Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Find the distance between \(a\) and \(b\).\(a=9.34, b=-5.65\)

Step-by-Step Solution

Answer

The distance between \(a\) and \(b\) is 15.

Step 1: Identify the given numbers

The numbers to be used in this exercise are \(a = 9.34\) and \(b = -5.65\).

Step 2: Subtract the numbers

Subtract one number from the other. You can take them in any order, it won't affect the final result. Let's subtract \(b\) from \(a\): \(a - b = 9.34 - (-5.65)\).

Step 3: Simplify the expression

Solve the expression in step 2: \(a - b = 9.34 + 5.65 = 15\).

Key Concepts

Absolute Value

The term absolute value refers to the non-negative value of a number on a standard number line, regardless of its sign. It can be visually understood as the distance of a number from zero on this number line. For example, the absolute value of both 3 and -3 is 3, because both are three units away from zero.

When calculating the distance between two points, like in our exercise with values 9.34 and -5.65, the absolute value becomes particularly important. After subtracting one number from the other, we apply the absolute value to ensure the distance is a positive number, since distance cannot be negative. So, even if the subtraction yields a negative number, taking the absolute value will provide us with the actual distance between the points.

Number Line

A number line is an essential concept in understanding many mathematical topics, including the distance between two points. It is a straight line with numbers placed at equal intervals along its length. Positive numbers are usually to the right of zero and negative numbers to the left.

By picturing the values of 9.34 and -5.65 on a number line, we see that they lie to the right and left of zero, respectively. To find the distance between them, we imagine walking along the number line from one point to the other. This mental image supports the subtraction step in our exercise, as we are effectively measuring how far apart the two points lie on the number line.

Subtraction of Real Numbers

Subtraction is one of the four fundamental arithmetic operations and is the process of determining the difference between two numbers. When working with real numbers, which include both positive and negative numbers, we follow specific rules to find this difference. For the exercise at hand, where we have 9.34 (a positive number) and -5.65 (a negative number), the subtraction operation is not simply taking away one quantity from another; it is the addition of a positive number and the inverse of a negative number.

Thus, subtracting -5.65 from 9.34 becomes the same as adding the opposite of -5.65, which is +5.65, to 9.34. The subtraction of real numbers helps us compute the literal difference between them, which corresponds to the distance on a number line. In other words, when we subtract -5.65 from 9.34, we are not just finding a numerical difference, but the actual space separating these two points.