Opposite

Definition

The opposite of a number or quantity is the value that is equidistant from zero on the number line but in the opposite direction. In the context of addition and subtraction of integers, the opposite of a number represents the additive inverse, which when combined with the original number, results in a sum of zero.

5 Must Know Facts For Your Next Test

1. The opposite of a positive integer is a negative integer, and the opposite of a negative integer is a positive integer.
2. Adding the opposite of a number to the original number always results in a sum of zero, as the opposite cancels out the original value.
3. Subtracting a number is equivalent to adding its opposite, as subtraction can be rewritten as adding the additive inverse.
4. The opposite of the opposite of a number is the original number, as the two opposite values cancel each other out.
5. Understanding the concept of opposites is crucial for accurately adding and subtracting integers, as it allows for efficient mental calculations and problem-solving.

Review Questions

• Explain how the concept of opposite relates to adding and subtracting integers.
  • The concept of opposite is fundamental to adding and subtracting integers. When adding integers, the opposite of a number can be used to cancel out the original value, resulting in a sum of zero. For example, adding a positive integer and its opposite (a negative integer) will always result in zero. Similarly, subtracting a number can be rewritten as adding the opposite (or additive inverse) of that number, which simplifies the calculation. Understanding the relationship between a number and its opposite is crucial for efficiently performing operations with integers.
• Describe how the properties of opposites can be used to solve problems involving integer operations.
  • The properties of opposites can be leveraged to solve problems involving the addition and subtraction of integers. For instance, when adding a positive and a negative integer, one can recognize that the two values are opposites and cancel each other out, resulting in a sum equal to the absolute value of the difference between the two numbers. Similarly, when subtracting integers, the problem can be rewritten as adding the opposite (or additive inverse) of the subtrahend, simplifying the calculation. Understanding the concept of opposites allows students to apply efficient mental strategies and shortcuts when working with integer operations.
• Evaluate how a deep understanding of the opposite concept can impact a student's overall proficiency in performing integer arithmetic.
  • A comprehensive understanding of the opposite concept is essential for developing proficiency in performing integer arithmetic. By recognizing the relationship between a number and its opposite, students can more easily navigate addition and subtraction of integers, often without the need for cumbersome algorithms or step-by-step procedures. This deeper conceptual understanding allows students to efficiently solve problems, make connections between different operations, and develop flexible problem-solving skills. Mastering the opposite concept empowers students to approach integer arithmetic with confidence and adaptability, ultimately leading to greater success in their studies and future mathematical endeavors.

Related terms

Additive Inverse: The additive inverse of a number is the value that, when added to the original number, results in a sum of zero.

Negative Number: A negative number is a number less than zero on the number line, representing a value in the opposite direction from positive numbers.

Absolute Value: The absolute value of a number is the distance of that number from zero on the number line, regardless of its sign (positive or negative).