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Digital root is a fascinating mathematical concept that has been studied for centuries. It is also known as the repeated digital sum, and is a value obtained by repeatedly summing the digits of a given number until a single-digit number is obtained. The digital root is a useful tool that can help simplify calculations, and can provide insights into patterns in numbers. In this article, we will discuss the definition of digital root, its history, and various applications.

The definition of digital root is straightforward: given a non-negative integer n, the digital root of n is defined as follows:

1. If n is a single-digit number, its digital root is itself.
2. Otherwise, the digital root of n is the same as the digital root of the sum of its digits.

For example, the digital root of 123 is calculated as follows: 1 + 2 + 3 = 6 The sum is not a single-digit number, so we repeat the process: 6 = 6 Therefore, the digital root of 123 is 6.

The digital root is not a new concept; it has been studied for centuries. The ancient Greek mathematician Pythagoras is credited with the discovery of the digital root, and it was a topic of interest among medieval numerologists. The concept of digital root has been used in various cultures throughout history, including in Chinese numerology and Indian Vedic mathematics.

The digital root is a useful tool that can simplify calculations in various areas of mathematics. For example, it can be used to quickly check the validity of multiplication problems. To use digital root for this purpose, simply take the digital roots of the numbers being multiplied, multiply them together, and take the digital root of the product. If the digital root of the product is the same as the digital root of the original problem, the multiplication problem is likely correct.

In addition to its practical applications, digital root is also a topic of interest among recreational mathematicians. There are many interesting properties and patterns associated with digital roots. For example, every multiple of 9 has a digital root of 9. This property can be explained by the fact that any multiple of 9 can be expressed as a sum of digits that add up to a multiple of 9. When we take the digital root of this sum, we get 9. Similarly, every power of 10 has a digital root of 1. This is because any power of 10 is a 1 followed by a certain number of zeros, and the sum of digits of any number consisting of only ones is equal to the number of digits. Therefore, the digital root of any power of 10 is equal to the digital root of 1.

There are also many interesting patterns associated with digital roots in prime numbers. For example, if we take the digital root of the sum of the digits of any prime number greater than 3, we get either 1, 4, 7, or 9. This property is known as the prime digital root conjecture, and has been proven to be true for all primes up to 10^13. There are also many other interesting patterns and properties associated with digital roots in prime numbers, and they continue to be an area of active research among mathematicians.

In addition to its use in number theory and algebra, digital root is also useful in geometry. For example, digital root can be used to classify triangles based on the lengths of their sides. A triangle with sides a, b, and c has a digital root of 1 if a, b, and c are all congruent mod 9, a digital root of 2 if two sides are congruent mod 9 and the third is not, and a digital root of 3 if no two sides are congruent mod 9. This classification scheme can be extended to other polygons as well.