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Chapter 1: Problem 54

Find the distance between the given numbers on a number line. $$ 0 \text { and }-33 $$

Short Answer

Expert verified
The distance is 33 units.

Step by step solution

01

Identify the Given Numbers

We need to calculate the distance between two numbers on a number line: 0 and -33.
02

Understand the Concept of Distance on a Number Line

The distance between any two numbers on a number line is the absolute value of their difference. This means we need to find \(|0 - (-33)| \).
03

Calculate the Difference

Subtract the smaller number from the larger one: \(0 - (-33) = 0 + 33 = 33\).
04

Apply the Absolute Value

The absolute value of a number is its non-negative value. Here, the absolute value of 33 is simply 33, as it is already positive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
Understanding the concept of absolute value is crucial in mathematics, especially when dealing with distances on a number line. The absolute value of a number is like asking, "How far are you from zero?" It doesn't matter if you're on the positive or negative side; it's all about distance.
  • If the number is already positive or zero, the absolute value is the same as the number itself. For example, the absolute value of 5 is simply 5.
  • If the number is negative, you drop the negative sign to find the absolute value. For instance, the absolute value of -5 is 5.
When you look at our example of finding the distance between 0 and -33, you're really asking for \( |0 - (-33)| \), or how far -33 is from 0 on the number line. The absolute value guarantees this distance is always a non-negative number.
Negative Numbers
Negative numbers are an integral part of the number system. They are numbers less than zero and are represented with a minus sign (e.g., -33). On a number line, negative numbers are found to the left of zero.
Let's break it down further:
  • Negative numbers are used to represent values lower than zero, such as debts, temperatures below freezing, or elevations below sea level.
  • When calculating, treating negative numbers correctly is important, particularly when performing operations like addition and subtraction.
In the context of finding distance, seeing -33 simply indicates a leftward movement from zero. However, distance itself can never be negative. That's why we use absolute values to ensure our measurement is non-negative.
Mathematical Operations
Mathematical operations like addition and subtraction can be straightforward, but when negative numbers come into play, things can get a bit tricky. Let's simplify the operations involved in our example.
When we are tasked with finding the distance between 0 and -33:
  • First, deal with subtraction: When subtracting a negative number, you actually are adding its positive counterpart. For example, \(0 - (-33)\) becomes \(0 + 33\).
  • Next, handle addition: The sum in this case results in a positive number, reflecting the distance as an absolute value.
As you can see, understanding how to manipulate mathematical signs correctly is vital. Knowing when to add or subtract can change an equation鈥檚 result from confusing to completely clear, making seemingly difficult problems much easier to solve.

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