Problem 32 - Precalculus Mathematics for Calculus

Question

Place the correct symbol \((<,>, \text { or }=)\) in the space. \(\begin{array}{llllll}\text { (a) } \frac{2}{3} & 0.67 & \text { (b) } \frac{2}{3} & -0.67 & \text { (c) }|0.67| & |-0.67|\end{array}\)

Step-by-Step Solution

Answer

(a) <, (b) >, (c) =

Step 1: Convert the Fraction to Decimal

Convert \(\frac{2}{3}\) into a decimal. Divide 2 by 3 to get approximately 0.6666... (a repeating decimal).

Step 2: Compare Decimal Values for (a)

Now compare 0.6666... to 0.67. Since 0.6666... is slightly less than 0.67, we can place \(<\) between them.

Step 3: Compare Fraction and Negative Decimal for (b)

Compare \(\frac{2}{3} (approximately 0.6666...)\) with -0.67. Since any positive number is greater than a negative number, place \(>\) between them.

Step 4: Simplify Absolute Value Expressions for (c)

Calculate \(|0.67|\) which is 0.67, and \(|-0.67|\) which is also 0.67. Since they are equal, place \(=\) between them.

Key Concepts

Fractions to Decimals

Fractions are a way to express numbers as parts of a whole. But sometimes it’s useful to convert fractions into decimals, especially when comparing them to other numbers.
To turn a fraction into a decimal, you simply divide the numerator (the top number) by the denominator (the bottom number). For example, to convert \(\frac{2}{3}\) into a decimal, divide 2 by 3.
The result is approximately 0.6666… This decimal repeats indefinitely and is called a repeating decimal.

• Example: \(\frac{1}{2} = 0.5\) (terminating decimal)
• Example: \(\frac{1}{3} = 0.333...\) (repeating decimal)

Understanding these conversions is crucial for mathematical operations where decimals are more convenient to handle than fractions. Having this skill makes it easier to compare, add, subtract, and visualize numbers in various contexts.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It’s always non-negative. This concept helps simplify expressions and solve equations more easily.
To find the absolute value, simply remove any negative sign in front of a number. For instance, both \(|0.67|\) and \(|-0.67|\) equal 0.67. The absolute value disregards any directionality indicated by the sign.

• Key Fact: \(|a| = a\) if \(a\) is positive or zero
• Key Fact: \(|a| = -a\) if \(a\) is negative

This principle is particularly useful when dealing with numbers in equations or inequalities, as it allows for standardized comparison without worrying about signs.

Repeating Decimals

Repeating decimals are numbers that have digits that repeat in a specific pattern after the decimal point. This is a common occurrence when converting fractions into decimals.
For example, when you convert \(\frac{2}{3}\), you get 0.6666..., where '6' is the repeating digit. Such decimals can be written in a shorter form by placing a bar over the repeating digits: \(0.\overline{6}\).

• Recognizing: Look for repeating patterns in decimals.
• Notation: Use a bar to indicate repetition (e.g., \(0.\overline{3}\) for \(0.333...\)).

Understanding repeating decimals is essential for accurately performing arithmetic operations and comparisons. It allows one to recognize underlying patterns and ensure precision in calculations and measurements.