Real Numbers: The set of all rational and irrational numbers.
Rational Numbers: Numbers that can be expressed as a/b where a and b are integers and b ≠ 0.
Number Line: A visual representation where larger numbers are to the right of smaller numbers.
Ordering: Arranging numbers from smallest to largest (ascending) or largest to smallest (descending).
- Convert to common form: Convert all numbers to decimals or fractions
- Identify signs: Separate positive and negative numbers
- Compare absolute values: For same signs, use magnitude
- Apply rules: Negative numbers are always less than positive numbers
- Order systematically: From left to right on number line
-2.5 = -2.5
3/4 = 0.75
-1 = -1.0
0.8 = 0.8
-3/2 = -1.5
Negative numbers: -2.5, -1.0, -1.5
Positive numbers: 0.75, 0.8
Among -2.5, -1.5, -1.0: -2.5 < -1.5 < -1.0
Among 0.75, 0.8: 0.75 < 0.8
Negative numbers first, then zero (if present), then positive numbers
Result: -2.5, -1.5, -1.0, 0.75, 0.8
-2.5 is farthest left, -1.5 is next, then -1.0, then 0.75, finally 0.8
From least to greatest: -2.5, -3/2, -1, 3/4, 0.8
• Decimal conversion: Convert fractions to decimals for easier comparison
• Negative comparison: For negative numbers, larger absolute value means smaller number
• Ordering principle: On number line, left is smaller, right is larger
Inequality: A mathematical statement comparing two expressions using <, >, ≤, or ≥.
Inequality Properties: Rules that govern how inequalities behave under operations.
Multiplication Property: How multiplication affects inequality direction.
Given: a > b and c < 0
This means a - b > 0 (positive difference)
a > b
ac > bc (if c > 0)
But since c < 0, we need to analyze what happens
Since c < 0, multiplying by c is like multiplying by -|c|
a > b means a - b > 0
c(a - b) < 0 (since c < 0 and (a-b) > 0)
c(a - b) = ca - cb = ac - bc
So ac - bc < 0
Therefore ac < bc
When multiplying by a negative number, the order reverses:
Example: 3 > 2, but 3(-1) = -3 < -2 = 2(-1)
This is because multiplying by -1 reflects numbers across zero
Let a = 5, b = 3, c = -2
Is a > b? Yes: 5 > 3 ✓
Is c < 0? Yes: -2 < 0 ✓
Is ac < bc? 5(-2) = -10 and 3(-2) = -6, so -10 < -6 ✓
If a > b and c < 0, then ac < bc. The inequality sign flips because multiplying by a negative number reverses the order.
• Multiplication property: Multiplying by positive preserves inequality, negative reverses it
• Order preservation: a > b iff a - b > 0
• Sign multiplication: Positive × Negative = Negative
Absolute Value: |a| = distance from a to 0 on the number line, always non-negative.
Distance: The non-negative value representing separation between two points.
Opposites: Two numbers that are the same distance from 0 but on opposite sides.
|−7| = distance from -7 to 0 = 7
|3| = distance from 3 to 0 = 3
7 > 3, so |−7| > |3|
|−3| = distance from -3 to 0 = 3
|−7| = 7 and |−3| = 3
So |−7| > |−3|
Absolute value measures distance from zero, regardless of direction
|-7| = 7 means -7 is 7 units from 0
|3| = 3 means 3 is 3 units from 0
So |-7| > |3| because -7 is farther from 0 than 3 is
|−7| = 7, |3| = 3, so |−7| > |3|. Also |−7| > |−3| since |−3| = 3.
• Absolute value definition: |a| = a if a ≥ 0, |a| = -a if a < 0
• Distance interpretation: Absolute value represents distance from zero
• Comparison principle: Greater absolute value means farther from zero
Real Numbers: All rational and irrational numbers that can be placed on a number line.
Rational Numbers: Numbers expressible as a/b where a, b are integers and b ≠ 0.
Irrational Numbers: Numbers that cannot be expressed as a/b (like √2, π, e).
Number Line: A visual representation where larger numbers are positioned to the right of smaller numbers.
- Convert to common form: Convert all numbers to decimals or fractions
- Identify signs: Separate positive and negative numbers
- Compare magnitudes: Use absolute values for same-signed numbers
- Apply properties: Use inequality rules and properties
- Verify results: Check with number line or alternative method
Rational Number: A number that can be expressed as a/b where a, b are integers and b ≠ 0.
Irrational Number: A number that cannot be expressed as a/b, with non-terminating, non-repeating decimal expansion.
Perfect Square: A number that is the square of an integer.
√2 cannot be expressed as a fraction of integers
√2 ≈ 1.41421356... (non-terminating, non-repeating)
Therefore, √2 is irrational
3/4 is already in the form a/b where a=3, b=4 (integers, b≠0)
3/4 = 0.75 (terminating decimal)
Therefore, 3/4 is rational
√9 = 3 = 3/1 (expressible as fraction)
Therefore, √9 is rational
π is a fundamental mathematical constant
π ≈ 3.14159265... (non-terminating, non-repeating)
Therefore, π is irrational
0.333... = 1/3 (convertible to fraction)
This is a repeating decimal, which is always rational
Therefore, 0.333... is rational
√8 = √(4×2) = 2√2
Since √2 is irrational, 2√2 is also irrational
Therefore, √8 is irrational
Rational: 3/4, √9, 0.333...
Irrational: √2, π, √8
• Rational definition: Expressible as a/b with integer a, b and b ≠ 0
• Perfect squares: √n is rational if n is a perfect square
• Repeating decimals: Always rational (convertible to fraction)
Descending Order: Arranging numbers from largest to smallest.
Estimation: Finding approximate values to enable comparison.
Number Line Positioning: Larger numbers are positioned to the right.
-√2 ≈ -1.414
1.5 = 1.5
-3/4 = -0.75
√3 ≈ 1.732
-1.2 = -1.2
Positive: 1.5, √3
Negative: -√2, -3/4, -1.2
√3 ≈ 1.732 > 1.5
Among -0.75, -1.2, -1.414: -0.75 > -1.2 > -1.414
So: -3/4 > -1.2 > -√2
Positive numbers first (from largest), then negative numbers (from largest)
Result: √3, 1.5, -3/4, -1.2, -√2
√3 ≈ 1.732, 1.5, -3/4 = -0.75, -1.2, -√2 ≈ -1.414
On number line: -1.414 < -1.2 < -0.75 < 1.5 < 1.732
Descending: 1.732 > 1.5 > -0.75 > -1.2 > -1.414
From greatest to least: √3, 1.5, -3/4, -1.2, -√2
• Decimal conversion: Convert to common form for easy comparison
• Sign separation: Positive numbers are always greater than negative numbers
• Negative comparison: Among negatives, the one closer to zero is greater