# Zero is not a Perfect Happy Number

## Happy Numbers

A Happy Number is a number that satisfies the following condition. If you square its digits, add them together, and then take the result and square its digits and add them together, and keep doing that over and over again, you come down to the number 1. eg. 32.

32: 3^{2} + 2^{2} = 9 + 4 = 13

13: 1^{2} + 3^{2} = 1 + 9 = 10

10: 1^{2} + 0^{2} = 1 + 0 = 1

The happy numbers below 500 are

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496.

## Perfect Numbers

What is a Perfect Number?

One that is the sum of all its divisors excluding itself.

For example, consider the number 6. Its divisors are 1, 2, 3 and 6.

Leaving 6, sum of all its divisors = 1 + 2 + 3 = 6. So 6 is a perfect number.

Examples: 6 (1+2+3), 28 (1+2+4+7+14), 496, 8128, etc., are perfect numbers.

A perfect number is a number that is half the sum of all of its positive divisors.

## Number Theory

## Zero

* *0 is a number to represent nothing. 0 is an integer between -1 and 1. 0 is neither positive nor negative. 0 is an even number as it is divisible by 2. 0 is used as a placeholder in place value system. 0 is the additive identity of Integers and Reals in Algebra.

The concept of zero as a number and not merely a symbol for separation is attributed to India. The Sanskrit word *Shunya *was used in Indian Texts to refer the concept of void. The oldest known text using a zero is the Jain text *Lokavibhaga* from India dated 458 AD. An early documented use of the zero by Indian Mathematician Brahmagupta in his work dates to 628 AD. He treated zero as a number and explained operations involving it, including division. In 9th century AD, practical calculations were carried out using zero, treating like any other number, even in the operation division. The first known use of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuja Temple at Gwalior in India, dated 876 AD.

The Hindu-Arabic numerals and the positional number system were introduced in 825 AD by a Persian Mathematician Muḥammad ibn Mūsā al-Khwārizmī** **from Baghdad in his book titled *al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala*. This book summarised Greek and Hindu knowledge and also contained his contributions to mathematics including an explanation of the use of zero. So we can conclude that zero was introduced by Indians.

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