Question

Consider the verbal phrase: the difference of 7 and a number \(\boldsymbol{n}\). What operation does the word difference indicate?

Short answer

The word 'difference' indicates the operation of subtraction.

Step-by-step solution

Understanding the Verb 'Difference'

In mathematics, the term 'difference' is generally used to denote the operation of subtraction

Formulating the Mathematical Expression

The verbal phrase is 'the difference of 7 and a number n'. In mathematical language, this translates into the operation 7 - n

Key concepts

Difference in Math

When we encounter the term 'difference,' it's easy to think about distinctions or comparisons between things, but in the mathematical context, this word has a very specific meaning. The 'difference' in math refers to the result you get when you subtract one number from another. It's the answer to a subtraction problem.

For example, if we have the numbers 10 and 4, and we want to find the difference between them, we compute 10 - 4 to get the difference, which is 6. It is imperative to remember that the difference is always found by taking the larger number and subtracting the smaller number, which can sometimes cause confusion. For instance, the difference between 4 and 10 is also 6, despite the order of the numbers being reversed, because the calculation is effectively finding how much greater one value is compared to the other.

Additionally, in mathematical problems, the difference can be a negative number if we subtract a larger number from a smaller one. So, if we have two numbers, a and b, the difference when we say 'the difference of b from a' would be expressed as a - b. In cases where b is larger than a, the difference results in a negative value.

Subtraction

Subtraction is one of the four basic arithmetic operations, along with addition, multiplication, and division. It's symbolized using a minus sign (-). The purpose of subtraction is to find the difference between numbers or to decrease a quantity by another.

One critical aspect of subtraction is understanding the terms used. The number from which you subtract something is called the 'minuend', and the number you are subtracting is the 'subtrahend'. The result of this operation, as we've learned earlier, is called the 'difference'.

Order Matters

In subtraction, unlike addition, the order of the numbers is important. For example, 7 - 3 will not give the same result as 3 - 7. The former equals 4, while the latter equals -4, demonstrating that flipping the order can sometimes lead to a negative result. This is why subtraction is not considered a commutative operation.

Translating Verbal Phrases to Mathematical Expressions

Understanding the translation of verbal phrases into mathematical expressions is a fundamental skill in algebra. It involves converting words into symbols and numbers, which can then be manipulated mathematically. A good understanding of key vocabulary words such as 'difference,' 'sum,' 'product,' and 'quotient' is crucial, as these words indicate specific operations: subtraction, addition, multiplication, and division respectively.

Let's dissect an example: the phrase 'the difference of 7 and a number n'. This phrase indicates that we are looking for the result of subtracting 'a number n' from 7. In mathematical terms, that translates into the expression 7 - n. Here, 'the difference of' signals us to use the subtraction operation, while the 'and' helps us identify the order of the terms involved in the subtraction.

Practice is key to mastering this translation process. Regularly converting verbal phrases to mathematical expressions will help you recognize patterns and common terminology, making it easier to approach and solve algebraic problems.