(iii) A number smaller than 5

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Points to Remember:

  • The question requires identifying numbers smaller than 5.
  • The approach is factual, focusing on the mathematical definition of “smaller than.”

Introduction:

The question asks us to identify numbers that are less than 5. This is a fundamental concept in mathematics dealing with numerical comparisons and ordering. The set of numbers includes both positive and negative numbers, integers and fractions, and even irrational numbers. Understanding the concept of “smaller than” is crucial for various mathematical operations and applications.

Body:

Defining “Smaller Than”: A number x is considered “smaller than” 5 if x < 5. This inequality means that x lies to the left of 5 on a number line.

Examples of Numbers Smaller Than 5:

  • Integers: 4, 3, 2, 1, 0, -1, -2, -3, and so on. There are infinitely many negative integers smaller than 5.
  • Fractions: 4.9, 4.5, 3.7, 2.1, 0.5, -1.2, and so on. There are infinitely many fractions smaller than 5.
  • Decimal Numbers: 4.999, 3.14159, 0.001, -2.5, etc. Again, infinitely many decimal numbers are smaller than 5.
  • Irrational Numbers: π (approximately 3.14159) is smaller than 5, as is √2 (approximately 1.414). There are infinitely many irrational numbers smaller than 5.

Illustrative Number Line:

A number line can visually represent this concept:

-3 -2 -1 0 1 2 3 4 5 6 7
|-----------------|
Numbers smaller than 5

The Infinite Nature of the Solution Set: It’s crucial to understand that there is an infinite number of values smaller than 5. We can always find a number that is smaller than any given number smaller than 5.

Conclusion:

In summary, the question’s simple premise – identifying numbers smaller than 5 – reveals a fundamental mathematical concept with an infinite number of solutions. The set includes all integers, fractions, decimals, and irrational numbers less than 5. Understanding this concept is foundational for more complex mathematical operations and problem-solving. Further exploration into number systems and inequalities will build upon this basic understanding, promoting a deeper appreciation for mathematical principles.

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