QuestionIdentify which numbers in the set $E=\{\sqrt{14}, \pi, \sqrt{14}+1, \pi+\frac{1}{14}\}$ are (a) Natural, (b) Integers, (c) Rational, (d) Irrational, (e) Real.

Studdy Solution

STEP 1

Assumptions1. The set contains the following numbers $\sqrt{14}$ , $\pi$ , $\sqrt{14}+1$ , $\pi+\frac{1}{14}$ . We need to categorize these numbers into natural numbers, integers, rational numbers, irrational numbers, and real numbers.
3. Natural numbers are positive integers starting from1.
4. Integers include all whole numbers, both positive and negative, including zero.
5. Rational numbers can be expressed as a fraction of two integers.
6. Irrational numbers cannot be expressed as a fraction of two integers.
7. Real numbers include all rational and irrational numbers.

STEP 2

First, let's identify the natural numbers in the set. Natural numbers are positive integers starting from1. Looking at the set, we see that there are no natural numbers because none of the numbers in the set are positive integers.
$Natural\, numbers = \emptyset$

STEP 3

Next, let's identify the integers in the set. Integers include all whole numbers, both positive and negative, including zero. Looking at the set, we see that there are no integers because none of the numbers in the set are whole numbers.
$Integers = \emptyset$

STEP 4

Now, let's identify the rational numbers in the set. Rational numbers can be expressed as a fraction of two integers. Looking at the set, we see that there are no rational numbers because none of the numbers in the set can be expressed as a fraction of two integers.
$Rational\, numbers = \emptyset$

STEP 5

Next, let's identify the irrational numbers in the set. Irrational numbers cannot be expressed as a fraction of two integers. Looking at the set, we see that all the numbers in the set are irrational numbers because they cannot be expressed as a fraction of two integers.
$rrational\, numbers = \left\{\sqrt{14}, \pi, \sqrt{14}+1, \pi+\frac{1}{14}\right\}$

STEP 6

Finally, let's identify the real numbers in the set. Real numbers include all rational and irrational numbers. Looking at the set, we see that all the numbers in the set are real numbers because they include all the rational and irrational numbers.
$Real\, numbers = \left\{\sqrt{14}, \pi, \sqrt{14}+1, \pi+\frac{1}{14}\right\}$

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