Problem 36 - Algebra and Trigonometry

Question

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\\{-7,-0 . \overline{6}, 0, \sqrt{49}, \sqrt{50}\\}$$

Step-by-Step Solution

Answer

a. There are no natural numbers. b. The whole numbers are 0, \(\sqrt{49}\) (=7). c. The integers are -7, 0. d. The rational numbers are -7, -0.666..., 0, \(\sqrt{49}\) (=7). e. The irrational number is \(\sqrt{50}\). f. All the given numbers are real numbers.

Step 1: Identification of Natural Numbers

Natural numbers start from 1 and go up to infinity. They include all positive counting numbers. None of the numbers in the set (-7, -0.666...,0, \(\sqrt{49}\), \(\sqrt{50}\)) are natural numbers because all of them are either negative, fractional or zero.

Step 2: Identification of Whole Numbers

Whole numbers start from 0 and go up to infinity. They include all positive counting numbers along with zero. The number 0, and \(\sqrt{49}\) (which is 7) from the given set are whole numbers.

Step 3: Identification of Integers

Integers are all the natural numbers, their negatives and zero. The numbers -7 and 0 from the given set are integers.

Step 4: Identification of Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction p/q of two integers, p and q, with the denominator q not equal to zero. All integers are rational, but the converse is not true. The numbers -7, -0.666... (it's a repeating decimal so can be expressed as a fraction: -2/3), 0, and \(\sqrt{49}\) (which is exactly 7, thus can be expressed as 7/1) in the set are rational.

Step 5: Identification of Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a fraction p/q for any two integers p and q, and their decimal representation does not terminate or repeat. The number \(\sqrt{50}\) in the set is an irrational number because it could not be expressed as a fraction and it can't be square rooted exactly into a decimal.

Step 6: Identification of Real Numbers

Real numbers include all the rational and irrational numbers. They include every number which can be put on the number line. So all the numbers in the set (-7, -0.666..., 0, \(\sqrt{49}\), \(\sqrt{50}\)) are real numbers.