Modulo in the Order of Operations
Understand how modulo operations fit into PEMDAS/BODMAS order of operations
Modulo Calculator
The number being divided
The number to divide by (must be non-zero)
Modulo Result
Division by zero is undefined
Modulo in Order of Operations
PEMDAS/BODMAS Order
1. Parentheses/Brackets - Expressions in parentheses first
2. Exponents/Orders - Powers and roots
3. Multiplication, Division, Modulo - Left to right (same precedence)
4. Addition, Subtraction - Left to right (same precedence)
Key Points about Modulo
• Modulo (%) has the same precedence as multiplication and division
• Operations of equal precedence are evaluated left to right
• 2 * 3 % 4 = (2 * 3) % 4 = 6 % 4 = 2
• 3 % 4 * 2 = (3 % 4) * 2 = 3 * 2 = 6
What is Modulo?
The modulo operator (%) returns the remainder after division.
Formula:
a mod n = r
where a = b × n + r
Examples:
• 21 mod 5 = 1 (because 21 = 4 × 5 + 1)
• 23 mod 10 = 3 (because 23 = 2 × 10 + 3)
• 3 mod 10 = 3 (because 3 = 0 × 10 + 3)
Programming vs Mathematics
Programming Languages:
Modulo (%) has same precedence as * and /
2 * 3 % 4 → 6 % 4 → 2
Mathematics:
Sometimes modulo has higher precedence
3 mod 4 * 2 might mean 3 in mod 8
Use parentheses to avoid confusion!
Common Examples
15 % 7 + 2
= 1 + 2 = 3
20 - 8 % 3
= 20 - 2 = 18
(5 + 3) % 4
= 8 % 4 = 0
12 / 3 % 5
= 4 % 5 = 4
Understanding Modulo in Order of Operations
What is the Modulo Operator?
The modulo operator returns the remainder of a division operation. If we divide number 'a' by number 'n', the modulo operation gives us the remainder 'r' where 0 ≤ r < n.
Is Modulo the Same as Division?
No, modulo is related to division but different:
- •7 ÷ 2 = 3.5 (regular division)
- •7 // 2 = 3 (integer division)
- •7 % 2 = 1 (modulo - remainder)
Where is Modulo in PEMDAS?
PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) doesn't explicitly include modulo, but most programming languages place it at the same level as multiplication and division.
Precedence Rules
- 1. Parentheses: Expressions in parentheses first
- 2. Exponents: Powers and roots
- 3. Multiplication, Division, Modulo: Equal precedence, left to right
- 4. Addition, Subtraction: Equal precedence, left to right