:

\[\begin{align} (x+y)^2 &= (x+y)(x+y) \\ &= x^2+xy+xy+y^2 \\ &= x^2+2xy+y^2. \end{align} \nonumber\]

Although we display the equation in three lines, they together form *one* equation. The equal signs at the beginning of the second and third lines indicate that they are the continuation of the previous line. Since this is actually one long equation, we only need to say \((x+y)^2\) once, namely, at the beginning.

When part of the right-hand side extends beyond the margin, you may want to balance the look of the entire equation by repositioning the left-hand side:

\[ \begin{array}{l}

{(x^2+2xy+y^2) (x^2+2xy+y^2)} \\

= x^4+2x^3y+x^2y^2 + 2x^3y+4x^2y^2+2xy^3 + x^2y^2+2xy^3+y^4 \\

= x^4+4x^3y+6x^2y^2+4xy^3+y^4.

\end{array} \nonumber\]

In the multi-line display format, always write the equal signs at the *beginning* of the lines. Do not forget to align the equal signs.

When part of the right-hand side is too long to display as a single piece, we may split it into multiple pieces:

\[\begin{align} (x+y)^5 &= (x+y)^2 (x+y)^3 \\ &= (x^2+2xy+y^2) (x^3+3x^2y+3xy^2+y^3) \\ &= x^5+3x^4y+3x^3y^2+x^2y^3+2x^4y+6x^3y^2+6x^2y^3+2xy^4 \\ & \quad {} +x^3y^2+3x^2y^3+3xy^4+y^5 \\ &= x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+y^5. \end{align} \nonumber\]

It is a common practice to use indentation to indicate the continuation of part of a line into the next.

There will be more discussion as we continue. Let us not forget: the best way to learn is to watch and observe how others do it. Reading is a must! Reading and analyzing technical papers will surely improve your mathematical knowledge as well as your writing.