Arithmetic Basics

Cheatsheet Content

### Basic Operations - **Addition (+):** Combining two or more numbers. - Example: $5 + 3 = 8$ - **Subtraction (-):** Finding the difference between two numbers. - Example: $8 - 3 = 5$ - **Multiplication (* or x):** Repeated addition. - Example: $5 \times 3 = 5 + 5 + 5 = 15$ - **Division (/ or $\div$):** Splitting a number into equal parts. - Example: $15 \div 3 = 5$ (15 split into 3 equal parts is 5) ### Order of Operations (PEMDAS/BODMAS) The sequence in which mathematical operations should be performed. - **P**arentheses (or **B**rackets) - **E**xponents (or **O**rders) - **M**ultiplication and **D**ivision (from left to right) - **A**ddition and **S**ubtraction (from left to right) Example: $10 + 2 \times (6 - 3)^2 \div 3$ 1. Parentheses: $10 + 2 \times (3)^2 \div 3$ 2. Exponents: $10 + 2 \times 9 \div 3$ 3. Multiplication/Division (left to right): - $2 \times 9 = 18 \implies 10 + 18 \div 3$ - $18 \div 3 = 6 \implies 10 + 6$ 4. Addition/Subtraction (left to right): - $10 + 6 = 16$ ### Number Types - **Natural Numbers ($\mathbb{N}$):** $\{1, 2, 3, ...\}$ (Counting numbers) - **Whole Numbers:** $\{0, 1, 2, 3, ...\}$ (Natural numbers + zero) - **Integers ($\mathbb{Z}$):** $\{..., -2, -1, 0, 1, 2, ...\}$ (Whole numbers + negative numbers) - **Rational Numbers ($\mathbb{Q}$):** Numbers that can be expressed as a fraction $p/q$, where $p, q$ are integers and $q \neq 0$. - Example: $1/2, -3/4, 5$ (which is $5/1$) - **Irrational Numbers:** Numbers that cannot be expressed as a simple fraction. Their decimal representation is non-repeating and non-terminating. - Example: $\pi \approx 3.14159...$, $\sqrt{2} \approx 1.41421...$ - **Real Numbers ($\mathbb{R}$):** All rational and irrational numbers. ### Fractions - **Definition:** A part of a whole, represented as $\frac{\text{Numerator}}{\text{Denominator}}$. - **Equivalent Fractions:** $\frac{a}{b} = \frac{a \times c}{b \times c}$ - **Adding/Subtracting:** Must have a common denominator. - Example: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$ - **Multiplying:** Multiply numerators and denominators. - Example: $\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$ - **Dividing:** Multiply by the reciprocal of the second fraction. - Example: $\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$ ### Decimals - **Place Value:** Each digit's position relative to the decimal point determines its value. - Example: $123.45 = 1 \times 100 + 2 \times 10 + 3 \times 1 + 4 \times 0.1 + 5 \times 0.01$ - **Adding/Subtracting:** Align decimal points. - Example: $1.23 + 4.5 = 5.73$ - **Multiplying:** Multiply as whole numbers, then count total decimal places in factors to place decimal in product. - Example: $1.2 \times 0.3 = 0.36$ (1 decimal place + 1 decimal place = 2 decimal places) - **Dividing:** Move decimal point in divisor and dividend to make divisor a whole number. - Example: $1.2 \div 0.3 = 12 \div 3 = 4$ ### Percentages - **Definition:** A fraction out of 100. Denoted by %. - **Decimal to Percent:** Multiply by 100. - Example: $0.25 = 0.25 \times 100\% = 25\%$ - **Percent to Decimal:** Divide by 100. - Example: $25\% = 25 \div 100 = 0.25$ - **Finding a Percentage of a Number:** Convert percent to decimal and multiply. - Example: $20\%$ of $50 = 0.20 \times 50 = 10$