Problem 8 - Precalculus

Question

Determine whether the number is a natural number, an integer, a rational number, or an irrational number. (Some numbers fit in more than one category.) The following facts will be helpful in some cases: Any number of the form \(\sqrt{n},\) where \(n\) is a natural number that is not a perfect square, is irrational. Also, the sum, difference, product, and quotient of an irrational number and a nonzero rational are all irrational. (For example, the following four numbers are irrational: \(\sqrt{6}\), \(\sqrt{10}-2,3 \sqrt{15},\) and \(-5 \sqrt{3} / 2,\) \((3-\sqrt{2})+(3+\sqrt{2})\)

Step-by-Step Solution

Answer

The number is a natural number, an integer, and a rational number.

Step 1: Evaluate the Expression

Examine the expression \((3 - \sqrt{2}) + (3 + \sqrt{2})\). Notice that this involves adding two terms.

Step 2: Combine Like Terms

Simplify by adding the two terms within the parentheses: \((3 - \sqrt{2}) + (3 + \sqrt{2}) = 3 + 3 - \sqrt{2} + \sqrt{2}\). This combines to: \(6\), as the \(\sqrt{2}\) terms cancel each other out.

Step 3: Determine Number Category

Since the result of the expression is \(6\), identify its categories. \(6\) is a natural number (since it is a positive whole number), an integer (since it does not have a fractional part), and a rational number (since it can be expressed as a fraction \(\frac{6}{1}\)). It is not irrational, as irrational numbers cannot be expressed as a simple fraction.

Key Concepts

Natural Numbers

Natural numbers are the basic counting numbers that you start learning as a child. They are positive integers that begin from 1 and go upwards like 1, 2, 3, and so on.
Natural numbers do not include zero, negative numbers, or fractions. These numbers are used in everyday counting tasks, such as counting apples or listing steps.
Whenever you encounter a number problem, identifying if a number is a natural number is often the simplest step since it only involves checking if it’s a positive whole number.

Integers

Integers expand upon natural numbers by including zero and the negatives of natural numbers. So, numbers like -3, -2, -1, 0, 1, 2, 3 are all integers.
There is no fractional or decimal part in an integer. You can think of integers as numbers on a number line that stretch infinitely in both positive and negative directions.

• Positive Integers: These are exactly what natural numbers are.
• Negative Integers: These numbers are simply the negatives of natural numbers, like -1, -2.
• Zero: This is the neutral integer between positive and negative integers.

Integers come in handy for situations involving gains and losses or temperatures below zero.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, like a fraction \(\frac{a}{b}\) where a and b are integers, and b is not zero.
This means every integer is also a rational number, since it can be written as itself divided by 1. For example, 5 is rational because it can be written as \(\frac{5}{1}\).
In general, if you can represent a number as a fraction with integer values in the numerator and denominator, it qualifies as a rational number.

• Whole numbers: These are rational numbers with a denominator of 1.
• Fractions: Any fraction where both the top and bottom are whole numbers is rational.

Rational numbers encompass natural numbers, integers, and simple fractions.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. They have non-repeating, non-terminating decimal expansions.
Common examples include \(\sqrt{2}\), \(\pi\), and decimal numbers that seem to go on forever without any repeating sequence.
Unlike rational numbers, you can't write irrationals as a neat fraction. Instead, they often appear as square roots of non-perfect squares or constants like \(\pi\).

• Non-Perfect Square Roots: Any square root that doesn't simplify to a whole number is irrational, like \(\sqrt{7}\).
• Extraordinary constants: Numbers like \(\pi\) or e have unique decimal patterns that never end or repeat.

Understanding which numbers are irrational helps in recognizing patterns not typically associated with simple arithmetic operations.