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Chapter 1: Problem 121

Explain the different uses and meanings of the "-" sign in the expression \(3-(-2)^{-1}\)

Short Answer

Expert verified
The different meanings of the minus sign are subtraction and indicating a negative number; the final simplified value is 3.5.

Step by step solution

01

Identify and Define Components

First, identify the parts of the expression. The expression given is \(3-(-2)^{-1}\). Note that there are three components: 3, \(-(-2)^{-1}\), and the operation minus (-) between them.
02

Meaning of the Minus Sign

The minus sign (-) has different meanings in this expression. The first minus sign between 3 and \((-2)^{-1}\) denotes subtraction. The second minus sign in front of 2 indicates that 2 is negative.
03

Evaluate the Expression Inside the Parentheses

Before subtracting, simplify \((-2)^{-1}\). Here, the minus sign with the exponent (-1) indicates the reciprocal of \(-2\). The reciprocal of \(-2\) is \(-1/2\).
04

Simplify Further

Now substitute back into the expression: \(3 - (-1/2)\). The expression inside the parentheses simplifies to \(-1/2\).
05

Apply Subtraction

Subtracting a negative is the same as adding its positive. So, \(3 - (-1/2)\) becomes \(3 + 1/2\).
06

Final Simplification

Finally, perform the addition: \(3 + 1/2 = 3.5\). Thus, the simplified value of the expression is 3.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subtraction and the Minus Sign
In math, the minus sign (-) is commonly used to indicate subtraction. For instance, in the expression 3 - 2, the minus sign tells us to take away 2 from 3, leaving us with 1. In the given problem, we see this when we have 3 - (-1/2). Initially, this step shows us the operation of subtracting a negative number, which requires a deeper understanding to solve correctly.
Understanding Negative Numbers
Negative numbers are numbers less than zero and are denoted by a minus sign in front, such as -1, -2, and so on. In the context of our problem, (-2)^{-1}, the -2 is a negative number. It's important to remember that a negative sign tells us direction on a number line, usually indicating something below zero. When dealing with a negative in subtraction, you鈥檙e often adding the positive equivalent during the process.
Learning the Reciprocal
The reciprocal of a number is another number which, when multiplied with the original, yields 1. For instance, the reciprocal of 2 is 1/2 because 2 * (1/2) equals 1. When dealing with negative numbers, such as (-2)^{-1}, you must recognize that the reciprocal of -2 is -1/2. This means that (-2)^{-1} equals -1/2, and any operations involving this must accurately reflect this property.
Simplification of Expressions
Simplification involves reducing an expression to its simplest form. In the problem, we move from 3 - (-2)^{-1} to 3 - (-1/2). Firstly, identifying the reciprocal correctly is crucial. Then, knowing that subtracting a negative number translates into adding the positive counterpart simplifies our task further. Hence, 3 - (-1/2) converts to 3 + 1/2 seamlessly.
Addition of Fractions
Adding fractions involves making sure that you combine the numerators correctly after possibly converting them to a common denominator. In our case 3 + 1/2 doesn't require a common denominator since 3 is a whole number. Converting 3 to a fraction (6/2), we can readily add: 6/2 + 1/2 equals 7/2. Thus, when converted back, we get 3.5, our resultant value, making it easy to add fractions with consistent simplification.

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