# Essential Maths for Proofers

### Introduction.

Projects which have only a few mathematical formulas don't really need the full power of LaTeX.

Our major problem in proofreading mathematical formulas is that we need to convert what may have been printed in a form that breaks out of the line (such as the big fraction in the examples below) into an equivalent form that can be represented on a single line but is still mathematically correct. We also want to be able to do this without having a degree in mathematics.

To meet this need here are some guidelines on handling any formulas that you may meet. Please note that you don't need to be a mathematician to use the information here. We will have some discussion on mathematical syntax but we will avoid doing sums!

Variable letters and functions are shown in this article as italic characters. This is simply to ensure that they can't be confused with the normal text.

Warning: The markup on this page currently has no basis for implementation into any form that's finally useful for publishing, either as plain-text or html. (Although it's better than just leaving a [**Maths] note in the text.) It might possibly be used for further work to develop such transformation utilities, perhaps into LaTeX to create graphics to be included in HTML, but that doesn't help plain-text versions.

In particular, don't use this as a guide for proofing Encyclopedia Britannica - they have no way to use it. Also DO NOT under any circumstance use these instructions for projects that bear the {LaTeX} tag in the title. Special guidelines for proofing LaTeX projects may be found here.

## Definitions.

### Formula.

A formula is simply a statement of the relationship between two or more symbolic quantities. It's written in just another language that mathematicians use to describe a relationship and can be translated into plain English, French or any other language. One of the most famous formulas is probably E = mc². "E equals m multiplied by c squared". Notice how much more concise the formula is.

### Expression.

An expression is a term for a combination of mathematical symbols. An algebraic expression is only a phrase, not a whole sentence, so it cannot contain an equals sign. (a + bc) is an example of an expression.

### Function.

A function is a relation between a set of elements, (values, numbers) and another set of elements. So imagine a function called Mult2, which returns the result of multiplying the input value by 2. If 4 is passed as an argument (input), the Mult2 function will return 8 (output). As a formula, this process is written: Mult2(4) = 8 or Mult2(x) = 2x.

## Precedence.

### Why we need to know about it.

The basic operations that we will meet are:

Exponentiation ^, multiplication ×, division ÷, addition + and subtraction −.

For a formula to 'work' those operations need to be carried out in the correct order. Formulas are evaluated left to right if there is no precedence but the order of the list above is important because it will change the order in which the formula is evaluated. Operators have a default order of operation with ^ having the highest precedence, i.e. it's done first, then multiplication and division and lastly addition and subtraction, that is the same order in which they are listed above.

An example may help. Let's take a × b + c × d where a = 2, b = 3, c = 4 and d = 5

So we have 2 × 3 + 4 × 5. Working strictly left to right:
2 × 3 = 6
6 + 4 = 10
10 × 5 = 50.

Now taking into account the precedence of the operators:
2 × 3 = 6
Do the next multiplication as multiplication takes precedence over addition
4 × 5 = 20
6 + 20 = 26,

So precedence is important.

### Parentheses, brackets and braces.

Sometimes even precedence isn't enough to ensure that operations are carried out in the right sequence. Then you will find parentheses (), brackets [] and braces {}. These take the highest precedence and are evaluated from the most deeply nested outwards.

### Vincula

There is one other grouping convention you may come across. That is a vinculum or horizontal line placed over an expression to indicate that it is to be considered as a group. Here is an example of a vinculum used in this way:

means you will have to add b and c first and then subtract the result from a.

We would need to proof that by replacing the vinculum with another pair of parentheses. (a − (b + c))

The long line above the expression after a square root symbol is a vinculum. You'll meet it in the example on square roots below.

## Proofing the formula.

We know how to proof that, E = mc^2 using the normal Proofreading Guidelines but it can, and usually does, get a little more tricky.

### Blank spaces within formulas.

Let's start with blank spaces or whitespace. In general we place blank spaces around the equals sign and operators like + and −.

### The equals sign.

Use a single = from the keyboard for equals. It should always have a blank space each side of it. In very old texts you may come across a very long equals sign, we still use the short character for that one too.

Use the keyboard characters and ensure that there's a space before and after.

So, a + b + c = d is correct

a+b+c=d is wrong.

### Multiplication and division.

If the multiplication sign is used please use the correct symbol × from the drop-down list in the proofing interface, the characters x or X are not correct. Also note that, if the × is used, it has a blank space before and after.

Commonly, multiplication is shown by placing the two or more variables close together so ab is interpreted as a × b. However, just proof what's on the page: don't introduce a multiplication symbol. Sometimes, you may find a full-stop or period (.) or a middle dot (·) used between the variables to indicate multiplication: proof it as printed, with a blank space before and after.

Division follows very similar rules. If the division symbol ÷ is used then it has a space before and after, if the forward slash / symbol is used then there are no spaces left either side of the symbol.

A right parenthesis indicates division with the denominator on the left, the numerator under a vincula on the right, and the quotient below the numerator if the numerator is a sum as it is here or above the vincula if not. For example for this table from an original page image: ```           ------
.44 1/6  ) \$80.28
------
182 da.```

### Fractions.

In the DP Guidelines you have seen the instruction, "Proofread fractions as follows: ¼ becomes 1/4, and 2½ becomes 2-1/2." This is likely to cause us some problems within formulas.

Notice that 2½ × 2 = 5 but, because of the precedence of multipication over subtraction we would have, interpreting standardly-proofed text literally:

2-1/2 × 2 = 2 − (1/2 × 2) = 2 − 1 = 1.

Even allowing for the fact that the lack of blank spaces around the - character tells us that this is not a normal expression, it would be much safer for the 2½ to be represented simply by 2 1/2 (or as {2 1/2} with braces if necessary to reduce ambiguity). However, as long as you remember to put spaces around the operator I suppose we can live with the Guidelines form.

### Decimal separator.

Books of DP vintage often use a middle dot · as a decimal point: please use the drop-down menu for this. In some old texts the decimal separator is printed rather low, and looks like a full stop or period. We will still use the traditional · character unless the PM has specifically asked you to use a full stop for the decimal point.

### Exponentiation.

Exponentiation is indicated in the printed text by using a superscript number, letter or expression. A simple example is that c² in Einstein's formula. We simply follow the normal Guidelines practice and use ^ with no space between the variable and the ^, c^2.

### Positive and negative quantities.

Sometimes a + or − symbol doesn't mean addition or subtraction. For example in talking about positive and negative electric charges. In these cases there is no space between the variable and the sign so we may find −a or +b. Quite often these signed variables are placed within parentheses when used in a formula.

Look at the following correctly spaced formulas:

```(+a) × (-b) = -(ab)
```

and

```(+a)/(-b) = -(a/b)
```

but

```(+a) ÷ (-b) = -(a ÷ b)
```

### Decorated variables.

If a formula contains closely-related variables, sometimes the same letter will be used but with decorations. The most common decoration is probably a subscript, so just follow the normal rules for proofing subscripts: a_{1}, a_{2} etc. Another common decoration is the "prime" symbol a′ which is often seen in calculus and in geometry; the nearest thing we have in the drop-down proofing interface menu is an acute accent which should be used, with two acute accents for a double prime, three accents for a triple prime, etc. You will never find a superscripted number as decoration. That's reserved for exponentiation.

## Functions.

There are many function names and symbols you may come across in a text but we can use some general rules for dealing with them.

Normally a named function, (cos, exp etc) will have a blank space before and after it. This is not an invariable rule though, where the following expression is in parentheses then the space is suppressed. Here's an example where the function itself is within parentheses and a variable i is placed immediately before the function name.

This would be proofed as:

```e^{a + bi} = e^a(cos b + i sin b)
```

There are a few things to notice here.

1. We use braces {} to surround the exponent {a + bi} to be consistent with the general practice for sub/superscripts.

2. Notice carefully where whitespace appears in our formula. The printed page can sometimes confuse us by using a half-space or no space. We will use spaces to help make the formula more readable.

### Summation, the Big Σ.

Summation, or Sigma Notation, uses a large Greek capital Sigma. We will represent that character with [sum]. The limits of the summation are written below and above the [sum] character, if present these are proofread like sub/superscripts, the lower one first. Here is an example of the summation character proofed.

```e^z = [sum]_{n = 0}^[infinity] z^n/n!
```

Or, in English:
"e to the z equals the sum from n = 0 to infinity of z to the n over n factorial", I told you it was just a different language!

Note that here we have used [infinity] to represent the infinity symbol. Also note that {n = 0} is a subscript so we have used the normal proofing convention and preceeded the expression with an underline character and braces {} to enclose the expression.

### Roots.

We reserve a special markup [sqrt] for the square root as it is the most common root you are likely to come across. We can use the Taylor series as an example:

```[sqrt](1 + x) = 1 + (1/2)x - (1/8)x^2 + (1/16)x^3 - ...
```

Notice the vinculum over the 1 + x has needed to be replaced with parentheses.

For roots of higher power, cube roots, fourth roots and the like, we can use a general form of [nroot] where n represents the power of the root. So we have:

```[3root]x = [3root]r exp((1/3)i[theta]).
```

There are a few things to notice in this proofed version.

1. There is a blank space before the exp function. (See the general rule for named functions above.)

2. We have used the Greek name for the θ variable and placed it in brackets [] but without specific instructions to use Greek letter names transliterate as [Greek: th].

3. There are no spaces in exp((1/3)i[theta]). This is an example of a function, exp, being followed by an expression in parentheses ((1/3)i[theta])

### List of Mathematical symbols not available in Latin-1.

Symbol Description Proofing transliteration
Identically equal to, equivalent `[equiv]`
Not equal to `[/=]`
Nearly equal to `[approx]`
Greater than or equal to `[>=]`
Less than or equal to `[<=]`
Therefore `[therefore]`
Because `[because]`
Δ Change in (calculus) `[Delta]`
Triangle (geometry) `[triangle]`
Angle `[L]`
Parallel to `||`
Perpendicular to `[perp]`
Element of `[in]`
Not an element of `[/in]`
Infinity `[infinity]`
Integral `[integral]`
Sum of terms `[sum]`
Product of terms `[product]`
Prime `Acute accent from drop-down list`
Partial derivative `[part]`

### Other functions, operators and symbols.

These rules can be extended to almost all other mathematical notation you are likely to come up against. But don't forget that the PM may have added specific instructions for any which this article hasn't addressed and that a question in the relevant project thread will prompt a reply from either the PM or a Project Facilitator.

There's one that deserves a special mention as you're almost bound to fall over it at some time. The old-fashioned way of showing proportionality. It takes the form a : b :: c : d. Or, In English; "a is to b as c is to d". Notice the spaces around them and notice, no spaces in the 'as' symbol.

The factorial ! is just a short-hand way of saying 'Multiply that integer by all the lower integers in turn', So 4! = 4 × 3 × 2 × 1 = 24 In proofing the n! is treated as a single entity so no space between them.

## Examples.

Here are some examples of correctly proofed formulas.

Starting with a simple example.

```a(x + y)
```

Notice that there are no spaces around the (.

This needs no work in proofing but note that it is asking for the x + y to be evaluated first and then the multiplication to be carried out. I've introduced it here because it will help us sort out this next monster.

Our object is to get this into a single line and make sure that, when evaluated, it still 'works'.

Starting with the innermost parentheses.

```(x - y)
```

Now multiply by z

```(x - y)z
```

Divide by w gives us

```(x - y)z/w
```

Now we can bracket that together and multiply by 3

```3[(x - y)z/w]
```

```3[(x - y)z/w] + v = k
```

Note particularly where there are spaces and where they are missing.

Finally here's a scary formula to show you how to handle nested roots, vincula and parentheses.

Proofed according to our rules:

```[sqrt](x + iy) = [sqrt]((r + x)/2) + i(y/[sqrt](2(r + x)))
```

You'll need to count the opening and closing parentheses to make sure that each opened parenthesis has a matching closing one.