Integers and Rational Numbers
1.1 Integers and Rational Numbers
🧭 Overview
🧠 One-sentence thesis
Real numbers form a hierarchy where rational numbers (including integers, whole numbers, and counting numbers) can be expressed as fractions, ordered, compared, and manipulated using properties like absolute value and opposites.
📌 Key points (3–5)
- Real number hierarchy: Real numbers split into rational (can be written as a/b) and irrational (non-repeating, non-terminating decimals); integers, whole numbers, and counting numbers are all subsets of rational numbers.
- Fractions and equivalence: Proper fractions (numerator < denominator) represent values less than one; improper fractions (numerator ≥ denominator) represent values ≥ one; equivalent fractions have the same value when simplified.
- Ordering and comparing: Rational numbers can be ordered by finding common denominators; integers can be compared on a number line where rightmost is greatest.
- Common confusion: Absolute value always yields a positive result or zero—it measures distance from zero, not direction; don't confuse |−7| with −7.
- Opposites and additive inverse: Every number has an opposite at the same distance from zero in the other direction; a number plus its opposite always equals zero.
🔢 The real number hierarchy
🌳 How real numbers are organized
The excerpt presents real numbers as the most generic category, which then branches into two major types:
| Category | Definition | Examples |
|---|---|---|
| Rational numbers | Numbers that can be written as a/b where a and b are integers and b ≠ 0 | Fractions, integers, terminating decimals |
| Irrational numbers | Non-repeating, non-terminating decimals | π, √2 |
🪜 Subsets within rational numbers
Each category is a subset of the one above it:
Counting Numbers: the natural numbers from 1 to infinity, i.e. {1, 2, 3, 4, 5...}
Whole Numbers: all Counting Numbers and the number zero, i.e. {0, 1, 2, 3, 4, 5...}
Integers: all the whole numbers, zero and the negatives of the whole numbers, i.e. {...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...}
- All counting numbers are whole numbers.
- All whole numbers are integers.
- All integers are rational numbers.
- All rational and irrational numbers are real numbers.
🍰 Thinking about rational numbers as fractions
The excerpt uses a cake analogy: if you cut a cake into b slices, your share is a of those slices.
Example: The rational number 1/2 means cutting the cake into two parts and taking one part.
🧩 Types and properties of fractions
📏 Proper vs improper fractions
- Proper fractions: numerator is less than the denominator; represents a number less than one; "you always end up with less than a whole cake."
- Improper fractions: numerator is greater than or equal to the denominator; represents a number ≥ one; can be rewritten as a mixed number (an integer plus a proper fraction).
♻️ Equivalent fractions and simplification
Equivalent fractions: two fractions that give the same numerical value.
The excerpt explains that 1/2 and 2/4 are equivalent because they represent the same visual portion.
How to simplify (reduce) a fraction:
- Write out the prime factors of both numerator and denominator.
- Cancel matching factors that appear in both.
- Re-multiply the remaining factors.
Example from the excerpt: (2/4) = (2·1)/(2·2·1) = 1/2
- Reducing does not change the value; it simplifies how we write it.
- When all common factors are canceled, the fraction is in simplest form.
📝 Classification examples
The excerpt provides three examples:
- 3/7: both are prime, so already in simplest form (proper fraction).
- 9/3: improper fraction; simplifies to 3.
- 50/60: proper fraction; simplifies to 5/6 by canceling common factors.
📊 Ordering and comparing rational numbers
🔢 Ordering rational numbers
Ordering rational numbers: arranging them according to directions such as ascending (lowest to highest) or descending (highest to lowest).
The excerpt notes this is useful for determining which unit cost is cheapest.
Example scenario: Three can sizes with different prices—find cost per ounce, then arrange in ascending order: 0.040, 0.061875, 0.07375.
⚖️ Comparing fractions
To compare fractions like 3/7 and 4/9:
- Create a common denominator (7 × 9 = 63).
- Convert both fractions: 3/7 = 27/63 and 4/9 = 28/63.
- Compare numerators: because 28 > 27, therefore 4/9 > 3/7.
📍 Graphing and comparing integers
- To graph an integer on a number line, place a dot above the number.
- Greatest number is farthest to the right; least is farthest to the left.
- Use symbols: > means "greater than"; < means "less than".
Example: 2 is farther right than -5, so 2 > -5 (or -5 < 2).
🎯 Absolute value and opposites
🔄 Opposites and the additive inverse
Opposite: represents the same distance from zero but in the other direction.
Additive Inverse Property: For any real number a, a + (−a) = 0.
- Every number has an opposite.
- Adding a number to its opposite always gives zero.
Example: 7 and -7 are opposites; 7 + (−7) = 0.
📏 Absolute value definition
Absolute value: represents the distance from zero when graphed on a number line.
- Written as |x|, read as "the absolute value of x."
- Both 7 and -7 are 7 units away from zero, so |7| = 7 and |−7| = 7.
⚠️ Key rules for absolute value
- Treat absolute value expressions like parentheses: evaluate operations inside first.
- The absolute value is always positive or zero—it cannot be negative.
- We only care about distance from zero, not direction.
Don't confuse: |−7| = 7 (positive), but −|−7| = −7 (negative sign outside the absolute value).
🧮 Worked examples
The excerpt provides four examples:
- |5 + 4| = |9| = 9
- 3 − |4 − 9| = 3 − |−5| = 3 − 5 = −2
- |−5 − 11| = |−16| = 16
- −|7 − 22| = −|−15| = −(15) = −15
Notice in the last example: the negative sign outside the absolute value makes the final answer negative.