Problem 103 - Prealgebra

Question

Add. $$-\frac{3}{4}+\frac{3}{4}$$

Step-by-Step Solution

Answer

The sum of \\(-\frac{3}{4}\\) and \\(+\frac{3}{4}\\) is zero.

Step 1: Understand the Concept of Adding Opposites

When you add a positive number and its negative counterpart (opposite), the sum is always zero. In this exercise, you are adding \(-\frac{3}{4}\) and \(+\frac{3}{4}\).

Step 2: Recognize the Numbers Involved

Identify that there are two fractions: \(-\frac{3}{4}\) is the negative fraction and \(+\frac{3}{4}\) is the positive fraction.

Step 3: Add the Fractions

When adding \(-\frac{3}{4}\) and \(+\frac{3}{4}\), observe that the negative and positive fractions cancel each other out, resulting in a sum of zero. Mathematically, \(-\frac{3}{4} + \frac{3}{4} = 0\).

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are often used to represent quantities like debt or temperatures below freezing. In mathematics, they are usually denoted with a minus sign. For example, in the fraction \(-\frac{3}{4}\), the minus sign indicates that the fraction is negative.
Negative numbers are essential in various mathematical operations and concepts, including:

• Representing opposite quantities, such as a loss in finance or a drop in temperature.
• Balancing equations by using their properties to cancel out positive numbers.
• Being integral in calculating differences, where the result might be less than zero.

When working with fractions, knowing the sign is crucial as it directly affects the result of any operation. For instance, when adding fractions, a negative fraction can nullify a positive fraction of equal magnitude.

Positive Numbers

Positive numbers are values greater than zero. They are the numbers we often use in daily life to count objects, measure lengths, or assess gains. In arithmetic, they are written without any sign or with a plus sign. For example, the fraction \(+\frac{3}{4}\) is a positive number.
Understanding positive numbers helps in:

• Adding quantities to increase value, such as adding more units or increasing temperature.
• Counting natural numbers, which are all positive.
• Performing operations where the result naturally increases, like savings or income.

In the mathematical notation, positive numbers readily combine with negative numbers, where they can offset each other. This is particularly useful in algebra where sums of opposites result in zero.

Zero

Zero is a unique and significant number in mathematics. It is neither positive nor negative, and it plays a vital role as a neutral element in arithmetic operations. Zero is the number that represents no quantity or null value.
Some key aspects of zero include:

• Being the result when equal magnitude positive and negative numbers are added, such as in \(-\frac{3}{4} + \frac{3}{4} = 0\).
• Acting as an identity element in addition, meaning any number added to zero remains unchanged.
• Being the baseline or reference point in many measurements, like GPS coordinates or temperature scales.

In arithmetic and algebra, zero helps maintain balance and neutrality in equations. Understanding zero's role helps when solving equations and in various real-world applications.