The expression d(x,y) = |x - y| defines the distance between two points x and y on the real number line using absolute value. This formula reflects how far apart the two numbers are, regardless of their order, since the absolute value ensures that the result is always non-negative. Understanding this concept is crucial for grasping more complex ideas related to distance, metric spaces, and real analysis.
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The expression d(x,y) is symmetric, meaning that d(x,y) = d(y,x) for any real numbers x and y.
The distance function d satisfies the triangle inequality, which states that d(x,z) ≤ d(x,y) + d(y,z) for any points x, y, and z.
d(x,y) = 0 if and only if x = y, indicating that two points are the same if their distance is zero.
This distance formula applies not only to real numbers but also serves as a foundational concept for defining distances in more complex mathematical spaces.
Graphically, d(x,y) represents the length of the line segment connecting points x and y on the number line.
Review Questions
How does the distance formula d(x,y) = |x - y| illustrate the properties of absolute value in terms of symmetry?
The distance formula d(x,y) = |x - y| demonstrates symmetry because it shows that the distance between two points is independent of their order. Specifically, if you swap x and y, you get d(y,x) = |y - x|, which equals |x - y|. This means that it doesn't matter which point you consider first; the measured distance remains the same.
Discuss how the triangle inequality relates to the distance function d(x,y) = |x - y| and why it is significant in mathematical analysis.
The triangle inequality is a key property of the distance function d(x,y) = |x - y|. It states that for any three points x, y, and z, the relationship d(x,z) ≤ d(x,y) + d(y,z) holds true. This property is significant because it reinforces the idea that direct paths are always shorter or equal to any indirect routes connecting three points. This principle underlies many aspects of mathematical analysis and helps to establish the structure of metric spaces.
Evaluate the implications of defining a metric using d(x,y) = |x - y| in terms of establishing a framework for understanding mathematical continuity.
Defining a metric with d(x,y) = |x - y| lays the groundwork for understanding continuity in a mathematical context. Continuity can be assessed by examining how small changes in input (x values) lead to small changes in output (function values), which is directly linked to this distance metric. The formal definition of continuity states that a function f is continuous at point c if for every ε > 0, there exists a δ > 0 such that when d(x,c) < δ, then d(f(x),f(c)) < ε. This framework is vital for analyzing functions and behaviors within calculus and real analysis.
The absolute value of a number is its distance from zero on the real number line, disregarding its sign.
Metric Space: A set along with a distance function that defines how distances between points are measured, providing a framework for analysis in mathematics.
Real Number Line: A one-dimensional continuum of real numbers, typically represented as a straight line where each point corresponds to a unique real number.