Opposite numbers are two integers that are the same distance from zero on the number line, but in opposite directions. They have the same absolute value but opposite signs, with one being positive and the other being negative.
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Opposite numbers always have a sum of zero when added together.
The product of two opposite numbers is always negative.
Opposite numbers can be used to represent quantities that move in opposite directions, such as gains and losses, or positive and negative charges.
Finding the opposite of a number is a key skill in working with integers and performing operations such as addition and subtraction.
Opposite numbers are important in understanding the concept of the number line and how integers are represented and compared.
Review Questions
Explain how opposite numbers are represented on the number line and how this relates to their properties.
Opposite numbers are represented on the number line as two numbers that are the same distance from zero, but in opposite directions. For example, the numbers +5 and -5 are opposite numbers because they are both 5 units away from zero, with +5 being to the right of zero and -5 being to the left. This symmetry around zero gives opposite numbers the property that when added together, they always sum to zero. Additionally, the product of two opposite numbers is always negative, as one number is positive and the other is negative.
Describe the relationship between opposite numbers and the concept of additive inverse.
The additive inverse of a number is the number with the same absolute value but the opposite sign. This is the definition of opposite numbers. For example, the additive inverse of 7 is -7, and the additive inverse of -3 is 3. Opposite numbers are closely linked to the idea of additive inverse because they represent numbers that, when added together, result in a sum of zero. This property of opposite numbers is fundamental to understanding operations with integers, such as addition and subtraction.
Analyze how the concept of opposite numbers is used to represent and compare quantities that move in opposite directions.
Opposite numbers are often used to represent quantities that move in opposite directions, such as gains and losses, or positive and negative charges. For example, a person's net worth could be represented as a positive number for assets and a negative number for liabilities, with the opposite numbers indicating the movement of these quantities in opposite directions. Similarly, in physics, positive and negative charges are represented using opposite numbers to indicate their directional movement. Understanding the properties of opposite numbers, such as their additive inverse relationship and the fact that their product is always negative, is crucial for interpreting and analyzing these types of situations involving quantities that move in opposite directions.