What is behind this idea?

[Guest]

Hi!

So, I know basically nothing about math beyond basic algebra from school. I’m sure this is totally simple, but I don’t know the terminology well enough to find what it is I’m looking for on my own.
I’m really unsure how to explain what I want to talk about, so here’s how I noticed it:

I enjoy a variety of metaphysical interests, and one is numerology. PLEASE DON’T RUN AWAY.

In numerology, you find “power numbers” that correspond with different energies in much the same way astrology works (sort of a jungian thing- right?). If you search for your number online, it will probably present a program that breaks your birth date down into one digit.

For example, I was born on the 19th of the month. This number is reduced to one digit by adding its digits, so 1+9=10; 1+0=1.

So, my numerology number is 1. This means a lot of things that are completely irrelevant right now.

One can assign these numbers to other things in other ways. In sigil craft, a practice that abstracts words into symbols for various purposes, one can employ a cipher that turns words into numbers.

1 2 3 4 5 6 7 8 9
a b c d e f g h i
j k l m n o p q r
s t u v w x y z

Such that “cat” = (c)3+(a)1+(t)2= 6
And “dog” = (d)4+(o)6+g(7)= 17; 1+7= 8
If it interests you how to get from there to a symbol- search some sigil craft yo!

So after all this, I wondered if numbers could just reduce like this and still function in some way, so I wrote

63
+49
112

So then 63 became 9, 49 became 13 which then became 4.
9+4= 13, 1+3= 4
and then saw that for 112, 1+1+2= 4.

So the sum of the digits of the original numbers in an addition equation are the same as the sum of the digits of the sum of the equation itself- does that make sense?

I tried again with
87 15 6
+425 11 2
512 8
8

Then I said “you’re being crazy” so I tried 3 big numbers added together-

18753742 37 10 1
82203420 21 3
+76632111 27 9
177589273 13
49 4
13
4

And they still reduce! What is this called? I know like nothing about math, but I love this kind of stuff and I’m excited that I stumbled into it. I would love to know what the principle behind it is so I can learn more.

Thanks for sticking with me!
(I hope the numbers above made sense! Let me know if not and I’ll try to clarify!)

[Guest]

You've stumbled on the fact that digital roots https://en.wikipedia.org/wiki/Digital_root tell you the remainder of the number when divided by 9.

You can write 2674
= 2x1000 + 6x100 + 7x10 + 4
= 2x(999+1) + 6x(99+1) + 7x(9+1) + 4
= 2x999+2 + 6x99+6 + 7x9+7 + 4
= (2x999 + 6x99 + 7x9) + 2+6+7+4
= (9x2x111 + 9x6x11 + 9x7) + 2+6+7+4
= 9x(2x111+6x11+7) + 2+6+7+4
So the digital root of 2674 which is 2+6+7+4=19 differs from 2674 by some multiple of 9 so they must have the same remainder when divided by 9.
Similarly 19 divided by 9 must have the same remainder as 1+9=10
and 10 must have the same remainder as 1+0=1
So in fact 2674 must have remainder 1 when divided by 9 [which it does].
Hopefully you see from the example why this always works.

This means you get things like digital root(a) + digital root(b) = digital root (a+b) because remainders of a and b add to give the remainder of a+b
Similarly subtraction and multiplication work
34 x 17 = 578 --> 5+7+8=20 --> 2+0=2
3+4=7, 1+7=8
7x8=56 --> 5+6=11 --> 1+1=2