Problem 4 - A Graphical Approach to Precalculus with Limits

Question

For each set, list all elements that belong to the (a) natural numbers, (b) whole numbers, (c) integers. (d) rational numbers, (e) irrational numbers, and (f) real numbers. $$\\{-\sqrt{49},-0.405,-0 . \overline{3}, 0.1,3,18,6 \pi, 56\\}$$

Step-by-Step Solution

Answer

Natural: 3, 18, 56. Whole: 3, 18, 56. Integers: -7, 3, 18, 56. Rational: -7, -0.405, -0.̅3, 0.1, 3, 18, 56. Irrational: 6π. Real: All elements.

Step 1: Understand Set Elements

Look closely at each element of the set \(\{-\sqrt{49},-0.405,-0 . \overline{3}, 0.1,3,18,6 \pi, 56\}\). We need to classify each element based on the different number types: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Step 2: Identify Natural Numbers

Natural numbers are the set of positive integers starting from 1. From the given set, identify these types of numbers. Here, \(3, 18,\) and \(56\) are natural numbers.

Step 3: Identify Whole Numbers

Whole numbers include all natural numbers and zero. Check which elements from the set fit this category. Again, \(3, 18,\) and \(56\) qualify as whole numbers because negative numbers and numbers with decimal parts are not included.

Step 4: Identify Integers

Integers are all positive and negative whole numbers, including zero. From the given set, \(-\sqrt{49}\) which equals \(-7\), \(3\), \(18\), and \(56\) are integers.

Step 5: Identify Rational Numbers

Rational numbers are numbers that can be expressed as the fraction of two integers. Negative and decimal numbers that terminate or repeat are rational. Thus, \(-\sqrt{49}\), \(-0.405\), \(-0 . \overline{3}\), \(0.1\), \(3\), \(18\), and \(56\) are rational numbers.

Step 6: Identify Irrational Numbers

Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimals. From our set, \(6\pi\) is irrational because the decimal form of \(\pi\) never repeats.

Step 7: Identify Real Numbers

Real numbers include all rational and irrational numbers. List all elements from the set: \(-\sqrt{49}\), \(-0.405\), \(-0 . \overline{3}\), \(0.1\), \(3\), \(18\), \(6\pi\), and \(56\).

Key Concepts

Natural Numbers

Natural numbers are the simplest set of numbers you'll encounter. They start from 1 and increase in whole steps like 1, 2, 3, and so on. These numbers do not include zero, negative numbers, fractions, or decimals. They are essentially the numbers we use for counting everyday objects. For example, if you're counting apples, you would use natural numbers because you start from one apple, not zero or negative apples. In the given set \{-\sqrt{49}, -0.405, -0 . \overline{3}, 0.1, 3, 18, 6 \pi, 56\}, the natural numbers are \(3\), \(18\), and \(56\) because these are positive without any fractional or decimal parts.

Whole Numbers

Whole numbers are just like natural numbers, but they also include the number 0. This means the sequence of whole numbers is 0, 1, 2, 3, and so on. They do not include any negative numbers, fractions, or decimals. Whole numbers are great for counting things where you might have "none" or a starting point. Understanding whole numbers is crucial since they form a foundation for many mathematical concepts. In our set \{-\sqrt{49}, -0.405, -0 . \overline{3}, 0.1, 3, 18, 6 \pi, 56\}, the whole numbers are still \(3\), \(18\), and \(56\) as these elements also fall under natural numbers.

Integers

Integers expand the number horizon to include negative numbers, but still keep it simple with no fractions or decimals. Integers include negative numbers, zero, and positive numbers like \{...,-3,-2,-1,0,1,2,3,...\}. An integer can represent things like temperature, where you might say it is -3 degrees to indicate below zero conditions. In our set \{-\sqrt{49}, -0.405, -0 . \overline{3}, 0.1, 3, 18, 6 \pi, 56\}, the integers are \(-\sqrt{49}\) which simplifies to \(-7\), \(3\), \(18\), and \(56\). These numbers include both positive and negative whole numbers.

Rational Numbers

Rational numbers are numbers that can be written as a fraction using an integer for the numerator and a non-zero integer for the denominator. These numbers can appear as decimals, fractions, or whole numbers as long as the decimal part either terminates or repeats predictably. A simple way to spot rational numbers is if you can turn the number into a fraction, it is rational. An example is 0.75, which equals \(\frac{3}{4}\). In the set \{-\sqrt{49}, -0.405, -0 . \overline{3}, 0.1, 3, 18, 6 \pi, 56\}, the rational numbers are \(-\sqrt{49}\), \(-0.405\), \(-0 . \overline{3}\), \(0.1\), \(3\), \(18\), and \(56\) as they can all be presented as fractions or have repeating decimal parts.

Irrational Numbers

Irrational numbers cannot be neatly expressed as fractions of two integers. Unlike rational numbers, their decimal form goes on forever without repeating a pattern. Perhaps the most famous irrational number is \(\pi\), which is essential in calculations involving circles. You might visually encounter it as 3.141592..., continuing infinitely without a predictable sequence. From the set \{-\sqrt{49}, -0.405, -0 . \overline{3}, 0.1, 3, 18, 6 \pi, 56\}, only \(6\pi\) falls under irrational numbers. This happens because \(\pi\) itself is non-repeating and non-terminating, making \(6\pi\) the same.