The number 1 is among the happiest numbers, 4 is sad, and both are narcissistic
There are many different types of numbers, some of which you may remember from school: natural, rational, irrational, imaginary, calculable and incalculable numbers. Today, however, we are going to talk about something cheerful, namely, the “happy numbers.” Yes, they do indeed appear in mathematics, and that really is their technical name.
There’s a quick way to find out. To do this, you first need to check just how large the sum of squared digits of a number can become. Suppose you have a one-digit number, say, 9. Its square, 81, is larger than itself. The same is true for two-digit numbers such as 99: 92 + 92=162. This is not true, however, for numbers with three or more digits. Even for 999, the sum of the squares of its digits is smaller than the number itself, namely, 243.
But experts were not satisfied with this result. Mathematicians have also wondered, for example, what percentage of numbers are happy. Do they become rarer with increasing size, like the prime numbers, or do they always appear with about the same frequency? Such behavior is found, for example, in the set of all numbers beginning with a 1. Among the first nine numbers , there is exactly one that starts with a 1 , which corresponds to a density of 1⁄9. Among the first 19 numbers , there are 11 that start with a 1, giving a density of 11⁄19. And among the first 99 numbers, there are still 11 that start with a 1, so you have a density of 11⁄99=1⁄9 in this number interval.
Yet another puzzle is considering how many times the happy calculation is needed to bring a happy number to 1. This quantity can be used to define the overall happiness of a number. The fewer iterations, the happier the number. So 1, 10, 100, and so on are extremely happy, while 13 is slightly less so.
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