Digital root

# Digital root

Discussion

Encyclopedia
The digital root of a number is the (single digit) value obtained by an iterative process of summing digits
Digit sum
In mathematics, the digit sum of a given integer is the sum of all its digits,...

, on each iteration using the result from the previous iteration to compute a digit sum. The process continues until a single-digit number is reached.

For example, the digital root of is , because and

Digital roots can be calculated with congruence
Congruence relation
In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...

s rather than by adding up all the digits, a procedure that can save time in the case of very large numbers.

Digital roots can be used as a sort of checksum
Checksum
A checksum or hash sum is a fixed-size datum computed from an arbitrary block of digital data for the purpose of detecting accidental errors that may have been introduced during its transmission or storage. The integrity of the data can be checked at any later time by recomputing the checksum and...

. For example, since the digital root of a sum is always equal to the digital root of the sum of the summands' digital roots. A person adding long columns of large numbers will often find it reassuring to apply casting out nines
Casting out nines
Casting out nines is a sanity check to ensure that hand computations of sums, differences, products, and quotients of integers are correct. By looking at the digital roots of the inputs and outputs, the casting-out-nines method can help one check arithmetic calculations...

to his or her result—knowing that this technique will catch the majority of errors.

Digital roots are used in Western numerology
Numerology
Numerology is any study of the purported mystical relationship between a count or measurement and life. It has many systems and traditions and beliefs...

, but certain numbers deemed to have occult significance (such as 11 and 22) are not always completely reduced to a single digit.

The number of times the digits must be summed to reach the digital sum is called a number's additive persistence
Persistence of a number
In mathematics, the persistence of a number is a term used to describe the number of times one must apply a given operation to an integer before reaching a fixed point, i.e...

; in the above example, the additive persistence of 65,536 is 2.

## Significance and formula of the digital root

It helps to see the digital root of any positive integer as the position holds with respect to the last multiple of nine to the left of . For example, the digital root of 11 is 2, which means that 11 is the second number after 9. The digital root of 23 is 5, this means that 23 is the fifth number after a multiple of nine to the left of 23; in this case, 18. The digital root of 2035 is 1 which means that 2035-1, that is 2034, is a multiple of nine.

The digital roots of {1,2,3,4,5,6,7,8} which are the same digits themselves, reveal their position with respect to 0. The digital roots of nine and all of its multiples are nine, however, they all play the same role that zero plays for the integers from 1 to 8. It helps to think of the number nine and all its multiples as a kind of zero or zeros, so that the other integers be able to reveal their position or digital roots with respect to them. This is in part the nature of the decimal system.

With this in mind we may think of the digital root of the positive integer as , defined by:
which precisely says that,

This formula will give the digital root of and will assign the value 0 to all which are multiples of nine.

## Abstract multiplication of digital roots

The table below shows the digital roots produced by the familiar multiplication table
Multiplication table
In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....

in the decimal system. You can see that for example, 2x5=1; that's because the digital root of 10 is 1 or
1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 1 3 5 7 9
3 3 6 9 3 6 9 3 6 9
4 4 8 3 7 2 6 1 5 9
5 5 1 6 2 7 3 8 4 9
6 6 3 9 6 3 9 6 3 9
7 7 5 3 1 8 6 4 2 9
8 8 7 6 5 4 3 2 1 9
9 9 9 9 9 9 9 9 9 9

The table shows a number of interesting patterns and symmetries and is known as the Vedic square
Vedic square
In ancient Indian mathematics, a Vedic square is a variation on a typical 9 × 9 multiplication table. The entry in each cell is the digital root of the product of the column and row headings i.e...

.

## Formal definition

Let denote the sum of the digits of . Eventually the sequence becomes constant. Let (the digital sum of ) represent this constant value.

### Example

Let us find the digital sum of .

Thus,

For simplicity let us agree simply that

### Proof that a constant value exists

How do we know that the sequence eventually becomes constant? Here's a proof:

Let , with (For all , is an integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...

greater than or equal to and less than ). Then, . This means that , unless , in which case is a one-digit number. Thus, repeatedly using the function would cause to decrease by at least 1, until it becomes a one-digit number, at which point it will stay constant, as .

## Congruence formula

The formula is:
or,

To generalize the concept of digital roots to other bases b, one can simply change the 9 in the formula to b - 1.

The digital root is the value modulo 9 because and thus so regardless of position, the value mod 9 is the same – – which is why digits can be meaningfully added. Concretely, for a three-digit number,

To obtain the modular value with respect to other numbers n, one can take weighted sums, where the weight on the kth digit corresponds to the value of modulo n, or analogously for for different bases. This is simplest for 2, 5, and 10, where higher digits vanish (since 2 and 5 divide 10), which corresponds to the familiar fact that the divisibility of a decimal number with respect to 2, 5, and 10 can be checked by the last digit (even numbers end in 0, 2, 4, 6, or 8).

Also of note is since and thus taking the alternating sum of digits yields the value modulo 11.

## Some properties of digital roots

• Digital root of a square
Square number
In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself...

is 1, 4, 7, or 9.
• Digital root of a perfect cube is 1, 8 or 9.
• Digital root of a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

(except 3) is 1, 2, 4, 5, 7, or 8.
• Digital root of a power of 2
Power of two
In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....

is 1, 2, 4, 5, 7, or 8. Digital roots of the powers of 2 progress in the sequence 1, 2, 4, 8, 7, 5, 1. This even applies to powers less than 1; for example, 2 to the power of 0 is 1; 2 to the power of -1 (minus one) is .5, with a digital root of 5; 2 to the power of -2 is .25, with a digital root of 7; and so on, ad infinitum in both directions.
• Digital root of an even perfect number
Perfect number
In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors i.e...

(except 6) is 1.
• Digital root of a star number
Star number
A star number is a centered figurate number that represents a centered hexagram, such as the one that Chinese checkers is played on.The nth star number is given by the formula 6n + 1...

is 1 or 4.
• Digital root of a nonzero multiple
Multiple (mathematics)
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...

of 9 is 9.
• Digital root of a nonzero multiple
Multiple (mathematics)
In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, we say that b is a multiple of a if b = na for some integer n , which is called the multiplier or coefficient. If a is not zero, this is equivalent to saying that b/a is an integer...

of 3 is 3, 6 or 9.
• Digital root of a triangular number
Triangular number
A triangular number or triangle number numbers the objects that can form an equilateral triangle, as in the diagram on the right. The nth triangle number is the number of dots in a triangle with n dots on a side; it is the sum of the n natural numbers from 1 to n...

is 1, 3, 6 or 9.
• Digital root of a factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

≥ 6! is 9.
• Digital root of Fibonacci Series is a repeating pattern of 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9.
• Digital root of the product of twin primes
Twin prime
A twin prime is a prime number that differs from another prime number by two. Except for the pair , this is the smallest possible difference between two primes. Some examples of twin prime pairs are , , , , and...

, other than 3 and 5, is 8. The digital root of the product of 3 and 5 (twin primes) is 6.
• Digital root of a non-zero number is 9 if and only if the number is itself a multiple
Multiple
The word multiple can refer to:*Multiple , multiples of numbers*List of multiple discoveries, instances of scientists, working independently of each other, reaching similar findings...

of 9