Expressions vs Sentences and Conventions in the Mathematical Language

An expression is a name given to an object of interest [4]. Generally, a mathematical expression has the following characteristics:

• It is equivalent to a noun in the common language, that is, it is a name given to a quantity or a mathematical object. It could also have synonyms, i.e., other mathematical expressions that have the same meaning.

• It does not have a complete thought.

For example, the name given to the number of apples below is "3".

There are also other ways to express "3". For example, "1 + 2", "1 + 1 + 1", and "6 ÷ 2" are all other names for 3. An expression does not convey a complete thought but instead, only represents an object. Whereas a sentence conveys a complete thought. Other examples of expressions:

A mathematical sentence is a statement that can be identified as either true or false (even if only conditional). Mathematical sentences have the following characteristics:

• They have a complete thought.

• They state relationships between two or more mathematical expressions. By relationships we mean =, <, >, ≤, and ≥.

For example,

1 + 1 = 2

A mathematical sentence is a statement that can be identified as either true or false (even if only conditional). Mathematical sentences have the following characteristics:

• They have a complete thought.

• They state relationships between two or more mathematical expressions. By relationships we mean =, <, >, ≤, and ≥.

For example,

1 + 1 = 2

is a mathematical sentence because:

• it can be identified as either true or false (in fact, it is true);

• it has a complete thought telling us that "adding two 1's yields 2;" and,

• it states the relationship (equal or "=") between two expressions "1 + 1" and "2".

The conventions in a language are composed of spelling, grammar, and punctuation [7]. In the mathematical language, the conventions are symbols we use and the rules we follow in writing them.

Mathematicians use a wide range of "alphabet" in mathematical literature. This alphabet is composed of numbers, letters, and special symbols that denote a specific meaning.

Numbers

Of course, the most basic set of symbols that we use in mathematics are numbers. We will agree to limit our scope only to Hindu-Arabic numbers. It has 10 digits namely:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

It is assumed that the reader already knows how the Hindu-Arabic numeral system works including the decimal system and the negatives.

The decimal point is used to divide fractional values from whole numbers. For example, the number

1.2

means there is a whole number 1 and the fraction 2/10. That is why, the number 1.2 can be read as

"one and two-tenths".

However, for simpler reading especially when we have long strings of decimals, most teachers would allow reading it as "one-point-two".

An ellipsis ". . ." (or the three dot symbol) denotes that the decimal continues to infinity. If a pattern of repeating digits appears before the ellipsis, then the ellipsis signifies that the decimal is non-terminating and repeating.

0.333. . .

0.121212. . .

If there is no repeating pattern before the ellipsis, then it means the number is non-terminating and non-repeating (i.e., an irrational number).

3.1415926. . .

1.41421. . .

Some authors are using the vinculum symbol (the bar placed on top of the numbers) instead of the ellipsis but we will not use this convention in this course.

Fractions like

are read as "one half", "three fifths", and "seven halves". But, we also accept reading them as "one over two", "three over five", and "seven over two".

Letters

Mathematics also borrows letters from different alphabets to symbolize variables, constants, operations, or other ideas. Remember that in mathematical writing, letters are always CASE SENSITIVE. This means that capital letters may have totally different meanings from small letters. For example, in statistics, the capital letter N conventionally denotes population size while the small letter n denotes sample size.

We also use Greek letters for different purposes:

Generally, anyone can use any English or Greek letter to symbolize anything. However, some of these characters have already been popularly used for specific roles. For example, the letter π is already popular for the constant 3.14159. . . , the letter i is used to denote the imaginary number

Sometimes, these letters are stylized to signify even more specific meanings. For example, the "blackboard bold" style letters ℕ, ℤ, ℚ, ℝ, and ℂ denote the set of natural numbers, integers, rational numbers, real numbers, and complex numbers, respectively.

Even the font face may be important in writing symbols. For example the following symbols may have different meanings:

Operations, Relations and Grouping Symbols

We use the following symbols and techniques to denote the fundamental operations in arithmetic:

Meanwhile, the following are the mathematical relationships we commonly use:

Grouping symbols are used by mathematicians to effectively indicate which operations to treat first. We have,

( ) - curved brackets/parentheses

[ ] - square brackets/brackets

{ } - curly brackets/braces

By convention, parentheses are used as the "innermost" grouping symbol, followed by brackets, then lastly by braces. Other authors just use nesting parentheses for all levels of groupings.

There are also other notations that while acting as functions, may also be considered as grouping symbols:

| | - single vertical bar grouping: absolute value

‖ ‖ - double vertical bar grouping: norms

---- - fractional bar/vinculum

Special Symbols

There are also special symbols used for various purposes in mathematics. Some of them are given below.

There are many more notations and special symbols used in mathematics to denote expressions, operations, sentences, or any other mathematical statement aside from those given above.

There are rules in writing mathematical text. There are so many and may vary from author to author. Hence, we only note the most common and most important few.

• When writing expressions or sentences in a single line, the vertical alignment should be centered. This is especially emphasized when there are fractions, summations or larger notations: simple symbols should by aligned in the vertical middle.

• When writing multiple lines of equations (or inequalities), align the equal signs.

• Do not write ambiguously. Make use of grouping symbols or other tools to signify the statement. The viral 2019 problem is an example of an ambiguous mathematical expression.

• Use the equal sign as how it is defined: two expressions being equal. Many students think of the equal sign to represent "result" or "answer". For example, the following are wrong:

• Both sides of the relationship symbols should be numbers. We cannot write

because the left hand side is a set (the empty set) while the right hand side is a number. So they can NEVER be equal even if it is true that the empty set has zero elements. Another example is when a student means to say "the area of the triangle ABC is 1 square unit", he tends to write:

This is wrong because the left hand side of the equation is a triangle (which is a geometric figure) while the right hand side is a number. One way to correct both of these equations is:

For more rules on mathematical writing, check out these papers: Ten Simple Rules for Mathematical Writing by D. Bertsekas and Rules and Tips for Writing Mathematics by A. Crannell.

We will only be translating algebraic statements. That is, we will only work with operations, constants and variables, and the relations =, <, >, ≤, and ≥.

Operations

We can do this by translating first the operations.

A common mistake students commit in reading "=" is saying "is equals to". Either "is equal to" or "equals" is the correct way.

Variables

There are three ways to read variables:

1. as the letter itself;

2. as "a number" or "the number" or the class of measures it represents such as "the angle" or "the length"; and

3. the physical meaning of the variable.

1. The quotient of the square of ten and the cube of three

2. Velocity is the ratio of distance to time.

3. Seven less than the product of four and five

4. The square root of any positive real number is never negative.

5. The product of two consecutive integers

6. Sum of two cubes

7. The difference of the squares of two numbers over their sum is equal to their difference.

8. The area of a circle equals π times the square of its radius.

9. The Fahrenheit measure of temperature is equal to nine times its Celsius measure divided by five then increased by thirty-two.

10. The sum of the first ten positive integers