Question

Determine which numbers in the set are (a) natural numbers, (b) integers, (c) rational numbers, and (d) irrational numbers.$$ \left\\{\sqrt{5},-7,-\frac{7}{3}, 0,3.12, \frac{5}{4}\right\\} $$

Short answer

Natural numbers: none, Integers: -7, 0, Rational numbers: -7, \( -\frac{7}{3} \), 0, 3.12, \( \frac{5}{4} \), Irrational numbers: \( \sqrt{5} \).

Step-by-step solution

Understanding the Categories

To categorize the numbers, first understand the definitions of each category. Natural numbers are positive, non-zero numbers without any decimal part, like 1,2,3 and so on. Integers include all natural numbers, their negatives, and zero. Rational numbers are any numbers that can be expressed as a fraction \( \frac{a}{b} \) where 'a' and 'b' are integers and 'b' is not equal to zero. Irrational numbers cannot be expressed as a fraction and have non-repeating, non-terminating decimal parts.

Categorize \( \sqrt{5} \)

\( \sqrt{5} \) is an irrational number because it cannot be expressed as a fraction and its decimal part is non-repeating and non-terminating.

Categorize -7

-7 is an integer, but it doesn't belong to natural numbers as natural numbers are always positive. It's also a rational number as it can be expressed as \( \frac{-7}{1} \).

Categorize \( -\frac{7}{3} \)

\(-\frac{7}{3}\) is a rational number, as it's represented as a fraction where the numerator and denominator are integers. However, it's not an integer or natural number.

Categorize 0

0 is an integer and a rational number (as it can be expressed as \( \frac{0}{1} \)). It's not a natural number, as natural numbers start from 1.

Categorize 3.12

3.12 is a rational number, because it can be expressed as a fraction \( \frac{312}{100} \). It's not an integer or a natural number as it has a decimal part.

Categorize \( \frac{5}{4} \)

\( \frac{5}{4} \) is a rational number, because it is a fraction where the numerator and denominator are integers. It's not an integer or a natural number as it is not a whole number.

Key concepts

Natural Numbers

Natural numbers are considered the set of counting numbers that start from 1 and extend infinitely upwards. They include numbers like 1, 2, 3, and so on.
These are the numbers we naturally use for counting objects in everyday life, like counting apples or fingers of our hand.
Natural numbers do not include zero, negative numbers, fractions, or decimals.

• Example: 1, 25, 100
• Non-examples: 0, -5, 1/2, 3.14

Understanding the boundary of natural numbers is crucial when classifying a set of numbers, like the one in our example. Since none of the numbers in the given set is both positive and without a fractional or decimal part, this set contains no natural numbers.

Integers

Integers expand upon the set of natural numbers by including zero and all negative whole numbers. This makes integers a much broader category.
Simply put, integers encompass all whole numbers without any decimal or fractional parts but can be both positive and negative.

• Example: -3, 0, 7
• Non-examples: 3.5, 1/2, \( \sqrt{5} \)

Within the given set, 0 and -7 are recognized as integers. It’s important to note that being an integer also means being a rational number, but not necessarily all rational numbers are integers.

Rational Numbers

Rational numbers are numbers that can be written as a fraction \( \frac{a}{b} \), where both 'a' (the numerator) and 'b' (the denominator) are integers, and 'b' is not zero.
This means any number that can be represented as a simple fraction – including whole numbers and repeating or terminating decimals – fits into this category.

• Example: 4, \( \frac{-7}{3} \), 0.25
• Non-examples: \( \sqrt{2} \), \( \pi \)

In the set provided, -3/7, 0, 3.12, and 5/4 are classified as rational numbers. They are all expressible as simple fractions, making them part of this extensive category of numbers.

Irrational Numbers

Irrational numbers are a bit more elusive. These numbers cannot be written as simple fractions. They have non-terminating, non-repeating decimal parts, making them a unique set on their own.
Often, irrational numbers are square roots of non-perfect squares or numbers like \( \pi \).

• Example: \( \sqrt{3} \), \( \pi \)
• Non-examples: 1/2, 4.5, -3

In the original set, \( \sqrt{5} \) is an irrational number. Its decimal representation goes on forever without repeating, and it can't be expressed neatly as a fraction, perfectly illustrating what it means to be irrational.