Problem 1 - Algebra 2

Question

Name the sets of numbers to which each number belongs. $$ -4 $$

Step-by-Step Solution

Answer

-4 is an integer, rational, and real number.

Step 1: Identify the Set of Integers

First, consider if the number -4 belongs to the set of integers. The set of integers includes all positive and negative whole numbers as well as zero. Since -4 is a negative whole number, it is part of this set.

Step 2: Check Real Numbers

Next, determine if -4 is a real number. The set of real numbers includes all rational and irrational numbers. Since -4 is a rational number (as it can be expressed as a fraction, -4/1), it is also a real number.

Step 3: Consider Rational Numbers

Confirm if -4 belongs to the set of rational numbers. Rational numbers can be expressed as a fraction of two integers, where the denominator is not zero. Since -4 can be expressed as -4/1, it belongs to the set of rational numbers.

Key Concepts

Integers

Integers are a set of numbers that includes all positive whole numbers, negative whole numbers, and zero. They do not include fractions or decimals. Each integer is either a positive number, a negative number, or zero.
For instance:

• Positive integers: 1, 2, 3, etc.
• Negative integers: -1, -2, -3, etc.
• Zero: 0

In the case of the number -4, it is an integer because it is a whole number, but it is negative. Understanding that an integer can be negative or positive is crucial in identifying them within the real world of mathematics.
Identifying integers means simply picking out numbers from a set that don't have decimal or fractional parts, and recognizing that they can also be less than zero.

Real Numbers

Real numbers comprise a vast set of numbers including every number you can think of: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Essentially, they cover all possible numbers you might graph on a line that stretches from negative infinity through zero to positive infinity.
Real numbers are inclusive of:

• Rational numbers, like integers and fractions.
• Irrational numbers, like \( \pi \) and \( \sqrt{2} \).

When considering if a number like -4 is a real number, ask yourself: Can it be placed on a number line? If the answer is yes, then it falls into the category of real numbers. Since -4 clearly can be plotted on the number line, it stands as a real number.
Each real number can be represented on a continuous line without skips or gaps, including those naughty numbers that can't be written as fractions!

Rational Numbers

Rational numbers are mathematically friendly! They include all numbers that can be expressed as the fraction of two integers, meaning it can be written in the form of \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \) is not zero.
Examples of rational numbers:

• Whole numbers like 6 (expressed as \( \frac{6}{1} \)).
• Negative numbers like -4 (expressed as \( \frac{-4}{1} \)).
• Fractions like \( \frac{1}{3} \).

The number -4 is comfortably seated in the group of rational numbers because it can be expressed as \( \frac{-4}{1} \), a clear fraction where both numerator and denominator are integers and the denominator is not zero.
Rational numbers are reassuring in their predictability because they can always be expressed in terms of fractions, making them easily relatable and easy to understand.