Additive Inverse

Definition

The additive inverse of a number is the value that, when added to the original number, results in a sum of zero. It represents the opposite or negative value of a given number.

5 Must Know Facts For Your Next Test

1. The additive inverse of a number is denoted by the negative sign (-) placed before the number.
2. For any number $a$, the additive inverse of $a$ is written as $-a$.
3. Adding a number and its additive inverse always results in the additive identity, $0$.
4. The additive inverse property is crucial in the context of adding and subtracting integers, as well as solving equations using the Division and Multiplication Properties of Equality.
5. Recognizing and applying the additive inverse property is essential for manipulating and simplifying algebraic expressions involving signed numbers.

Review Questions

• Explain how the additive inverse property is used when adding and subtracting integers.
  • The additive inverse property is fundamental in the operations of adding and subtracting integers. When adding integers, the additive inverse of a number can be used to simplify the expression. For example, to add $5$ and $-3$, we can rewrite the expression as $5 + (-3)$, which is equal to $5 - 3 = 2$. Similarly, when subtracting integers, we can think of the subtraction as adding the additive inverse of the second number. For instance, $7 - 4$ can be rewritten as $7 + (-4)$, which is equal to $3$.
• Describe how the additive inverse property is applied in the context of the Properties of Real Numbers.
  • The additive inverse property is one of the key properties of real numbers. It states that for any real number $a$, there exists a unique real number $-a$, called the additive inverse of $a$, such that $a + (-a) = 0$. This property is crucial in simplifying algebraic expressions and equations, as it allows for the cancellation of terms. For example, when solving an equation like $x + 5 = 12$, we can subtract $5$ from both sides to isolate the variable $x$ by using the additive inverse property: $x + 5 - 5 = 12 - 5$, which simplifies to $x = 7$.
• Analyze how the additive inverse property is utilized when solving equations using the Division and Multiplication Properties of Equality.
  • The additive inverse property is essential in applying the Division and Multiplication Properties of Equality when solving equations. When solving an equation like $3x + 2 = 11$, we can isolate the variable $x$ by subtracting $2$ from both sides, using the additive inverse property: $3x + 2 - 2 = 11 - 2$, which simplifies to $3x = 9$. Then, we can divide both sides by $3$ using the Division Property of Equality, again relying on the additive inverse property, to find that $x = 3$. The additive inverse property allows for the cancellation of terms, enabling us to manipulate equations and isolate the variable of interest.

Related terms

Additive Identity: The additive identity, also known as the zero element, is the number 0 which, when added to any other number, leaves that number unchanged.

Opposites: Opposites are two numbers that, when added together, result in the additive identity of 0.

Signed Numbers: Signed numbers are numbers that have a positive or negative value, indicating their direction or position on the number line.