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Mathematical Sentences

Introduction

The language of mathematics is universally recognized. It can also be used to express and share ideas with people without having to translate them into other languages. The ideas that mathematicians want to express are simple to articulate because of the language of mathematics. The language of mathematics is precise, concise, and powerful.

In this article, we will learn about mathematical sentences and how to write them as part of our study of the mathematical language.

What is a mathematical sentence?

Definition

A mathematical sentence is made up of numbers, variables, or combinations of numbers and variables that can only have one of the two outcomes—true or false—but not both.

The following are examples of mathematical sentences.

( a ) The numbers 3, 5, 7, 11, and 17 are all prime numbers. – This statement is true.
( b ) 4 + 5 = 3 × 3. – This is a true statement.
( c ) 7 + 7 = 12 – 4. – This is a false statement.
( d ) If n is an odd number, then 2n is an even number. – This is a true statement.

Open Mathematical Sentence vs. Closed Mathematical Sentence

Open Mathematical Sentence

An open mathematical sentence depends on one or more unknown variables to determine whether it is true or false. It is neither true nor false until specific values have replaced the variables.

For example, let us use the mathematical sentence 4x + 2 = 34. The table below shows the calculations when x is replaced with { 2, 3, 4, 5, 6, 7, 8 }

x4x + 2 = 34True or False
24 ( 2 ) + 2 ≠ 34False
34 ( 3 ) + 2 ≠ 34False
44 ( 4 ) + 2 ≠ 34False
54 ( 5 ) + 2 ≠ 34False
64 ( 6 ) + 2 ≠ 34False
74 ( 7 ) + 2 ≠ 34False
84 ( 8 ) + 2 = 34True

Since x = 8 makes the equation true, the solution to the equation 4x + 2 = 34 is 8. 

Closed Mathematical Sentence

A sentence that can be determined to be true or to be false is referred to as a closed mathematical sentence. It has no variables, and no more numbers can be added.

The following are examples of closed mathematical sentences:

( a ) 2 + 4 + 5 = 11. – This sentence is true.
( b ) 36 ÷ 9 = 4. – This statement is true.
( c ) 100 – 25 = 50. – This statement is false.
( d ) 25 > 35. – This statement is false.
( e ) The square root of 25 is 5. – This is a true statement.

Mathematical Sentence vs. Mathematical Expression

Mathematical Expression

A mathematical expression does not contain a complete thought; it is a combination of symbols that are well-defined according to rules that depend on the context.

The following are examples of mathematical expressions:

( a ) n + 8
( b ) n – 25
( c ) 5n
( d ) 35 ÷ 7
( e ) 4 × 12 

Notice that the above examples do not have equal signs. The first three examples are called algebraic expressions, containing variables, numbers, and numerical operators. The last two examples are numerical expressions since they have only numbers and mathematical operators. A mathematical expression containing fractional numbers and mathematical operators is a fractional expression.  

Mathematical Sentence

A mathematical sentence states a complete thought and makes a statement about the expression, using either numbers, variables, or a combination of both.

The following are examples of mathematical sentences:

( a ) 2 + 5 + 9 = 16
( b ) 3x + 4 < 50
( c ) 2x + 6y > 20 + x
( d ) 3x = 54
( e ) 45 ÷ 9 = 5

Notice that the first and last examples are both true statements. The other three examples are only true for specific values of the variables.

How to write mathematical sentences?

Since a mathematical sentence represents information numbers, variables, and operations, we must know how to translate verbal phrases into mathematical phrases.

Mathematical Phrases

Below are verbal and mathematical phrases to help us write mathematical sentences. Addition

Verbal PhrasesExamplesMathematical Phrases
plus5 plus 85 + 8
sumthe sum of 3 and 63 + 6
increaseincrease 8 by 28 + 2
increased by10 increased by 510 + 5
added to20 added to 55 + 20
more than7 more than 99 + 7
the sum ofthe sum of 11 and 1211 + 12

Subtraction

Verbal PhrasesExamplesMathematical Phrases
minus50 minus 1550 – 15
differencethe difference between 10 and 610 – 6
subtracted from2 subtracted from 1414 – 2
subtract fromsubtract 8 from 1212 – 8
decreasedecrease 13 by 613 – 6
decreased by15 decreased by 915 – 9
diminished by18 diminished by 1318 – 13
less30 less 1730 – 17
less than22 less than 3535 – 22
fewer than16 fewer than 2424 – 16

Multiplication

Verbal PhrasesExamplesMathematical Phrases
times5 times y5y
twicetwice 42 × 4
thricethrice a number n3 × n or 3n
multiplymultiply 5 by 75 × 7
multiplied by2 multiplied by 82 × 8
productthe product of 9 and 79 × 7

Division

Verbal PhrasesExamplesMathematical Phrases
divided by20 divided by 520 ÷ 5
dividedivide 100 by 25100 ÷ 25
quotientthe quotient of 15 and 315 ÷ 3
distribute equallydistribute 30 pens equally among 5 students30 ÷ 5

Mathematical Symbols

The following are the most common mathematical symbols when translating an English sentence into a mathematical sentence.

Addition ( + )
English phrase keywords:  the sum of, added to, increased by, plus, more than, greater than, exceeds by, more.

Subtraction ( – )
English phrase keywords: the difference of, decreased by, diminished by, minus, subtracted from, less than, take away, less.

Multiplication ( × )
English phrase keywords: times, twice, thrice, the product of, multiplied by, of.

Division ( ÷ )
English phrase keywords: divide by, the quotient of, all over, ratio, split into.

Exponents 
English phrase keywords: square, cube.

Equal sign ( = )
English phrase keywords: equals, is equal to, is.

Less than ( < )
English phrase keywords: is less than, is fewer than.

Greater than ( > ) 
English phrase keywords: is greater than, is more than.

Less than or equal to ( ≤  )
English phrase keywords: no more than, not more than, is at most, a maximum of.

Greater than or equal to ( ≤  )
English phrases keywords: is at least, a minimum of, no less than, not less than.

Translating an English Sentence into a Mathematical Sentence

Translate each into mathematical sentences and let m represent a certain number. 

( a ) The sum of 3 and a number is 16.
( b ) The sum of thrice a number and 12 is 100. 
( c ) The product of 5 and a number decreased by 6 is 26.
( d ) The quotient of thrice a number and 4 is 6.
( e ) The product of 6 and a number increased by 4 is at least 25.
( f ) Five times the square of a number diminished by 10 is less than 4.

Answer:

( a ) The sum of 3 and m is 16.

Since we have the keywords sum of and is, we will use the plus sign ( + ) and equal sign ( = ). Hence, the mathematical sentence is written as, 

3 + m = 16.

( b ) Thrice a number increased by 12 is 100. 

We will use the plus sign ( + ), the multiplication symbol ( × ), and an equal sign ( = ) since we have the keywords thrice, increased by, and is. Therefore, the mathematical sentence is written as, 

3m + 12 = 100

( c ) The product of 5 and a number decreased by 6 is 26.

We will use the multiplication symbol ( × ), subtraction symbol ( – ), and equal sign ( = ) since we have the keywords product of, decreased by, and is. Therefore, the mathematical sentence is written as, 

( 5 × m ) – 6 = 26 or 5m – 6 = 26

( d ) The quotient of thrice a number and 4 is 6.

We will use the division symbol ( ÷ ),  multiplication symbol ( × ), and an equal sign ( = ) since we have the keywords product of, the quotient of, twice, and is. Thus, the mathematical sentence is written as, 

( 3 × m ) ÷ 4 = 6 or 3m ÷ 4 = 6

( e ) The product of 6 and a number increased by 4 is at least 25.

We will use the multiplication symbol ( × ), plus sign ( + ), and equal sign ( = ) since we have the keywords product of, increased by, and is. Hence, the mathematical sentence is written as, 

( 6 × m ) + 4 ≥ 25 or 6m + 4 ≥ 25

( f ) Five times the square of a number diminished by 10 is less than 4.

We will use the multiplication symbol ( × ), minus sign ( – ), less than ( < ), and exponent since we have the keywords times, square of, diminished by, and less than. Therefore, the mathematical sentence is written as, 

( 5 × m2 ) – 10 < 4 or 5m2 – 10 < 4

Application of Mathematical Sentences

Word problems can be used to represent mathematical sentence problems. The examples that follow demonstrate how to translate word problems into mathematical sentences. 

Example 1

Marivic has 4 apples. If Maricar gives her three apples, how many does Marivic have?

Solution

Since the word problem has the keyword total, the mathematical sentence involves addition, and we must use the plus sign. The mathematical sentence to represent and answer the word problem is written as,

4 + 3 = 7

Therefore, Marivic has a total of 7 apples.

Example 2 

The perimeter of a rectangle is 54 when the length is twice the width. Find the following: mathematical sentence, width, and length.

Solution

Let us say that x is the width. We may represent the length as 2x. 

x = width of the rectangle
2x = length of the rectangle

The formula of the perimeter of a rectangle is P = 2L + 2W. Therefore, the mathematical sentence for the problem is written as,

54 = 2 ( 2x ) + 2x or 54 = 4x + 2x

To find the width of the rectangle, let us use the mathematical sentence and solve for x.

54 = 4x + 2x
54 = 6x
54 ÷ 6 = 9
9 = x

Therefore, the width of the rectangle is 9 units, and the length is 18 units.

Example 3

If we add two consecutive integers, the sum is 31. Write a mathematical sentence to represent the statement.

Solution

Let us say that the first number is p. Since the two numbers are consecutive; the next number must be p + 1. 

1st number = p
2nd number = p + 1

Therefore, the mathematical sentence is written as

p +  p + 1 = 31 or 2p + 1 = 31.

Summary 

A mathematical sentence is made up of numbers, variables, or combinations of numbers and variables that can only have one of the two outcomes—true or false—but not both.

An open mathematical sentence depends on one or more unknown variables to determine whether it is true or false. It is neither true nor false until specific values have replaced the variables.

A sentence that can be determined to be true or to be false is referred to as a closed mathematical sentence. It has no variables, and no more numbers can be added.

A mathematical expression does not contain a complete thought; it is a combination of symbols that are well-defined according to rules that depend on the context.

The following are the most common mathematical symbols when translating an English sentence into a mathematical sentence.

Addition ( + )
English phrase keywords:  the sum of, added to, increased by, plus, more than, greater than, exceeds by, more.

Subtraction ( – )
English phrase keywords: the difference of, decreased by, diminished by, minus, subtracted from, less than, take away, less.

Multiplication ( × )
English phrase keywords: times, twice, thrice, the product of, multiplied by, of.

Division ( ÷ )
English phrase keywords: divide by, the quotient of, all over, ratio, split into.

Exponents 
English phrase keywords: square, cube.

Equal sign ( = )
English phrase keywords: equals, is equal to, is.

Less than ( < )
English phrase keywords: is less than, is fewer than.

Greater than ( > ) 
English phrase keywords: is greater than, is more than.

Less than or equal to ( ≤  )
English phrase keywords: no more than, not more than, is at most, a maximum of.

Greater than or equal to ( ≤  )
English phrase keywords: is at least, a minimum of, no less than, not less than.

Frequently Asked Questions on Mathematical Sentences ( FAQs )

What is the other term for a mathematical sentence?

A mathematical sentence is also known as a mathematical statement or proposal, a statement that may be composed of numbers, variables, or a combination of numbers and variables that can be true or false, but not both.

Why is mathematical sentence fluency essential?

Mathematical sentences offer versatility in problem-solving. Students can analyze the numbers using language to determine the significance of each digit. They can use additional techniques to construct and break down numbers based on place value as they develop their logical thinking and mental math abilities.

The ability of students to translate the language (phrases or statements) into mathematical expressions and sentences is crucial to solving mathematical problems correctly. The correct equation and representation for solving a particular mathematical problem will result from adequately using variables, symbols, operations, and notations in translating language.

What are common mathematical symbols used to translate verbal sentences into mathematical sentences?

Expressions, equations, and mathematics problems frequently involve using words and symbols. The following are the most commonly used to translate verbal sentences into mathematical sentences.

NameSymbolRead asExample
Addition+plus2 + 5 = 7
Subtractionminus10 – 2 = 8
Multiplication×times10 × 4 = 40
Division÷divided by20 ÷ 5 = 4
Equality=is equal toequals2 + 8 =10
Approximately Equalis approximately equal to√10 ≈ 3.16
Strict Inequality<is less than10 < 15
>is greater than20 > 12
Inequalityis less than or equal to7 ≤ 8
8 ≤ 8
is greater than or equal to13 ≥ 10
13 ≥ 13
Inequationis not equal to10 – 8 ≠ 5

What is the difference between a mathematical expression and a mathematical sentence?

A mathematical expression does not contain a complete thought; it is a combination of symbols that are well-defined according to rules that depend on the context. Mathematical expressions can be numerical, fractional, or algebraic. 

The following are examples of mathematical expressions:

( a) 5 + 6 – 2 

This is a numerical expression since it contains numbers and a mathematical operator.

( b ) ½+¾-¼ 

This is a fractional expression since it contains fractional numbers and mathematical operators.

( c ) 5x – 3y + 6

This is an algebraic expression since it contains numbers, variables, and mathematical operators.

A mathematical sentence states a complete thought and makes a statement about the expression, using either numbers, variables, or a combination of both. A mathematical sentence can be either true or false but not both.

The following are examples of mathematical sentences.

( a ) The numbers 3, 5, 7, 11, and 17 are all prime numbers. – This statement is true.
( b ) 4 + 5 = 3 × 3. – This is a true statement.
( c ) 7 + 7 = 12 – 4. – This is a false statement.
( d ) If n is an odd number, then 2n is an even number. – This is a true statement.

What is the difference between open mathematical sentences from closed mathematical sentences?

An open mathematical sentence depends on one or more unknown variables to determine whether it is true or false. It is neither true nor false until specific values have replaced the variables.

For example, let us use the mathematical sentence 3m – 2 = 13. The table below shows the calculations when m is replaced with { 3, 4, 5, 6}

m3m – 2 = 13True or False
33 ( 3 ) – 2  ≠ 13False
43 ( 4 ) – 2  ≠ 13False
53 ( 5 ) – 2  = 13True
63 ( 6 ) – 2  ≠ 13False

A sentence that can be determined to be true or to be false is referred to as a closed mathematical sentence. It has no variables, and no more numbers can be added.

The following are examples of closed mathematical sentences:

( a ) 3 + 4 + 5 = 12. – This sentence is true.
( b ) 45 ÷ 9 = 5. – This statement is true.
( c ) 50 – 25 = 20. – This statement is false.
( d ) 15 > 40. – This statement is false.
( e ) The square root of 36 is 6. – This is a true statement.

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