Expressions vs Sentences and Conventions in the Mathematical Language
Lecture 05
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Just like it is possible to translate from English or Filipino to Spanish or Korean, it is also possible to translate statements to and from the language of mathematics. This skill is often neglected by many students but one of the secrets to ace in math is to learn how to convert mathematical statements into English (or other ordinary languages) and vice versa.
In this lecture, we will first differentiate mathematical expressions and mathematical sentences. Then, we will translate mathematical statements into English and vice versa. We will only focus on Algebraic expressions and sentences.
Expressions vs Sentences
Expressions vs Sentences
There are generally two kinds of mathematical statements: a mathematical expression and a mathematical sentence. A mathematical expression is equivalent to nouns in the common language while a mathematical sentence is equivalent to, of course, a sentence.
Expressions
Expressions
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:
Sentences
Sentences
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".
Exercise. Identify whether the following are expressions or sentences.
Conventions in the Mathematical Language
Conventions in the Mathematical Language
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.
Symbols
Symbols
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.
Rules in Mathematical Writing
Rules in Mathematical Writing
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.
2. When writing multiple lines of equations (or inequalities), align the equal signs.
3. 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.
This could have been improved by writing clearly:
4. 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:
5. 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.
Translating Mathematical Statements into English
Translating Mathematical Statements into English
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.
"one plus two"
"one added by two", "two added to one", "one added with two" or "two added with one"
"one increased by two"
"two more than one" (note that this is different from saying "two is more than one" because that would be written as 2 > 1.)
"the sum of one and two", "the total of one and two"
"ten minus five"
"the difference of ten and five" (used best only when the difference is positive)
"five subtracted from ten", "ten subtracted by five"
"ten less five", "five less than ten" (note that this is different from saying "five is less than ten" because that would be written as 5 < 10)
"ten diminished by five"
"two times three"
"the product of two and three"
"two multiplied by three", "three multiplied to two", "two multiplied with three" or "three multiplied with two"
"two tripled" or "three doubled", "twice three" or "thrice two"
"six divided by two"
"the quotient of six and two" (best used only when the numerator has a higher absolute value than the denominator)
"the ratio of six to two"
"half of six" or "six halved"
"six over two" (best used only when the division is expressed as a fraction)
"two raised to the fourth power" (saying "two raised to the power of four" is both grammatically and logically incorrect),
"two raised to the fourth" (some authors begin accepting "two raised to four")
"the fourth power of two"
second and third powers can be read as "squared" or "the square of" and "cubed" or "the cube of"
"square-root of two"
in general, with an index n, this can be read as "the nth root of..."
index 3 can be read as "cube-root"
Relations
Now let us translate relations.
Remember that relations complete a mathematical sentence. Hence, when they are present, the English translation should contain the verb such as "is" and should end with a period.
"The sum of one and two is equal to three."
"The sum of one and two equals three."
"One plus two yields three." OR "One plus two results to three."
"Represents" is used to declare or define variables. For example, "let x represent the sum of one and two."
A common mistake students commit in reading "=" is saying "is equals to". Either "is equal to" or "equals" is the correct way.
"One fourth is less than one third."
"One fourth is smaller than one third."
"One half is more than one third."
"One half is greater than one third."
"Fourteen squared is less than or equal to two hundred."
"The square of fourteen does not exceed two hundred."
"maximum" is best used only when the inequality involves a variable
aside from what are given above, "at most" is also a possible translation for this inequality
"The cube of four added with five is greater than or equal to fifty."
"The sum of four cubed and five is not less than fifty."
"minimum" is best used when the inequality involves a variable.
aside from what are given above, "at least" is also a possible translation for this inequality
"The square root of two is approximately one point four."
"The square root of two is close to one and four tenths."
"The square root of two is about one point four."
Variables
There are three ways to read variables:
as the letter itself;
as "a number" or "the number" or the class of measures it represents such as "the angle" or "the length"; and
the physical meaning of the variable.
Exercise. Translate the following expressions and sentences into English in at least two ways.
Exercise. Translate from English into mathematics.
The quotient of the square of ten and the cube of three
Velocity is the ratio of distance to time.
Seven less than the product of four and five
The square root of any positive real number is never negative.
The product of two consecutive integers
Sum of two cubes
The difference of the squares of two numbers over their sum is equal to their difference.
The area of a circle equals π times the square of its radius.
The Fahrenheit measure of temperature is equal to nine times its Celsius measure divided by five then increased by thirty-two.
The sum of the first ten positive integers
References
References
[1] Oxford Learner's Dictionary. Subset. from https://www.oxfordlearnersdictionaries.com/us/definition/english/subset?q=subset
[2] Rosen, K.H. (2012). Discrete mathematics and its applications 7ed. McGraw Hill.
[3] R.E. Jamison. Learning the Language of Mathematics. Clemson University
[4] C.F. Burns. The Language of Mathematics. One Mathematical Cat! Please.
[5] Literary Devices. Ambiguity. from https://literarydevices.net/ambiguity/
[6] University of Illinois Springfield. Writing Clear and Concise Sentences. from https://www.uis.edu/cas/thelearninghub/writing/handouts/grammar-mechanics-and-style/writing-clear-concise-sentences/
[7] National Assessment Program. Language Conventions. from https://www.nap.edu.au/naplan/language-conventions
Review Questions
Review Questions
What are the key differences between expressions and sentences?
What composes the conventions of the mathematical language?
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