Question

Find the distance between \(a\) and \(b\).\(a=-\frac{7}{2}, b=0\)

Short answer

The distance between \(a\) and \(b\) is \(\frac{7}{2}\).

Step-by-step solution

Subtraction

Firstly, compute the difference between \(b\) and \(a\). In this case, \(b - a = 0 - (-\frac{7}{2})\). So, \(b - a = \frac{7}{2}\).

Application of Absolute Value

Then take the absolute value of the result from the first step. This is denoted as \(|b - a|\). Therefore, \(|b - a| = |\frac{7}{2}|\).

Final Calculation

Since \(\frac{7}{2}\) is positive, its absolute value remains the same. Therefore, the distance between \(a\) and \(b\) is \(\frac{7}{2}\).

Key concepts

Absolute Value

Understanding the absolute value is crucial when dealing with distances because it represents the magnitude of a number regardless of its sign. The absolute value of a number, often written as |x|, is always nonnegative. This is because absolute value indicates 'how far' a number is from zero on the number line without considering direction.

For instance, both -5 and 5 are five units away from zero, and therefore, have an absolute value of 5, denoted as |-5| = 5 and |5| = 5. This becomes extremely useful in real-world scenarios, such as when you want to find the shortest distance between two points or when you're measuring something where only the size (not the direction) matters.

Distance Formula

The essence of finding the distance between two points lies in the distance formula. This formula is rooted in the Pythagorean theorem, but in its simplest form, when used on a number line, it is basically the absolute value of the difference between two points. To calculate the distance between two points, a and b, we use the formula |a - b|.

For example, if we want to find the distance from -2 to 3 on the number line, we compute |3 - (-2)|, which simplifies to |3 + 2|, yielding a distance of 5 units. Always remember: distance is a scalar quantity—it has magnitude but no direction—hence why the absolute value is so integral to the distance calculation.

Subtracting Negative Numbers

Working with negative numbers often trips students up, especially when it comes to subtraction. The rule to remember is that subtracting a negative number is the same as adding its positive counterpart. To put it another way, 'minus a negative becomes plus a positive'.

Problem Solving Tip

If you're given b - (-a), this simplifies to b + a. Whenever you see a double negative in the problem, just think of it as an addition problem. So in the context of our distance example, |0 - (-7/2)| becomes |0 + 7/2|, which simplifies to |7/2|. Recognizing this helps to avoid common errors and makes calculations involving negative numbers much smoother.