Additive Inverse

Definition

The additive inverse of a number is the opposite value that, when added to the original number, results in a sum of zero. It is a fundamental concept in the context of integers and other number systems.

5 Must Know Facts For Your Next Test

1. The additive inverse of a positive integer is the corresponding negative integer, and vice versa.
2. Adding a number and its additive inverse always results in a sum of zero, regardless of the number's sign.
3. The additive inverse plays a crucial role in the operations of addition and subtraction with integers.
4. Additive inverses are essential in solving equations with integers, as they allow for the isolation of variables.
5. The properties of additive inverses are foundational to understanding the behavior of decimals and the operations performed on them.

Review Questions

• Explain how the concept of additive inverse is used in the context of adding integers.
  • When adding integers, the additive inverse is used to simplify the process. If two integers have opposite signs, their sum can be found by subtracting the smaller absolute value from the larger absolute value and retaining the sign of the larger number. For example, the sum of 5 and -3 can be calculated as 5 + (-3) = 5 - 3 = 2, because -3 is the additive inverse of 3. This property allows for efficient addition of integers, regardless of their signs.
• Describe the role of additive inverse in the process of solving equations with integers.
  • The additive inverse is crucial in solving equations with integers. By adding the additive inverse of a term on both sides of an equation, it is possible to isolate the variable and find its value. For instance, to solve the equation $x + 5 = 12$, we can add the additive inverse of 5, which is -5, to both sides: $x + 5 + (-5) = 12 + (-5)$, simplifying to $x = 7$. This process of using additive inverses to eliminate terms and isolate variables is a fundamental technique in solving linear equations with integers.
• Analyze how the properties of additive inverse apply to the operations of multiplication and division with decimals.
  • The properties of additive inverse extend to the operations of multiplication and division with decimals. When multiplying a decimal by its additive inverse, the result is always zero, as the additive inverse of a number is the opposite value that, when added, results in a sum of zero. Similarly, when dividing a decimal by its additive inverse, the result is always 1, as the additive inverse of a number is the reciprocal that, when multiplied, results in a product of 1. These properties are fundamental to understanding the behavior of decimals and the operations performed on them, such as solving equations with decimal coefficients.

Related terms

Integers: The set of whole numbers, including positive, negative, and zero, that can be expressed without a fractional component.

Opposites: Two numbers that, when added together, result in a sum of zero. The additive inverse is the opposite of a given number.

Additive Identity: The number zero, which when added to any other number, leaves that number unchanged.