Numbers and Operations - Ninth Grade Math
1. Add, Subtract, Multiply, and Divide Integers
Addition of Integers
Rule 1: Same signs → Add absolute values, keep the sign
$(+) + (+) = +$ Example: $7 + 4 = 11$
$(-) + (-) = -$ Example: $(-7) + (-4) = -11$
$(+) + (+) = +$ Example: $7 + 4 = 11$
$(-) + (-) = -$ Example: $(-7) + (-4) = -11$
Rule 2: Different signs → Subtract absolute values, use sign of larger number
$(+) + (-) =$ sign of larger Example: $7 + (-4) = 3$
$(-) + (+) =$ sign of larger Example: $(-7) + 4 = -3$
$(+) + (-) =$ sign of larger Example: $7 + (-4) = 3$
$(-) + (+) =$ sign of larger Example: $(-7) + 4 = -3$
Subtraction of Integers
Key Formula: $a - b = a + (-b)$
Change subtraction to addition of the opposite
Change subtraction to addition of the opposite
Examples:
• $5 - 8 = 5 + (-8) = -3$
• $(-3) - 7 = (-3) + (-7) = -10$
• $(-6) - (-4) = (-6) + 4 = -2$
• $5 - 8 = 5 + (-8) = -3$
• $(-3) - 7 = (-3) + (-7) = -10$
• $(-6) - (-4) = (-6) + 4 = -2$
Multiplication of Integers
Rule 1: Same signs → Positive result
$(+) \times (+) = +$ Example: $5 \times 3 = 15$
$(-) \times (-) = +$ Example: $(-5) \times (-3) = 15$
$(+) \times (+) = +$ Example: $5 \times 3 = 15$
$(-) \times (-) = +$ Example: $(-5) \times (-3) = 15$
Rule 2: Different signs → Negative result
$(+) \times (-) = -$ Example: $5 \times (-3) = -15$
$(-) \times (+) = -$ Example: $(-5) \times 3 = -15$
$(+) \times (-) = -$ Example: $5 \times (-3) = -15$
$(-) \times (+) = -$ Example: $(-5) \times 3 = -15$
Division of Integers
Rule 1: Same signs → Positive quotient
$(+) \div (+) = +$ Example: $12 \div 3 = 4$
$(-) \div (-) = +$ Example: $(-12) \div (-3) = 4$
$(+) \div (+) = +$ Example: $12 \div 3 = 4$
$(-) \div (-) = +$ Example: $(-12) \div (-3) = 4$
Rule 2: Different signs → Negative quotient
$(+) \div (-) = -$ Example: $12 \div (-3) = -4$
$(-) \div (+) = -$ Example: $(-12) \div 3 = -4$
$(+) \div (-) = -$ Example: $12 \div (-3) = -4$
$(-) \div (+) = -$ Example: $(-12) \div 3 = -4$
Important Notes:
• Division by zero is undefined: $a \div 0$ = undefined
• Zero divided by any non-zero number is zero: $0 \div a = 0$
• Division by 1 doesn't change the value: $a \div 1 = a$
• Division by zero is undefined: $a \div 0$ = undefined
• Zero divided by any non-zero number is zero: $0 \div a = 0$
• Division by 1 doesn't change the value: $a \div 1 = a$
2. Evaluate Numerical Expressions Involving Integers
Order of Operations (PEMDAS/BODMAS):
1. Parentheses/Brackets: $( )$, $[ ]$, $\{ \}$
2. Exponents/Orders: Powers and roots
3. Multiplication and Division: Left to right
4. Addition and Subtraction: Left to right
1. Parentheses/Brackets: $( )$, $[ ]$, $\{ \}$
2. Exponents/Orders: Powers and roots
3. Multiplication and Division: Left to right
4. Addition and Subtraction: Left to right
Example: Evaluate $-3 + 5 \times 2 - 8 \div 4$
Step 1: $-3 + 10 - 2$ (multiplication and division first)
Step 2: $7 - 2$ (addition left to right)
Step 3: $5$ (final answer)
Step 1: $-3 + 10 - 2$ (multiplication and division first)
Step 2: $7 - 2$ (addition left to right)
Step 3: $5$ (final answer)
3. Convert Between Decimals and Fractions
Decimal to Fraction
Method:
1. Count decimal places
2. Write decimal as numerator (without decimal point)
3. Denominator = $10^n$ where $n$ = number of decimal places
4. Simplify the fraction
1. Count decimal places
2. Write decimal as numerator (without decimal point)
3. Denominator = $10^n$ where $n$ = number of decimal places
4. Simplify the fraction
Examples:
• $0.75 = \frac{75}{100} = \frac{3}{4}$
• $0.625 = \frac{625}{1000} = \frac{5}{8}$
• $2.4 = 2\frac{4}{10} = 2\frac{2}{5}$
• $0.75 = \frac{75}{100} = \frac{3}{4}$
• $0.625 = \frac{625}{1000} = \frac{5}{8}$
• $2.4 = 2\frac{4}{10} = 2\frac{2}{5}$
Fraction to Decimal
Method: Divide the numerator by the denominator
$\frac{a}{b} = a \div b$
$\frac{a}{b} = a \div b$
Examples:
• $\frac{3}{4} = 3 \div 4 = 0.75$ (terminating)
• $\frac{1}{3} = 1 \div 3 = 0.333...$ or $0.\overline{3}$ (repeating)
• $\frac{5}{8} = 5 \div 8 = 0.625$ (terminating)
• $\frac{3}{4} = 3 \div 4 = 0.75$ (terminating)
• $\frac{1}{3} = 1 \div 3 = 0.333...$ or $0.\overline{3}$ (repeating)
• $\frac{5}{8} = 5 \div 8 = 0.625$ (terminating)
4. Add and Subtract Rational Numbers
Adding/Subtracting Fractions
Same Denominator:
$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$
$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$
$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$
$\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}$
Different Denominators:
1. Find LCD (Least Common Denominator)
2. Convert fractions to equivalent fractions with LCD
3. Add/subtract numerators
$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ or use LCD
1. Find LCD (Least Common Denominator)
2. Convert fractions to equivalent fractions with LCD
3. Add/subtract numerators
$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ or use LCD
Example: $\frac{2}{3} + \frac{1}{4}$
LCD = 12
$= \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$
LCD = 12
$= \frac{8}{12} + \frac{3}{12} = \frac{11}{12}$
Adding/Subtracting Decimals
Rule: Align decimal points vertically, then add or subtract
Example: $12.5 + 3.75 = 16.25$
Example: $12.5 + 3.75 = 16.25$
5. Multiply and Divide Rational Numbers
Multiplying Fractions
Formula: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$
Multiply numerators, multiply denominators
Multiply numerators, multiply denominators
Example: $\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}$
Dividing Fractions
Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$
Multiply by the reciprocal (flip the second fraction)
Multiply by the reciprocal (flip the second fraction)
Example: $\frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} = \frac{10}{12} = \frac{5}{6}$
Multiplying/Dividing Decimals
Multiplication: Multiply ignoring decimals, then count total decimal places
Division: Move decimal point in divisor to make it whole, move same places in dividend
Division: Move decimal point in divisor to make it whole, move same places in dividend
6. Simplify Complex Fractions
Complex Fraction: A fraction where the numerator and/or denominator contains fractions
Example: $\frac{\frac{2}{3}}{\frac{4}{5}}$
Example: $\frac{\frac{2}{3}}{\frac{4}{5}}$
Method 1: Division Method
Steps:
1. Express numerator and denominator as single fractions
2. Divide: Multiply by reciprocal of denominator
3. Simplify
$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
1. Express numerator and denominator as single fractions
2. Divide: Multiply by reciprocal of denominator
3. Simplify
$\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
Method 2: LCD Method
Steps:
1. Find LCD of all denominators in the complex fraction
2. Multiply both top and bottom by this LCD
3. Simplify the result
1. Find LCD of all denominators in the complex fraction
2. Multiply both top and bottom by this LCD
3. Simplify the result
Example: Simplify $\frac{\frac{5}{2}}{\frac{2}{4}}$
Method 1: $= \frac{5}{2} \times \frac{4}{2} = \frac{20}{4} = 5$
Method 2: LCD = 4, multiply by $\frac{4}{4}$: $= \frac{10}{2} = 5$
Method 1: $= \frac{5}{2} \times \frac{4}{2} = \frac{20}{4} = 5$
Method 2: LCD = 4, multiply by $\frac{4}{4}$: $= \frac{10}{2} = 5$
7. Evaluate Numerical Expressions Involving Rational Numbers
Use PEMDAS with fractions and decimals:
1. Simplify parentheses first
2. Handle exponents
3. Multiply/divide left to right
4. Add/subtract left to right
1. Simplify parentheses first
2. Handle exponents
3. Multiply/divide left to right
4. Add/subtract left to right
Example: $\frac{1}{2} + \frac{3}{4} \times 2 - \frac{1}{3}$
Step 1: $\frac{1}{2} + \frac{6}{4} - \frac{1}{3}$ (multiply first)
Step 2: $\frac{1}{2} + \frac{3}{2} - \frac{1}{3}$ (simplify)
Step 3: $\frac{4}{2} - \frac{1}{3} = 2 - \frac{1}{3} = \frac{5}{3}$ or $1\frac{2}{3}$
Step 1: $\frac{1}{2} + \frac{6}{4} - \frac{1}{3}$ (multiply first)
Step 2: $\frac{1}{2} + \frac{3}{2} - \frac{1}{3}$ (simplify)
Step 3: $\frac{4}{2} - \frac{1}{3} = 2 - \frac{1}{3} = \frac{5}{3}$ or $1\frac{2}{3}$
8. Square Roots
Definition: $\sqrt{a} = b$ means $b^2 = a$
The square root of a number $a$ is the value that, when multiplied by itself, equals $a$
The square root of a number $a$ is the value that, when multiplied by itself, equals $a$
Properties of Square Roots:
• $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ where $b \neq 0$
• $\sqrt{a^2} = |a|$ (absolute value)
• $(\sqrt{a})^2 = a$
• $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ where $b \neq 0$
• $\sqrt{a^2} = |a|$ (absolute value)
• $(\sqrt{a})^2 = a$
Perfect Squares to Memorize:
$\sqrt{1} = 1$, $\sqrt{4} = 2$, $\sqrt{9} = 3$, $\sqrt{16} = 4$, $\sqrt{25} = 5$
$\sqrt{36} = 6$, $\sqrt{49} = 7$, $\sqrt{64} = 8$, $\sqrt{81} = 9$, $\sqrt{100} = 10$
$\sqrt{121} = 11$, $\sqrt{144} = 12$, $\sqrt{169} = 13$, $\sqrt{196} = 14$, $\sqrt{225} = 15$
$\sqrt{1} = 1$, $\sqrt{4} = 2$, $\sqrt{9} = 3$, $\sqrt{16} = 4$, $\sqrt{25} = 5$
$\sqrt{36} = 6$, $\sqrt{49} = 7$, $\sqrt{64} = 8$, $\sqrt{81} = 9$, $\sqrt{100} = 10$
$\sqrt{121} = 11$, $\sqrt{144} = 12$, $\sqrt{169} = 13$, $\sqrt{196} = 14$, $\sqrt{225} = 15$
Important:
• $\sqrt{a}$ is rational if $a$ is a perfect square
• $\sqrt{a}$ is irrational if $a$ is not a perfect square
• $\sqrt{a}$ is rational if $a$ is a perfect square
• $\sqrt{a}$ is irrational if $a$ is not a perfect square
9. Cube Roots
Definition: $\sqrt[3]{a} = b$ means $b^3 = a$
The cube root of a number $a$ is the value that, when cubed, equals $a$
The cube root of a number $a$ is the value that, when cubed, equals $a$
Properties of Cube Roots:
• $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$
• $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ where $b \neq 0$
• $\sqrt[3]{a^3} = a$ (no absolute value needed)
• $(\sqrt[3]{a})^3 = a$
• Cube roots can be negative: $\sqrt[3]{-8} = -2$
• $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$
• $\sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}$ where $b \neq 0$
• $\sqrt[3]{a^3} = a$ (no absolute value needed)
• $(\sqrt[3]{a})^3 = a$
• Cube roots can be negative: $\sqrt[3]{-8} = -2$
Perfect Cubes to Memorize:
$\sqrt[3]{1} = 1$, $\sqrt[3]{8} = 2$, $\sqrt[3]{27} = 3$, $\sqrt[3]{64} = 4$, $\sqrt[3]{125} = 5$
$\sqrt[3]{216} = 6$, $\sqrt[3]{343} = 7$, $\sqrt[3]{512} = 8$, $\sqrt[3]{729} = 9$, $\sqrt[3]{1000} = 10$
$\sqrt[3]{-8} = -2$, $\sqrt[3]{-27} = -3$, $\sqrt[3]{-64} = -4$
$\sqrt[3]{1} = 1$, $\sqrt[3]{8} = 2$, $\sqrt[3]{27} = 3$, $\sqrt[3]{64} = 4$, $\sqrt[3]{125} = 5$
$\sqrt[3]{216} = 6$, $\sqrt[3]{343} = 7$, $\sqrt[3]{512} = 8$, $\sqrt[3]{729} = 9$, $\sqrt[3]{1000} = 10$
$\sqrt[3]{-8} = -2$, $\sqrt[3]{-27} = -3$, $\sqrt[3]{-64} = -4$
10. Sort and Classify Rational and Irrational Numbers
Rational Numbers
Definition: Numbers that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$
Rational numbers include:
• All integers: $-3, 0, 7$
• All fractions: $\frac{2}{3}, \frac{-5}{7}, \frac{9}{1}$
• Terminating decimals: $0.75, 2.5, -1.25$
• Repeating decimals: $0.\overline{3}, 0.\overline{142857}, 1.\overline{6}$
• Square roots of perfect squares: $\sqrt{9} = 3, \sqrt{25} = 5$
• All integers: $-3, 0, 7$
• All fractions: $\frac{2}{3}, \frac{-5}{7}, \frac{9}{1}$
• Terminating decimals: $0.75, 2.5, -1.25$
• Repeating decimals: $0.\overline{3}, 0.\overline{142857}, 1.\overline{6}$
• Square roots of perfect squares: $\sqrt{9} = 3, \sqrt{25} = 5$
Irrational Numbers
Definition: Numbers that CANNOT be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers
Irrational numbers include:
• Non-terminating, non-repeating decimals
• Square roots of non-perfect squares: $\sqrt{2}, \sqrt{3}, \sqrt{5}$
• Special constants: $\pi \approx 3.14159...$, $e \approx 2.71828...$
• Examples: $\sqrt{7}, \sqrt{10}, \sqrt{15}, 0.101001000100001...$
• Non-terminating, non-repeating decimals
• Square roots of non-perfect squares: $\sqrt{2}, \sqrt{3}, \sqrt{5}$
• Special constants: $\pi \approx 3.14159...$, $e \approx 2.71828...$
• Examples: $\sqrt{7}, \sqrt{10}, \sqrt{15}, 0.101001000100001...$
| Property | Rational Numbers | Irrational Numbers |
|---|---|---|
| Form | $\frac{p}{q}$, $q \neq 0$ | Cannot be written as $\frac{p}{q}$ |
| Decimal Form | Terminating or repeating | Non-terminating, non-repeating |
| Examples | $\frac{3}{4}, 0.5, -7, 0.\overline{6}$ | $\sqrt{2}, \pi, \sqrt{11}$ |
11. Properties of Operations on Rational and Irrational Numbers
Operations with Rational Numbers
Closure Property: Operating on two rationals always gives a rational
• Rational + Rational = Rational
• Rational − Rational = Rational
• Rational × Rational = Rational
• Rational ÷ Rational = Rational (if not dividing by zero)
• Rational + Rational = Rational
• Rational − Rational = Rational
• Rational × Rational = Rational
• Rational ÷ Rational = Rational (if not dividing by zero)
Operations with Irrational Numbers
Key Results:
• Rational + Irrational = Irrational
• Rational × Irrational = Irrational (if rational $\neq 0$)
• Irrational + Irrational = May be rational or irrational
• Irrational × Irrational = May be rational or irrational
• Rational + Irrational = Irrational
• Rational × Irrational = Irrational (if rational $\neq 0$)
• Irrational + Irrational = May be rational or irrational
• Irrational × Irrational = May be rational or irrational
Examples:
• $2 + \sqrt{3}$ = Irrational
• $5 \times \sqrt{2}$ = Irrational
• $\sqrt{2} + (-\sqrt{2}) = 0$ = Rational
• $\sqrt{2} \times \sqrt{2} = 2$ = Rational
• $\sqrt{2} + \sqrt{3}$ = Irrational
• $2 + \sqrt{3}$ = Irrational
• $5 \times \sqrt{2}$ = Irrational
• $\sqrt{2} + (-\sqrt{2}) = 0$ = Rational
• $\sqrt{2} \times \sqrt{2} = 2$ = Rational
• $\sqrt{2} + \sqrt{3}$ = Irrational
Number Properties
| Property | Addition | Multiplication |
|---|---|---|
| Commutative | $a + b = b + a$ | $a \times b = b \times a$ |
| Associative | $(a + b) + c = a + (b + c)$ | $(a \times b) \times c = a \times (b \times c)$ |
| Identity | $a + 0 = a$ | $a \times 1 = a$ |
| Inverse | $a + (-a) = 0$ | $a \times \frac{1}{a} = 1$ ($a \neq 0$) |
| Distributive | $a(b + c) = ab + ac$ | |
12. Classify Numbers - The Number System
Number System Hierarchy:
Real Numbers = Rational Numbers + Irrational Numbers
Real Numbers = Rational Numbers + Irrational Numbers
Number Classifications
| Number Type | Definition | Examples |
|---|---|---|
| Natural Numbers (ℕ) | Counting numbers | 1, 2, 3, 4, 5, ... |
| Whole Numbers (W) | Natural numbers + zero | 0, 1, 2, 3, 4, 5, ... |
| Integers (ℤ) | Whole numbers + negatives | ..., -3, -2, -1, 0, 1, 2, 3, ... |
| Rational Numbers (ℚ) | $\frac{p}{q}$ form, $q \neq 0$ | $\frac{1}{2}, -3, 0.75, 0.\overline{3}$ |
| Irrational Numbers | Cannot be written as $\frac{p}{q}$ | $\sqrt{2}, \pi, \sqrt{5}, e$ |
| Real Numbers (ℝ) | All rational + irrational | All numbers above |
Classification Steps
Step 1: Check if it has a decimal or fraction. If no → Integer (and Rational)
Step 2: If integer and positive or zero → Also a Whole Number
Step 3: If integer and positive → Also a Natural Number
Step 4: If terminating or repeating decimal → Rational
Step 5: If non-terminating, non-repeating decimal → Irrational
Step 2: If integer and positive or zero → Also a Whole Number
Step 3: If integer and positive → Also a Natural Number
Step 4: If terminating or repeating decimal → Rational
Step 5: If non-terminating, non-repeating decimal → Irrational
Classification Examples:
• 7: Natural, Whole, Integer, Rational, Real
• 0: Whole, Integer, Rational, Real
• -5: Integer, Rational, Real
• $\frac{3}{4}$: Rational, Real
• $0.\overline{6}$: Rational, Real
• $\sqrt{16} = 4$: Natural, Whole, Integer, Rational, Real
• $\sqrt{7}$: Irrational, Real
• $\pi$: Irrational, Real
• 7: Natural, Whole, Integer, Rational, Real
• 0: Whole, Integer, Rational, Real
• -5: Integer, Rational, Real
• $\frac{3}{4}$: Rational, Real
• $0.\overline{6}$: Rational, Real
• $\sqrt{16} = 4$: Natural, Whole, Integer, Rational, Real
• $\sqrt{7}$: Irrational, Real
• $\pi$: Irrational, Real
Quick Reference Formulas
Integer Signs:
• Same signs: Add and keep sign (+ or -)
• Different signs: Subtract and use larger's sign
• Multiply/Divide same signs: Positive result
• Multiply/Divide different signs: Negative result
• Same signs: Add and keep sign (+ or -)
• Different signs: Subtract and use larger's sign
• Multiply/Divide same signs: Positive result
• Multiply/Divide different signs: Negative result
Fraction Operations:
• Add/Subtract: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$ (or use LCD)
• Multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
• Divide: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
• Add/Subtract: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$ (or use LCD)
• Multiply: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
• Divide: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$
Roots:
• $\sqrt{a} = b$ means $b^2 = a$
• $\sqrt[3]{a} = b$ means $b^3 = a$
• $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$
• $\sqrt{a} = b$ means $b^2 = a$
• $\sqrt[3]{a} = b$ means $b^3 = a$
• $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$
• $\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$
Remember: Practice these formulas regularly and always show your work step-by-step!
Master PEMDAS/BODMAS for all operations with integers, fractions, decimals, and roots.
Master PEMDAS/BODMAS for all operations with integers, fractions, decimals, and roots.