Problem 3 - 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 \(-5 \sqrt{3} / 2,\) (a) \(10^{6}\) (b) \(10^{6} / 10^{7}\)

Step-by-Step Solution

Answer

(a) Natural, integer, rational; (b) Rational.

Step 1: Understand the Types of Numbers

Before identifying the type of number, understand the categories: Natural numbers are positive integers starting from 1. Integers include all whole numbers, both positive and negative, and zero. Rational numbers can be expressed as fractions where both numerator and denominator are integers. Irrational numbers cannot be expressed as fractions, and their decimal representation is non-terminating and non-repeating.

Step 2: Analyze the Number in (a): \(10^6\)

\(10^6\) means the number 10 is multiplied by itself 6 times, which equals 1,000,000. This number is a positive whole number with no fractional or decimal part.

Step 3: Determine the Type for (a): \(10^6\)

Since 1,000,000 is a positive whole number, it is a natural number. Since it’s an integer without a fractional part, it’s also a rational number as it can be expressed as \(\frac{1,000,000}{1}\). As it is not a non-terminating or non-repeating decimal, it is not irrational.

Step 4: Analyze the Number in (b): \(\frac{10^6}{10^7}\)

The expression \(\frac{10^6}{10^7}\) simplifies to \(\frac{1}{10}\). This is a division of two integers resulting in a fraction.

Step 5: Determine the Type for (b): \(\frac{1}{10}\)

\(\frac{1}{10}\) is expressed as a fraction (\(\frac{1}{10}\)), meaning it is a rational number. It is neither a whole number nor a non-terminating decimal, so it is not an integer or irrational.

Key Concepts

Natural Numbers

Natural numbers are the simplest form of numbers we encounter in our daily lives. They start from 1 and go on to infinity, including numbers like 2, 3, 4, and so on.

• These numbers are always positive.
• They do not include fractions or decimals. They are whole numbers.
• Natural numbers are the building blocks of our counting system.

When you think about counting objects, like apples or books, you're using natural numbers! For example, in the exercise, the number 1,000,000 is a natural number because it is a positive whole number.

Integers

Integers include all natural numbers, but they also bring more into the picture. They comprise three main categories:

• Positive numbers (like 1, 2, 3...)
• Negative numbers (like -1, -2, -3...)
• The number zero

In essence, integers represent all whole numbers, whether they are above zero, below zero, or exactly zero. So, when looking at the number 1,000,000, it falls into the integer category because it's a whole number. Similarly, negative numbers such as -5 are also considered integers.

Rational Numbers

Rational numbers are extremely fascinating as they offer a bit more flexibility. A rational number can be any number that can be written as a fraction, \(\frac{a}{b}\)where both \(a\) and \(b\) are integers and \(b\) is not zero.

• This means they can be positive, negative, or zero.
• They can include fractions that you see every day, like \(\frac{1}{2}\) or \(\frac{3}{4}\).
• They can also be whole numbers, such as 5, expressed as \(\frac{5}{1}\).

An example from the exercise is \(\frac{1}{10}\), a clear representation of a rational number, as it is a fraction made up of two integers.

Irrational Numbers

Irrational numbers are quite intriguing! They cannot be expressed as simple fractions. Their decimal form is non-terminating and does not repeat.

• For instance, the square root of a non-perfect square (like \(\sqrt{2}\)) is irrational.
• Numbers like \(\pi\) and \(e\) are also irrational.

Irrational numbers often pop up in measurements involving curves, circles, or in more advanced mathematics. Unlike rational numbers, you won't find easy patterns in their decimals, making them both fascinating and challenging!