Review of Rational Numbers and Integers

TL;DR

Rational numbers are numbers that can be written as a fraction, including integers, decimals that stop, and repeating decimals. Integers are whole numbers, both positive and negative, including zero. Understanding these number types is crucial for subtracting rational numbers.

1. The Mental Model

Think of numbers as belonging to different families. The "rational number" family is quite big and includes the "integer" family as a smaller part of it. If you can express a number perfectly as a fraction of two integers, it's rational.

2. The Core Material

What's a Rational Number?

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A rational number is any number that can be written as a fraction, $\frac{a}{b}$, where 'a' and 'b' are integers, and 'b' is not zero.

This definition covers a lot of ground:
* Integers: Like 3, -5, 0. You can write 3 as $\frac{3}{1}$ or $\frac{6}{2}$.
* Fractions: Like $\frac{1}{2}$, $\frac{-3}{4}$, $\frac{7}{5}$.
* Terminating Decimals: Decimals that end, like 0.5 (which is $\frac{1}{2}$) or 2.75 (which is $\frac{11}{4}$).
* Repeating Decimals: Decimals that have a pattern that repeats forever, like 0.333... (which is $\frac{1}{3}$) or 0.142857142857... (which is $\frac{1}{7}$).

What's an Integer?

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Integers are all whole numbers, including positive numbers, negative numbers, and zero. They don't have any fractional or decimal parts.

Examples of integers: ..., -3, -2, -1, 0, 1, 2, 3, ...

Relationship Between Rational Numbers and Integers

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graph TD
    A["Numbers"] --> B["Rational Numbers"]
    B --> C["Integers"]
    B --> D["Non-Integer Rational Numbers"]
    C --> E["Positive Integers (Natural Numbers)"]
    C --> F["Negative Integers"]
    C --> G["Zero"]
    D --> H["Terminating Decimals"]
    D --> I["Repeating Decimals"]
    A --> J["Irrational Numbers"]
    J --> K["Pi (π)"]
    J --> L["Square root of 2 (√2)"]

This diagram shows you how integers are a subset of rational numbers. Every integer is a rational number, but not every rational number is an integer. For instance, $\frac{1}{2}$ is rational but not an integer.

Why does this matter for subtraction?

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When you subtract rational numbers, you might be dealing with any of these forms – fractions, decimals, or integers. The rules for subtraction will apply consistently. For example, subtracting a negative integer is the same as adding a positive integer. Subtracting decimals requires lining up decimal points. Understanding the nature of the numbers you're working with helps you choose the right approach.

3. Worked Example

Let's classify a few numbers:

• 7: This is an integer (positive) and also a rational number (can be written as $\frac{7}{1}$).
• -4.5: This is a terminating decimal. It's a rational number (can be written as $\frac{-9}{2}$), but it's not an integer.
• $\frac{3}{8}$: This is a fraction. It's a rational number, but not an integer. (Its decimal form is 0.375, which terminates).
• 0: This is an integer and a rational number ($\frac{0}{1}$).
• -0.666...: This is a repeating decimal. It's a rational number (can be written as $\frac{-2}{3}$), but it's not an integer.

4. Key Takeaways

• A rational number can always be expressed as a fraction of two integers, where the denominator isn't zero.
• Integers are whole numbers, including positive numbers, negative numbers, and zero, with no fractional or decimal parts.
• All integers are rational numbers, but not all rational numbers are integers.
• Terminating and repeating decimals are types of rational numbers.
• Understanding the type of number helps you apply the correct mathematical operations.

Common Mistakes to Avoid

• Don't confuse integers with just positive whole numbers; they include zero and negative whole numbers too.
• Don't think that all decimals are irrational; terminating and repeating decimals are rational.
• Forgetting that zero is an integer and a rational number.
• Assuming a number like $\sqrt{4}$ is irrational; it simplifies to 2, which is an integer and rational.

5. Now Try It

Spend 15 minutes classifying different numbers. Get a mix of fractions, decimals (both terminating and repeating), and positive/negative whole numbers. For each number, determine if it's an integer, a rational number (if not an integer), or possibly neither (though for this course, you'll mostly see rationals). Try writing rational numbers as fractions. Success means you can confidently identify and explain why each number fits its classification.