Calculate Digit Sum

Calculator for the digit sum (sum of digits) of a natural number

Digit Sum Calculator

What is the digit sum?

The digit sum (sum of digits) of a number is the sum of all its digits. It is computed by adding all individual digits together.

Digit sum of 12846 = ?
1 + 2 + 8 + 4 + 6 = 21
Enter number
Integer ≥ 0
Calculation result
Digit sum =
Calculation: Sum of all digits of the entered number

Digit Sum Info

Properties

Digit sum: Sum of all digits of a number

Sum of digits QS digital root

Rule: Add all digits of a number
Application: Divisibility tests, check digits

Quick examples
123: 1 + 2 + 3 = 6
456: 4 + 5 + 6 = 15
789: 7 + 8 + 9 = 24
1000: 1 + 0 + 0 + 0 = 1


Formulas and properties of the digit sum

Definition
\[Q(n) = \sum_{i=0}^{k} d_i\] \[\text{where } n = \sum_{i=0}^{k} d_i \cdot 10^i\]

Sum of all digits d₀, d₁, ..., dₖ of number n

Recursive computation
\[Q(n) = \begin{cases} n & \text{if } n < 10 \\ Q(\lfloor n/10 \rfloor) + (n \bmod 10) & \text{otherwise} \end{cases}\]

Recursive definition via digit extraction

Divisibility property
\[n \equiv Q(n) \pmod{9}\] \[n \text{ divisible by } 9 \Leftrightarrow Q(n) \text{ divisible by } 9\]

Digit-sum rule for divisibility by 9

Divisibility by 3
\[n \equiv Q(n) \pmod{3}\] \[3|n \Leftrightarrow 3|Q(n)\]

Digit-sum rule for divisibility by 3

Iterated digit sum
\[Q^*(n) = \begin{cases} n & \text{if } n < 10 \\ Q^*(Q(n)) & \text{otherwise} \end{cases}\]

Digital root via repeated digit sums

Properties
\[Q(a + b) \leq Q(a) + Q(b)\] \[Q(a \cdot b) \not= Q(a) \cdot Q(b) \text{ (in general)}\]

Digit sum is not multiplicative, but subadditive

Example calculations with digit sums

Example 1: Digit sum of 12846
Step 1: Decompose number into digits
1 2 8 4 6
12846 = 1×10⁴ + 2×10³ + 8×10² + 4×10¹ + 6×10⁰
Step 2: Add all digits
Q(12846) = 1 + 2 + 8 + 4 + 6
= 3 + 8 + 4 + 6 = 11 + 10 = 21
Digit sum = 21
\[Q(12846) = 1 + 2 + 8 + 4 + 6 = 21\]
Example 2: Divisibility test
Question: Is 12846 divisible by 3?
Digit sum: Q(12846) = 21
Check: 21 ÷ 3 = 7 → 21 is divisible by 3
Answer: Yes, 12846 is divisible by 3 (12846 ÷ 3 = 4282)
More examples
Q(123) = 6
1 + 2 + 3 = 6
Q(999) = 27
9 + 9 + 9 = 27
Q(1000) = 1
1 + 0 + 0 + 0 = 1
Q(9876) = 30
9 + 8 + 7 + 6 = 30
Divisibility checks
By 3: Is Q(n) divisible by 3?
By 9: Is Q(n) divisible by 9?
Examples:
123: Q=6, 6÷3=2 ✓
999: Q=27, 27÷9=3 ✓
Properties and applications
Divisibility
Test for 3 and 9
Check digits
ISBN, account numbers
Digital root
Iterated digit sum
Modular
n ≡ Q(n) (mod 9)

The digit sum simplifies divisibility checks and error detection

Applications of the digit sum

The digit sum has many practical and theoretical applications:

Divisibility tests
  • Test divisibility by 3 and 9
  • Quick checks for large numbers
  • Mathematical proofs and theorems
  • Simplify modular arithmetic
Check digits & error detection
  • ISBN numbers of books
  • Account numbers and IBANs
  • Credit card numbers (Luhn algorithm)
  • ID numbers
Computer Science & programming
  • Hash functions and checksums
  • Data validation and integrity
  • Algorithms for digit manipulation
  • Recursive programming exercises
Game theory & puzzles
  • Digital root and numerology
  • Mathematical puzzles and riddles
  • Card games and magic tricks
  • Gambling strategies

Digit sum: Simple concept with many applications

The digit sum is one of the simplest and most useful concepts in elementary number theory. What begins as simple addition of digits turns into a powerful tool for divisibility checks, error detection and mathematical proofs. The remarkable property n ≡ Q(n) (mod 9) makes the digit sum a key tool in modular arithmetic and reveals deeper mathematical structure. From primary education to cryptography, the digit sum exemplifies the elegance and utility of elementary mathematics.

Basic properties
  • Simple computation: add digits
  • Recursively definable
  • Modular equivalence: n ≡ Q(n) (mod 9)
  • Subadditivity: Q(a+b) ≤ Q(a)+Q(b)
Practical significance
  • Divisibility tests for 3 and 9
  • Check digits in identification numbers
  • Error detection in data transmission
  • Digital signature procedures
Mathematical connections
  • Digital root by iteration
  • Congruence arithmetic modulo 9
  • Understanding positional number systems
  • Learning recursive algorithms
Summary

The digit sum demonstrates how the simplest mathematical ideas give rise to powerful practical and theoretical applications. Adding digits leads to elegant divisibility criteria, enables error detection in critical systems and provides insight into the structure of our decimal system. From elementary education to modern computer science, the digit sum remains a compelling example of mathematical beauty and utility.