Problem 2 - Calculus with Concepts in Calculus

Question

Determine whether \(ab\). \(a=-\frac{1}{7}, b=-0.142857\)

Step-by-Step Solution

Answer

Neither \(a < b\) nor \(a > b\) is true; both are equal.

Step 1: Understanding the Problem

We are given two numbers, \(a = -\frac{1}{7}\) and \(b = -0.142857\). Our goal is to determine if \(a < b\) or \(a > b\).

Step 2: Conversion for Comparison

Compare both numbers in the same form. Convert \(a = -\frac{1}{7}\) into a decimal. Since \(\frac{1}{7} = 0.142857\) (repeating), converting it gives \(a = -0.142857\).

Step 3: Compare the Numerical Values

Since both \(a\) and \(b\) are now \(-0.142857\), compare them. Since both values are equal, neither \(a < b\) nor \(a > b\) is true.

Key Concepts

Fraction to Decimal Conversion

To compare numbers like \( a = -\frac{1}{7} \) and \( b = -0.142857 \), it's important to bring them both into the same format—either as fractions or as decimals. This makes it a lot easier to see which one is larger or if they are equal. In this case, let's focus on converting the fraction to a decimal.

- **Steps to Convert** 1. Identify the fraction: Here, we have \( \frac{1}{7} \). 2. Perform the division \( 1 \div 7 \): This means dividing 1 by 7. 3. The result of this division is a repeating decimal: \( 0.142857\ldots \).
So, \( -\frac{1}{7} \) becomes \( -0.142857 \) when converted to a decimal. This conversion allows us to directly compare \( a \) and \( b \) since they are now both in decimal form.

Negative Numbers

Understanding negative numbers is crucial when comparing values, especially in problems involving inequality like this one. Negative numbers are less than zero and are represented with a minus sign (-). When comparing two negative numbers, remember some key points:

- The closer a negative number is to zero, the greater its value. For example, \( -1 \) is greater than \( -2 \).
- If two negative numbers are the same, like \( -0.142857 \) and \( -0.142857 \), they are equal.

In our exercise, both \( a \) and \( b \) are \( -0.142857 \), meaning their values are exactly the same. So, there is no inequality establishing \( a < b \) or \( a > b \). It is important to recognize that equality among negative numbers means they are absolutely the same value on the number line.

Equality in Algebra

In algebra, determining equality isn't just about numbers seeming close; it's a precise statement about their mathematical equivalence. When we say two expressions or numbers are equal, they have exactly the same value.

In the exercise, after converting \( a = -\frac{1}{7} \) to \( -0.142857 \), both \( a \) and \( b \) turned out to be \( -0.142857 \).

- **Recognizing Equality** - Expressions like \( -\frac{1}{7} \) and its decimal form \( -0.142857 \) are different representations of the same value. - This demonstrates equality in algebra where two different-looking expressions can be essentially equal.

Understanding equality helps in identifying relationships, solving equations, and managing algebraic expressions in a variety of mathematical scenarios.