Problem 36 - College Algebra Essentials

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. Natural numbers: \( \sqrt{49} \). b. Whole numbers: 0, \( \sqrt{49} \). c. Integers: -7, 0, \( \sqrt{49} \). d. Rational numbers: -7, -0.\(\overline{6}\), 0, \( \sqrt{49} \). e. Irrational Numbers: \( \sqrt{50} \). f. Real numbers: -7, -0.\(\overline{6}\), 0, \( \sqrt{49} \), \( \sqrt{50} \).

Step 1: Identify Natural Numbers

Natural numbers are positive integers that start from 1. In the set, the only number that fits this definition is \( \sqrt{49} \), which equals 7.

Step 2: Identify Whole Numbers

Whole numbers are natural numbers including zero. In the given set, the numbers fitting this category are 0 and \( \sqrt{49} \) (which equals to 7).

Step 3: Identify Integers

Integers include all whole numbers and their negatives. In the set, the numbers that are integers are -7, 0 and \( \sqrt{49} \), which equals 7.

Step 4: Identify Rational Numbers

Rational numbers can be expressed as fractions (or ratios) of integers. In the given set -7, -0.\(\overline{6}\), 0, and \( \sqrt{49} \) (which equals 7) are rational numbers.

Step 5: Identify Irrational Numbers

Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal expansions. In the given set, \( \sqrt{50} \) is an irrational number.

Step 6: Identify Real Numbers

Real numbers include all rational and irrational numbers. Thus, all the given numbers fall under this category -7, -0.\(\overline{6}\), 0, \( \sqrt{49} \), \( \sqrt{50} \).

Key Concepts

Understanding Natural Numbers

Natural numbers are the simplest and most familiar type of numbers. They include all positive integers beginning from 1 and moving upwards without any limit. Simply put, when you count objects like apples or books, you are using natural numbers. Here's what's interesting about the natural number from the given set:

• \( \sqrt{49} \) is a natural number because \( \sqrt{49} = 7 \), and 7 belongs to the set of natural numbers as it is positive and a counting number.

Natural numbers do not include zero or any negative numbers. Remember, they are the numbers you learn to count first!

What Are Whole Numbers?

Whole numbers expand on natural numbers by including zero. They are the numbers you get starting from zero and then counting upwards like 1, 2, 3, and so on. The important point here is that whole numbers do not include fractions, decimals, or negative numbers.

• In the set, 0 and \( \sqrt{49} \) (which equals 7) are whole numbers. \( \sqrt{49} = 7 \) is a whole number because it's a positive integer, and zero is also included in whole numbers.

Whole numbers are useful in many everyday situations, like keeping track of quantities or scores.

Getting to Know Integers

Integers are a broader category of numbers that encompass whole numbers and their opposites—negative whole numbers. These include numbers like -3, -2, -1, 0, 1, 2, 3, and so forth.

• In the given set, the integers are -7, 0, and \( \sqrt{49} \) (which equals 7). As you can see, both positive and negative whole numbers, as well as zero, belong in this category.

Integers are easy to understand as they don't involve fractions or decimals. They're great for situations where difference and direction matter, like gains and losses, temperatures, elevations, etc.

Rational Numbers Explained

A rational number is any number that can be written as the fraction of two integers, where the denominator is not zero. If you can express it as a ratio of two integers, then it’s rational!

• For the numbers in the set: -7, -0.\( \overline{6} \), 0, and \( \sqrt{49} \) (as 7) can be written as fractions.
  • -7 is the same as \( \frac{-7}{1} \).
  • -0.\( \overline{6} \), a repeating decimal, is equivalent to \( \frac{-2}{3} \).
  • 0 can be written as \( \frac{0}{1} \), and 7 can be \( \frac{7}{1} \).

These simple fractions help clarify why these numbers are called "rational," originating from "ratio." If a number cannot be expressed as a simple fraction, it is not rational.

Discovering Irrational Numbers

Irrational numbers are a unique bunch, being numbers that cannot be exactly expressed as a fraction of two integers. They have infinite, non-repeating decimal expansions.

• From the set, \( \sqrt{50} \) is irrational. This is because there is no exact fraction that can represent \( \sqrt{50} \), and its decimal form goes on forever without a repeating pattern.

Irrational numbers are crucial in math, exemplifying numbers that don't fit neatly into traditional fraction formats, like \( \pi \) and \( e \). They often pop up in more advanced math, helping us understand concepts that go beyond simple counting.