Basic Calculator
Free online calculator for basic math operations. Supports keyboard input and maintains calculation history.
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Free Online Basic Calculator
Features
- • Basic arithmetic operations (+, -, ×, ÷)
- • Square and square root functions
- • Percentage calculations
- • Keyboard support for faster input
- • Calculation history with timestamps
- • Scientific notation for large numbers
- • Clear and Clear Entry functions
- • Mobile-friendly responsive design
Keyboard Shortcuts
0-9+ - * /Enter or =Escape or CBackspaceThis free online basic calculator performs essential mathematical operations with precision and ease. Perfect for students, professionals, and anyone needing quick calculations. The calculator maintains your calculation history and supports both mouse clicks and keyboard input for maximum convenience.
Understanding Order of Operations (PEMDAS/BODMAS)
One of the most fundamental concepts in arithmetic is the order of operations. Without a universally agreed-upon sequence, the same expression could yield different results depending on the order in which you perform calculations. The two most common mnemonics are PEMDAS (used primarily in the United States) and BODMAS (used in the United Kingdom and other countries).
PEMDAS: Parentheses, Exponents, Multiplication & Division (left to right), Addition & Subtraction (left to right)
BODMAS: Brackets, Orders, Division & Multiplication (left to right), Addition & Subtraction (left to right)
For example, the expression 8 + 2 x 5 equals 18, not 50, because multiplication is performed before addition. Similarly, 24 / (4 + 2) equals 4 because the parentheses force the addition to happen first. Always work from the innermost grouping outward, handle exponents next, then proceed with multiplication and division from left to right, and finally addition and subtraction from left to right.
Worked Example: Multi-Step PEMDAS Problem
Let us solve a multi-step expression step by step, strictly following PEMDAS rules. This type of problem is common in standardized tests and coding interviews.
Expression: 3 + 4 x 2 - (6 / 3) + 52
Step 1: Parentheses
Evaluate (6 / 3) = 2
Expression becomes: 3 + 4 x 2 - 2 + 52
Step 2: Exponents
Evaluate 52 = 25
Expression becomes: 3 + 4 x 2 - 2 + 25
Step 3: Multiplication and Division (left to right)
Evaluate 4 x 2 = 8
Expression becomes: 3 + 8 - 2 + 25
Step 4: Addition and Subtraction (left to right)
3 + 8 = 11, then 11 - 2 = 9, then 9 + 25 = 34
Final Answer: 34
Essential Arithmetic Properties
Basic arithmetic is governed by a handful of properties that make calculations more flexible and efficient. Understanding these properties helps simplify complex expressions and avoid mistakes.
- Commutative Property: The order of numbers does not change the result for addition and multiplication. For example, 3 + 7 = 7 + 3 and 4 x 5 = 5 x 4. This property does not apply to subtraction or division.
- Associative Property: When adding or multiplying three or more numbers, the grouping does not matter. For instance, (2 + 3) + 4 = 2 + (3 + 4). Again, this does not hold for subtraction or division.
- Distributive Property: Multiplication distributes over addition and subtraction. This means a x (b + c) = a x b + a x c. For example, 3 x (10 + 2) = 30 + 6 = 36.
- Identity Elements: Adding 0 to any number leaves it unchanged (additive identity), and multiplying any number by 1 leaves it unchanged (multiplicative identity).
Number Properties: Primes, Factors, and Divisibility
Understanding number properties is foundational to mathematics and useful in everyday calculations, from splitting bills evenly to solving coding problems.
Prime Numbers
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first 20 primes are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71. Note that 2 is the only even prime number. To check if a number is prime, test divisibility by all primes up to its square root.
Even and Odd Numbers
Even numbers are divisible by 2 (end in 0, 2, 4, 6, or 8). Odd numbers are not (end in 1, 3, 5, 7, or 9). Key rules: even + even = even, odd + odd = even, even + odd = odd. For multiplication: even x anything = even, odd x odd = odd.
Factors and Multiples
Factors of a number divide into it evenly. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24. The Greatest Common Factor (GCF) of two numbers is the largest factor they share. The Least Common Multiple (LCM) is the smallest number that both divide into evenly. For example, GCF(12, 18) = 6 and LCM(12, 18) = 36.
Quick Divisibility Rules
| Divisible By | Rule | Example |
|---|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) | 4,738 -- last digit 8, divisible by 2 |
| 3 | Sum of digits is divisible by 3 | 321: 3+2+1=6, divisible by 3 |
| 4 | Last two digits divisible by 4 | 1,324: 24/4=6, divisible by 4 |
| 5 | Last digit is 0 or 5 | 2,345 -- last digit 5, divisible by 5 |
| 6 | Divisible by both 2 and 3 | 132: even and 1+3+2=6, divisible by 6 |
| 9 | Sum of digits is divisible by 9 | 729: 7+2+9=18, and 1+8=9, divisible by 9 |
| 10 | Last digit is 0 | 5,430 -- last digit 0, divisible by 10 |
Mental Math Strategies
Developing strong mental math skills allows you to estimate answers, verify calculator results, and make quick decisions in everyday life. Here are proven strategies used by math experts:
1. Breaking Numbers Apart (Decomposition)
Split complex calculations into simpler parts using the distributive property.
16 x 25 = 16 x (20 + 5) = 320 + 80 = 400
47 + 38 = (47 + 40) - 2 = 87 - 2 = 85
2. Multiplying by Powers of 10
To multiply by 5, multiply by 10 and divide by 2. To multiply by 25, multiply by 100 and divide by 4.
48 x 5 = 480 / 2 = 240
36 x 25 = 3,600 / 4 = 900
3. Squaring Numbers Ending in 5
Take the tens digit, multiply it by the next integer, and append 25.
352: 3 x 4 = 12, append 25 = 1,225
752: 7 x 8 = 56, append 25 = 5,625
4. The 9s Trick for Multiplication
To multiply a single digit by 9: the tens digit of the answer is one less than the number, and the digits sum to 9.
7 x 9: tens digit = 6, ones digit = 9-6 = 3, answer = 63
8 x 9: tens digit = 7, ones digit = 9-7 = 2, answer = 72
5. Percentage Calculations
Remember that X% of Y equals Y% of X. Choose whichever is easier to compute.
8% of 50 = 50% of 8 = 4
4% of 75 = 75% of 4 = 3
Common Math Shortcuts Reference Table
Memorizing these common calculations saves time and reduces errors in everyday math.
| Shortcut | Formula / Rule | Example |
|---|---|---|
| 10% of any number | Move decimal one place left | 10% of 85 = 8.5 |
| 15% tip | 10% + half of 10% | 15% of $40 = $4 + $2 = $6 |
| 20% tip | 10% doubled | 20% of $65 = $6.50 x 2 = $13 |
| Multiply by 11 | Add digits, place sum in middle | 36 x 11 = 3(3+6)6 = 396 |
| Double and halve | Double one factor, halve the other | 14 x 15 = 7 x 30 = 210 |
| Difference of squares | (a+b)(a-b) = a2 - b2 | 23 x 17 = (20+3)(20-3) = 400-9 = 391 |
| Rule of 72 | 72 / interest rate = years to double | At 8% return: 72/8 = 9 years to double |
Powers and Roots Quick Reference
Knowing common squares, cubes, and roots helps you estimate answers and spot errors in calculations.
Perfect Squares (1-20)
| n | n2 | n | n2 |
|---|---|---|---|
| 1 | 1 | 11 | 121 |
| 2 | 4 | 12 | 144 |
| 3 | 9 | 13 | 169 |
| 4 | 16 | 14 | 196 |
| 5 | 25 | 15 | 225 |
| 6 | 36 | 16 | 256 |
| 7 | 49 | 17 | 289 |
| 8 | 64 | 18 | 324 |
| 9 | 81 | 19 | 361 |
| 10 | 100 | 20 | 400 |
Common Square Roots
| Value | Square Root | Approx. |
|---|---|---|
| 2 | 1.4142 | ~1.41 |
| 3 | 1.7321 | ~1.73 |
| 5 | 2.2361 | ~2.24 |
| 7 | 2.6458 | ~2.65 |
| 10 | 3.1623 | ~3.16 |
| 50 | 7.0711 | ~7.07 |
| 100 | 10.0000 | 10 |
Common Calculation Mistakes to Avoid
Even experienced students and professionals make arithmetic errors. Here are some of the most frequent pitfalls and how to prevent them:
- Ignoring the order of operations: Writing 2 + 3 x 4 = 20 instead of the correct answer, 14. Always perform multiplication and division before addition and subtraction unless parentheses dictate otherwise.
- Sign errors: Subtracting a negative number is equivalent to adding its positive counterpart. For example, 5 - (-3) = 8, not 2.
- Division by zero: Dividing any number by zero is undefined. Always check for a zero denominator before performing a division.
- Rounding too early: When working through multi-step problems, keep full precision until the final step to avoid accumulating rounding errors.
- Misplacing decimal points: A misplaced decimal can change a result by a factor of 10 or more. Double-check placement, especially in money and measurement calculations.
Practical Applications of Basic Math
Basic arithmetic is not just an academic exercise -- it underpins countless real-world decisions. Here are some everyday scenarios where strong math skills save you time and money:
Shopping and Discounts
Quickly calculate sale prices: a 30% discount on a $85 item means you pay 70% of $85 = $59.50. Stack discounts: a 20% off coupon on an already 25% off item is not 45% off -- it is 1 - (0.80 x 0.75) = 40% off.
Cooking and Recipes
Scaling recipes requires multiplying or dividing all ingredients. Doubling a recipe that calls for 3/4 cup of flour means using 3/4 x 2 = 1.5 cups. Converting units (tablespoons to cups, grams to ounces) uses multiplication by fixed ratios.
Budgeting and Finance
Monthly budgets rely on addition and subtraction. Calculating compound interest, loan payments, and investment returns all start with basic operations. Even understanding your paycheck stub requires division and percentage calculations.
Home Improvement
Measuring rooms for paint, flooring, or furniture requires area calculations (length x width). Estimating materials for a project requires multiplication and adding waste factors (typically 10-15% extra).