Problem 60 - Prealgebra

Question

Give the opposite of each of the following numbers. $$-200$$

Step-by-Step Solution

Answer

The opposite of \(-200\) is \(200\).

Step 1: Understand the Concept of Opposite Numbers

The opposite (or additive inverse) of a number is what you would add to it to get zero. For any number \(a\), its opposite is \(-a\). It changes the sign of the number: if the number is positive, its opposite is negative, and vice versa.

Step 2: Identify the Given Number

The number provided is \(-200\). This means it is a negative number because of the negative sign in front of it.

Step 3: Find the Opposite

Since the number given is negative \(-200\), its opposite will be the positive version of that number. To find the opposite, we negate the number, i.e., multiply it by \(-1\). So, the opposite of \(-200\) is \(-(-200) = 200\).

Key Concepts

Opposite Numbers

Opposite numbers are numbers that are the same distance away from zero, but in different directions on the number line. Imagine a number line with zero in the center; if you move to the right from zero, you have positive numbers, and moving to the left leads to negative numbers. When we talk about opposite numbers, we're essentially talking about numbers that cancel each other out.

For instance, the opposite of 5 is -5 and vice versa. If you add them together, you get zero \(5 + (-5) = 0\). Finding opposites is straightforward since it's just about flipping the sign, from positive to negative or from negative to positive. Therefore, in practice, finding the opposite of a number like -200 is as easy as changing it to 200.

Additive Inverse

The concept of an additive inverse is closely related to opposite numbers. The additive inverse of a number is the number that, when added to the original number, yields zero. It essentially 'negates' or 'neutralizes' that number.

For any integer \(a\), its additive inverse is \(-a\). This relationship is a key concept in integer operations. If you have -200, its additive inverse would be 200. The operation, \(-200 + 200 = 0\), demonstrates that the numbers successfully cancel each other out to zero.

This concept is essential in solving equations that require balancing, as it helps simplify expressions by eliminating terms.

Positive and Negative Numbers

Positive and negative numbers are foundational to understanding the number system. Positive numbers are greater than zero and located to the right of zero on a number line. Negative numbers, on the other hand, are less than zero and found to the left of zero.

These numbers have opposite properties; positive numbers represent quantities larger than or increasing from zero, while negative numbers represent quantities smaller than or decreasing from zero.

• Positive numbers are typically written with no sign, such as 20.
• Negative numbers are distinguished with a minus sign, such as -20.

Together, positive and negative numbers allow us to describe changes, elevations, debts, and many real-life situations where direction or deficiency is an essential aspect.