Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Tell whether each of the following is true or false. (a) \(-3<-7\) (b) \(-1>-17\) (c) \(-3<-\frac{22}{7}\)

Step-by-Step Solution

Answer

(a) False, (b) True, (c) False.

Step 1: Analyzing Statement (a)

Examine whether \(-3 < -7\). Compare the two numbers on the number line. Remember that on a number line, numbers to the right are greater. Since \(-3\) is further to the right than \(-7\), \(-3 > -7\). Therefore, \(-3 < -7\) is false.

Step 2: Analyzing Statement (b)

Examine whether \(-1 > -17\). Compare \(-1\) and \(-17\) on the number line. Since \(-1\) is to the right of \(-17\), it is indeed greater. Hence, \(-1 > -17\) is true.

Step 3: Analyzing Statement (c)

First, approximate the value of \(-\frac{22}{7}\), which represents a commonly used approximation of \(-\pi\) or approximately \(-3.14\). Now compare \(-3\) with \(-\frac{22}{7}\). Since \(-3\) is greater than \(-3.14\), \(-3 < -\frac{22}{7}\) is false.

Key Concepts

Number Line

The number line is an essential concept in math. It helps us visualize numbers and their relationships. Imagine a horizontal line that extends indefinitely in both directions. Each point on this line corresponds to a real number.

• Numbers to the right are larger.
• Numbers to the left are smaller.

This representation makes it easy to compare numbers. For instance, you can see how a number like even negative numbers interact on the line. When looking at a pair such as -3 and -7, viewing them on a line reveals that -3 is to the right of -7, indicating -3 is actually larger .

Negative Numbers

Negative numbers are a type of number less than zero. They are placed on the left side of a number line. Negative numbers often represent things like a loss or debt. For example, -7 indicates a more negative value than -3.

• As you move left on a number line, values decrease.
• -1 is greater than -17 because -1 lies to the right of -17 on the number line.

These concepts might seem counterintuitive at first. However, consistently using a number line aids in understanding negative values.

Comparison of Fractions

Comparing fractions involves understanding their exact values or approximate placements between whole numbers. When dealing with a negative fraction like -\(\frac{22}{7}\), it's helpful to approximate the value. This fraction is a common approximation for -π (pi), about -3.14.

• Negative fractions add layers to comparison tasks.
• Compare using approximations when exact value calculations are complex.

Visualizing a negative fraction on a number line is useful. You'll see negative 3 sits to the right of -\(\frac{22}{7}\), making -3 greater.

Approximation of Pi

Pi (π) is a crucial mathematical constant representing the ratio of a circle's circumference to its diameter. It's irrational, meaning it can't be precisely written as a fraction.A common approximation is \(\frac{22}{7}\), which equals roughly 3.14. When it comes to comparing negative approximations of pi like -\(\frac{22}{7}\),visualizing helps.

• Approximating pi assists in practical problem-solving.
• Negative pi values demand careful placement on number lines.

Understand the estimation rounds for crucial comparisons. This will guide you in topics that necessitate knowledge of pi, especially in complex scenarios where exactitude is less practical.