User blog:TSRITW/Rectangl'r R.A.N.T.S. 2wo: Common Math Misconceptions

Yeah, there are some things we need to make clear.

1: 2+2=4–1=3
I don’t think they’re using the notation right. Otherwise, we’d have to conclude that 2+2=3, because of the transitive property of equality: If a=b and b=c, then a=c. However, 2+2≠3. Perhaps what they really mean is this: 2+2=4 4–1=3 Two separate equations. Reminds me of a lyric from O Alfabeto (translated): “One plus two is three; add three and the result is six.” It does not mean “1+2=3+3=6”, but “1+2=3; 3+3=6”.

2: (a+b)2=a2+b2
This is just trying to use the distributive property without knowing what it really means. The distributive property only applies to decreasing subsequent hyperoperations, so you can distribute multiplication to addition, and you can distribute exponents to multiplication, but you can’t distribute exponents to addition. That’s now how it works. (a+b)2 = (a+b)(a+b) = a2+2ab+b2

3: |a–b|=a+b
Well, no. That’s not how the modulus function works. |a–b| cannot be simplified any further.

4: Base 1 = tally marks
Those dumb kids on the Otherbaseblocks Wiki actually believe that. Base 1 actually uses strings of 1s. 11 = 110 111 = 210 1111 = 310 11111 = 410 0 would either be an empty string or just 0. Negative numbers just have a negative sign added at the front (or they don’t exist). You can’t express non-integers in Base 1. Here’s why: imagine a number as a polynomial.

+++ ... +++

b is the base of the number, and a is a whole number that... So, when b=1, we can substitute b for 1:
 * is smaller than |b| if b is an integer and |b|>1
 * is smaller than the reciprocal of |b| if b is a unit fraction and 1/|b|>1
 * can only be 1 or 0 if |b|=1 or |b|=0

+++ ... +++

Since 1 is the multiplicative property, 1x=1, where x is a real number. And 1x = x, so...

+++ ... +++ = +++ ... +++

That makes the 0 digit kinda useless in Base 1. And if we’re crazy enough to include negative exponents... well, nothing changes much. So yeah, Base 1 and tally marks aren’t the same thing, they’re just similar.

Any more?
Feel free to leave a comment to give me some more factoids to debunk.