Problem 51 - Prealgebra

Question

Name all of the sets of numbers to which each real number belongs. Let \(N=\) natural numbers, \(\mathbf{W}=\) whole numbers, \(\mathbf{Z}=\) integers, \(\mathbf{Q}=\) rational numbers, and $\mathbf{I}=\text { irrational numbers. $$-5$$

Step-by-Step Solution

Answer

-5 is an integer (Z) and a rational number (Q).

Step 1: Identify Natural Numbers

Natural numbers (46N46) are the set of positive whole numbers starting from 1, such as 1, 2, 3, and so on. Since 4615 is negative, it does not belong to this set.

Step 2: Identify Whole Numbers

Whole numbers (46W46) begin from 0 and include all positive natural numbers. They do not include negative numbers. Since 4615 is negative, it does not belong to this set.

Step 3: Identify Integers

Integers (46Z46) include all positive and negative whole numbers, as well as zero. The number 4615 fits into this set because it is a whole number, even though it is negative.

Step 4: Identify Rational Numbers

Rational numbers (4610Q4610) are numbers that can be expressed as a fraction ▒a/b▒, where ▒a and ▒b are integers and ▒b  =3. In this case, 4615 can be written as 4615/1, making it a rational number.

Step 5: Identify Irrational Numbers

Irrational numbers (4631I) cannot be expressed as a simple fraction. Since 4615 is expressed exactly as a ratio of two integers, it is not an irrational number.

Key Concepts

Natural Numbers

Natural numbers are the most basic set of numbers we encounter in our daily lives. They start from 1 and include all the positive whole numbers. Here are some key points to remember about natural numbers:

• Natural numbers are used for counting items, like 1 apple, 2 oranges, etc.
• They do not include zero or negative numbers.

For example, the numbers 1, 2, 3, and so on are all natural numbers.
However, if a number is negative or not a whole number, like -5 or 0, it is not considered a natural number. This is essential to understanding where numbers belong in different sets.

Whole Numbers

Whole numbers expand on the concept of natural numbers by including 'zero'. While natural numbers only begin at 1, whole numbers start from 0 and increase in the same way. Here's what makes whole numbers unique:

• The inclusion of zero is the main difference between natural and whole numbers.
• Whole numbers are always non-negative.

You can think of whole numbers as a broader category than natural numbers, encompassing everything from zero upwards. Although 0, 1, 2, 3 are whole numbers, negative numbers like -5 do not belong to this group.

Integers

Integers are like the grand collection of whole numbers, natural numbers, and their negatives. This set includes:

• All positive and negative whole numbers.
• Zero, which acts as the neutral number in this set.

The integer set looks like this: \[ ... -3, -2, -1, 0, 1, 2, 3, ... \]
This spectrum of numbers allows room for negative values. So, the number -5 comfortably fits in the integers set. They do not allow for fractions or decimals, as they must be whole units.

Rational Numbers

Rational numbers include any number that can be expressed as a fraction, where both the numerator \( a \) and the denominator \( b \) are integers, and \( b eq 0 \). Here are defining features of rational numbers:

• Includes integers because they can be written as a fraction (e.g., \(-3 = \frac{-3}{1}\)).
• Also includes positive and negative fractions.

For example, the number -5 can be expressed as \( \frac{-5}{1} \), hence it is a rational number.
Rational numbers cover a broad range, including numbers we deal with every day, like 0.5, \(-2\), \( \frac{3}{4} \), and more. As long as it can be written as a fraction with integer parts, it's a rational number.