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Understanding Real Numbers: Rational and Irrational Explained

The set of real numbers encompasses all rational and irrational numbers. Rational numbers are expressed as fractions, such as a/b (where a and b are integers, and b ≠ 0). Examples include numbers like -6.8 and 0.3333. In contrast, irrational numbers cannot be expressed as simple fractions or repeating decimals, like √2 (approximately 1.414) and π (3.14159). This guide also covers whole numbers, natural numbers, and integers, clarifying their roles within the realm of real numbers and offering insights into classifying various numbers based on their types.

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Understanding Real Numbers: Rational and Irrational Explained

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  1. The Real Number System

  2. Real Numbers • The set of all rational and the set of all irrational numbers together make up the set of real numbers. • Any and all kinds of numbers fall under real numbers.

  3. Rational Numbers Rational numbers are numbers that can be written as fractions. That is, the form a/b where a and b are both integers and b ≠ 0.

  4. -6 8 2/5 .05 -2.6 5.3333333 -8.12121212… √16 Examples of Rational Numbers

  5. Irrational Numbers • Irrational Numbers – numbers that are not repeating or terminating decimals. • Examples: • .01001000100001… • √2 = 1.414213562… • 3.14159…

  6. Whole Numbers, Natural Numbers, and Integers • Whole Numbers include the following: • 0,1,2,3,4,5,6,7,8,9,10,….. • Natural Numbers include the following: • 1,2,3,4,5,6,7,8,9,10,….. Does not include 0. • Integers include the following: • …-3,-2,-1,0,1,2,3,…

  7. Classifying Real Numbers • Directions: Classify the following numbers as natural, whole, integer, rational, and/or irrational.

  8. 8 • This number is a natural number, a whole number, an integer, and a rational number. • 0.33333 • This repeating decimal is a rational number because it is equivalent to 1/3. • √17 • √17 = 4.123105… It is not the square root of a perfect square so it is irrational.

  9. -28/2 • Since -28/4 = -14, this number is an integer and a rational number. • -√121 • Since -√121 = -11, this number is an integer and a rational number.

  10. Solving Equations • a2 = 49 • To undo the square, take the square root of both sides. Then, you have this. • √a2 = √49 • a = √49 or a = -√49 • a = 7 or a = -7 • Hence, the solutions are 7 and -7.

  11. d2 = 55 • Take the square root of both sides. • √d2 = √55 • d = √55 or d = - √55 • d = 7.41 or d = - 7.41 • Hence, the solutions are 7.41 and -7.41.

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