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Representation of Real Numbers on a Line

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Learning objective
Plot real numbers on a number line and understand intervals and distances.

Understanding Representation of Real Numbers on a Line

The representation of real numbers on a line is a fundamental concept in algebra and mathematics. It provides a visual and intuitive way to understand numbers, their relationships, and operations involving them. This section covers the number line, plotting of real numbers, distances, intervals, and related set theory concepts essential for entrance exams like NDA.

1. Number Line Basics

A number line is a straight horizontal line where every point corresponds to a real number. The line extends infinitely in both directions. The center point is called the origin and is labeled as 0.

  • Numbers to the right of zero are positive and increase in value.
  • Numbers to the left of zero are negative and decrease in value.
  • The number line is continuous, meaning between any two numbers, there are infinitely many numbers.
-5 -4 -3 -2 -1 0 1 2 3

Figure: Number line showing integers from -5 to 3 with zero as the origin.

2. Plotting Real Numbers

Real numbers include integers, fractions, decimals, and irrational numbers. All these can be represented on the number line.

  • Integers: Whole numbers like -3, 0, 4 are plotted at equally spaced points.
  • Fractions and decimals: Numbers like \(\frac{1}{2}\), 0.75 lie between integers and can be precisely located.
  • Irrational numbers: Numbers like \(\sqrt{2}\), \(\pi\) cannot be expressed exactly as fractions but can be approximated and plotted.

Example: To plot \(\frac{3}{2}\) on the number line, locate the midpoint between 1 and 2 and mark it.

3. Distance Between Two Points

The distance between two points \(a\) and \(b\) on the number line is given by the absolute value of their difference:

\[\text{Distance} = |a - b|\]

This distance is always non-negative and represents the length of the segment between the two points.

4. Intervals on the Number Line

Intervals represent sets of real numbers lying between two endpoints. There are three main types:

  • Closed interval \([a, b]\): Includes endpoints \(a\) and \(b\).
    \( [a, b] = {x \in \mathbb{R} : a \leq x \leq b } \)
  • Open interval \((a, b)\): Excludes endpoints.
    \( (a, b) = {x \in \mathbb{R} : a < x < b } \)
  • Half-open intervals \([a, b)\) or \((a, b]\): Includes one endpoint but not the other.

Intervals can be represented graphically on the number line by shading the region between \(a\) and \(b\) and marking endpoints accordingly (solid dot for included, hollow for excluded).

5. Sets and Subsets on the Number Line

Sets of numbers can be represented on the number line. Important concepts include:

  • Null set \(\emptyset\): The empty set contains no elements and is a subset of every set.
  • Universal set \(U\): The set containing all elements under consideration, often the entire number line or a specified domain.
  • Power set \(\mathcal{P}(S)\): The set of all subsets of \(S\). If \(S\) has \(n\) elements, then \(\mathcal{P}(S)\) has \(2^n\) elements.

Set operations such as union \(\cup\), intersection \(\cap\), and difference \(-\) correspond to combining, overlapping, or subtracting regions on the number line.

6. Applications in Solving Inequalities

Inequalities like \(x > 3\) or \(x \leq -1\) can be represented as intervals on the number line. This visual representation helps in understanding solution sets and their unions or intersections.

For example, the solution to \(x > 2\) is the open interval \((2, \infty)\), shown as a ray starting just after 2 and extending to the right indefinitely.

7. Connection to Entrance Exam Questions

Many entrance exam questions test understanding of these concepts through problems involving:

  • Plotting points and intervals on the number line.
  • Calculating distances between points.
  • Working with sets and subsets, including power sets.
  • Solving inequalities and representing solutions graphically.

Mastering these concepts is essential for scoring well in NDA and other competitive exams.

Worked Examples

Example 1: Plotting a Fraction on the Number Line ★★

Problem: Plot the number \(\frac{5}{4}\) on the number line.

Solution:

  1. Locate integers 1 and 2 on the number line.
  2. Since \(\frac{5}{4} = 1.25\), it lies between 1 and 2.
  3. Divide the segment between 1 and 2 into 4 equal parts.
  4. Count 1 whole part plus 1 additional quarter part from 1 to reach \(\frac{5}{4}\).
  5. Mark the point and label it as \(\frac{5}{4}\).

Example 2: Distance Between Two Points ★★

Problem: Find the distance between points \(-3\) and \(4\) on the number line.

Solution:

Distance \(= |-3 - 4| = |-7| = 7\).

So, the distance between the points is 7 units.

Example 3: Representing an Interval ★★★

Problem: Represent the interval \([ -2, 3 )\) on the number line.

Solution:

  1. Mark the points \(-2\) and \(3\) on the number line.
  2. Use a solid dot at \(-2\) to indicate inclusion.
  3. Use a hollow dot at \(3\) to indicate exclusion.
  4. Shade the region between \(-2\) and \(3\).

Example 4: Number of Subsets of a Set ★★★

Problem: If a set \(S\) has 5 elements, find the number of subsets of \(S\).

Solution:

The number of subsets of a set with \(n\) elements is \(2^n\).

So, number of subsets \(= 2^5 = 32\).

Example 5: Solving and Representing Inequality ★★★

Problem: Solve the inequality \(x - 3 < 5\) and represent the solution on the number line.

Solution:

  1. Add 3 to both sides: \(x < 8\).
  2. The solution is all real numbers less than 8.
  3. On the number line, shade all points to the left of 8.
  4. Use a hollow dot at 8 since \(x\) is strictly less than 8.

Formula Bank

  • Distance between two points \(a\) and \(b\): \( |a - b| \)
  • Number of subsets of a set with \(n\) elements: \( 2^n \)
  • Closed interval: \( [a, b] = {x : a \leq x \leq b} \)
  • Open interval: \( (a, b) = {x : a < x < b} \)
  • Half-open intervals: \( [a, b) = {x : a \leq x < b} \), \( (a, b] = {x : a < x \leq b} \)
  • Union of sets: \( A \cup B = {x : x \in A \text{ or } x \in B} \)
  • Intersection of sets: \( A \cap B = {x : x \in A \text{ and } x \in B} \)
  • Difference of sets: \( A - B = {x : x \in A \text{ and } x \notin B} \)
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