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Arithmetic Calculator

Add, subtract, multiply & divide — plus expression evaluator, long division & LCM/GCD tools

Enter Two Numbers

Select an operation and see results instantly.

Number A

Number B

Operation

Result

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Expression Evaluator

Evaluates with correct BODMAS/PEMDAS order of operations. Supports parentheses, decimals, and negatives.

Enter Expression

Use × or * for multiply, ÷ or / for divide, − or - for subtract. Parentheses ( ) are supported.

Result

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LCM & GCD Calculator

Enter up to 4 positive integers. Shows prime factorization and step-by-step working.

Number 1

Number 2

Number 3 (opt)

Number 4 (opt)

Quick Formulas Reference

Addition A + B = Sum Subtraction A − B = Difference Multiplication A × B = Product Division A ÷ B = Quotient (B ≠ 0)

BODMAS Order: Brackets → Orders → Division/Multiply → Add/Subtract GCD: Euclidean algorithm (gcd(a,b) = gcd(b, a mod b)) LCM: LCM(a,b) = |a × b| ÷ GCD(a,b)

Long division: Dividend = Divisor × Quotient + Remainder | GCD also known as HCF (Highest Common Factor)

Worked Examples

Order of Operations

3 + 4 × 2 − 1

Step 1 (Multiply): 4 × 2 = 8 Expression: 3 + 8 − 1 Step 2 (Add left→right): 3 + 8 = 11 Expression: 11 − 1 Step 3 (Subtract): 11 − 1 = 10 Result = 10

Multiplication happens before addition/subtraction per BODMAS.

Long Division

156 ÷ 12 = 13

Step 1: 12 into 15 → 1 time 1 × 12 = 12; 15 − 12 = 3 Step 2: Bring down 6 → 36 12 into 36 → 3 times 3 × 12 = 36; 36 − 36 = 0 Quotient = 13, Remainder = 0

Each digit is processed from left to right.

LCM & GCD

LCM(12, 18) = 36 | GCD(12, 18) = 6

12 = 2² × 3 18 = 2 × 3² GCD = 2¹ × 3¹ = 6 (lowest powers) LCM = 2² × 3² = 36 (highest powers) Check: 12 × 18 = 216 = 6 × 36 ✓

LCM × GCD always equals the product of the two numbers.

What is Arithmetic?

Arithmetic is the oldest and most fundamental branch of mathematics. It encompasses the four basic operations — addition, subtraction, multiplication, and division — applied to numbers. Mastering arithmetic is the first step to all higher mathematics, from algebra and geometry to calculus and statistics. These operations appear everywhere in daily life: splitting a restaurant bill, calculating a discount, determining fuel consumption, or working out a timetable.

The Four Basic Operations

Addition (+) combines two or more quantities into a total called the sum. For example, 8 + 5 = 13. Addition is commutative (order doesn't matter: a + b = b + a) and associative (grouping doesn't matter).

Subtraction (−) finds the difference between two quantities by taking one away from another. For example, 13 − 5 = 8. Subtraction is neither commutative nor associative — order matters.

Multiplication (×) is repeated addition. Multiplying 6 × 4 means adding 6 four times: 6 + 6 + 6 + 6 = 24. The result is the product. Multiplication is commutative: a × b = b × a.

Division (÷) splits a quantity into equal parts. Dividing 24 ÷ 6 means how many groups of 6 fit into 24, giving a quotient of 4. Division by zero is undefined — it has no mathematical meaning.

BODMAS / PEMDAS — Order of Operations

When an expression contains multiple operations, the order in which you perform them matters enormously. The rules are captured by the acronyms BODMAS (used in the UK, India, and Australia) and PEMDAS (used in the USA):

Step	BODMAS	PEMDAS	Example
1	Brackets ( )	Parentheses ( )	2 × (3 + 1) = 2 × 4 = 8
2	Orders (powers, roots)	Exponents	2³ = 8
3	Division and Multiplication	Multiplication and Division	6 ÷ 2 × 3 = 9 (left to right)
4	Addition and Subtraction	Addition and Subtraction	10 − 3 + 2 = 9 (left to right)

Division and multiplication have equal precedence — they are evaluated left to right. Similarly, addition and subtraction are equal in precedence. Parentheses always override any other rule, which is why they are so useful when writing clear expressions.

Applications of Arithmetic

Arithmetic operations underpin virtually every quantitative task humans perform. In finance, they calculate interest, taxes, and profits. In engineering, they determine forces, measurements, and tolerances. In everyday cooking, arithmetic converts recipe quantities and scales serving sizes. In science, arithmetic is used to analyse data, compute ratios, and verify experimental results. Even digital computers ultimately reduce all computation to binary arithmetic at the hardware level.

The Long Division tool on this page is particularly useful for understanding how calculators themselves perform division internally — it makes the implicit steps of the algorithm visible. The LCM and GCD tools are valuable in reducing fractions, synchronising repeating events, and solving problems in number theory.

Frequently Asked Questions

What is arithmetic?

Arithmetic is the branch of mathematics dealing with the basic operations on numbers: addition, subtraction, multiplication, and division. It is the foundation of all higher mathematics. Addition combines two quantities into a sum, subtraction finds the difference between quantities, multiplication is repeated addition, and division splits a quantity into equal parts. Arithmetic operations follow a strict order of precedence known as BODMAS or PEMDAS.

What is BODMAS or PEMDAS?

BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) are mnemonics that define the correct order of operations in arithmetic expressions. When evaluating an expression like 3 + 4 × 2, you must multiply first (giving 8) then add 3, yielding 11 — not 14. Parentheses override any other precedence. Both BODMAS and PEMDAS describe the same rule set, just with different terminology.

How do I divide fractions?

To divide fractions, use the "keep, change, flip" method: keep the first fraction as-is, change the division sign to multiplication, then flip (take the reciprocal of) the second fraction. For example, (3/4) ÷ (2/5) becomes (3/4) × (5/2) = 15/8 = 1.875. This works because dividing by a fraction is the same as multiplying by its reciprocal.

What is the difference between LCM and GCD?

LCM (Least Common Multiple) is the smallest positive integer that is divisible by all given numbers. GCD (Greatest Common Divisor), also called HCF, is the largest positive integer that divides all given numbers without a remainder. For example, for 12 and 18: the GCD is 6 (since 6 divides both evenly) and the LCM is 36 (the smallest number both 12 and 18 divide into). They are related by the formula: LCM × GCD = product of the two numbers.

How does long division work?

Long division is a method for dividing large numbers step by step. Start from the leftmost digit of the dividend. Determine how many times the divisor fits into the current portion (this is the quotient digit). Multiply the divisor by that digit, write the result below, and subtract. Bring down the next digit of the dividend and repeat. Continue until no digits remain. Any leftover value is the remainder. For example, 156 ÷ 12: 12 goes into 15 once (12), subtract to get 3, bring down 6 to make 36, 12 goes into 36 three times exactly, giving quotient 13 remainder 0.

Can this calculator handle decimals?

Yes. The Basic calculator accepts decimal inputs for Number A and Number B. The Expression Evaluator also handles decimal numbers within expressions. Results are displayed up to 10 significant figures to avoid floating-point noise. The Long Division tool works with integers and displays the remainder separately. LCM and GCD tools work with positive integers.

What are the rules for negative numbers in arithmetic?

Key rules for negative numbers: Adding a negative is the same as subtracting (5 + (−3) = 2). Subtracting a negative is the same as adding (5 − (−3) = 8). Multiplying or dividing two numbers with the same sign gives a positive result (−4 × −3 = 12). Multiplying or dividing numbers with opposite signs gives a negative result (−4 × 3 = −12). Division by zero is always undefined regardless of sign.

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