Question

Complete the sentence. The absolute value of \(-10\) is _______________

Short answer

Answer: The absolute value of -10 is 10.

Step-by-step solution

Understand the concept of absolute value

The absolute value of a number is the non-negative value of that number, regardless of its sign. It is denoted by two vertical bars surrounding the number, like this: \(|-10|\).

Apply the absolute value rules

According to the rules of absolute value, the absolute value of a positive number is the number itself and the absolute value of a negative number is the negative of that number. In this case, we have a negative number \((-10)\), so the absolute value will be the negative of \(-10\).

Calculate the absolute value

The absolute value of \(-10\) can be calculated as follows: \(|-10| = -(-10) = 10\). Thus, the absolute value of \(-10\) is 10. So, the completed sentence is: The absolute value of \(-10\) is 10.

Key concepts

Non-Negative Value

When we talk about a 'non-negative' value in mathematics, we are referring to a number that is either greater than or equal to zero. This means all positive numbers, including zero itself, fall into the category of non-negative values. In our exercise, the absolute value of \( -10 \) is sought, which will indeed be a non-negative value.

To further clarify, any negative number, when processed through absolute value, will shed its negative sign and emerge as a non-negative entity. This arithmetic operation ensures that a number's distance from zero on the number line is represented without considering its direction - a crucial concept in various mathematical applications such as complex numbers, geometry, and solving inequalities. For instance, the number \( -10 \) is 10 units away from zero, hence its absolute value is \( 10 \) - a non-negative result.

Mathematical Notation

Mathematical notation is a system of symbols used to represent numbers, operations, relations, and other concepts within the language of mathematics. It serves the purpose of providing a concise way to express complex mathematical ideas. In our exercise, the notation for absolute value uses two vertical bars \( |...| \) surrounding the number whose absolute value is to be determined.

For example, the absolute value of -10 is written as \( |-10| \) in mathematical notation. This notation is widely recognized and understood in mathematical circles, allowing mathematicians and students alike to communicate about absolute values efficiently. Understanding such notations is key to grasping the language of math and excelling in its study.

Arithmetic Operation

An arithmetic operation is a fundamental mathematical process such as addition, subtraction, multiplication, or division. Absolute value can be regarded as another arithmetic operation, though it does not combine numbers but rather transforms a single number. It's like a function that outputs the distance a number is from zero on the number line, regardless of the direction.

Applying the absolute value operation to a negative number effectively 'flips' it to its positive counterpart. For the given number \( -10 \) in our exercise, the arithmetic operation of taking the absolute value changes it from a negative to a positive, resulting in \( |-10| = 10 \) as the solution. Arithmetic operations are the bedrock of math, enabling us to solve from the simplest to the most complex problems.