# sav2718

Submitted by Sav2718:

I saw that someone wrote as a comment on my business card:
“I saw something similar for fifth roots- getting the last digit right is a lot easier too!”.

So I just wanted to add a word about the rightmost digit of a perfect root in general:
The pattern for the rightmost digit is the same for every order of the form 4K+1 (the digit itself) and for 4K-1 (the same unique pattern as with cube roots). the same is true for roots of an even-order but an extra step of elimination is needed, two steps for even-order of the form 4K.

For example, lets take a look at some (perfect) square roots:

1^2=1
2^2=4
3^2=9
4^2=16
5^2=25
6^2=36
7^2=49
8^2=64
9^2=81
10^2=100
11^2=121
12^2=144
13^2=169
14^2=196
15^2=225
16^2=256
17^2=289
18^2=324
19^2=361
20^2=400
etc.

the pattern for the rightmost digit is always (0,)1,4,9,6,5,6,9,4,1(,0)
1 for 1 and 9 (the complement to 10)
4 for 2 and 8
9 for 3 and 7
6 for 4 and 6
5 for 5
a trail of 2N 0s (even number) in the right hand of the number beneath the root sign will be a trail of N 0s in right hand of the answer (ex: sqrt(15210000)=3900).

Now in order to calculate a given perfect square root lets look at an example:

sqrt(5329)

70^2(=4900)<5329<80^2(=6400)

or even more simplified:
7^2(=49)<53<8^2(=64)

so the leftmost digit of the answer is 7

5329 ends with a 9 so according to our pattern the rightmost digit is either 3 or 7 but how can we tell?

Let’s have a look at the trick for squaring a number that ends with 5:
(10a+5)^2=100a^2+2*10*5*a+5^2=100a^2+100a+25=100a*(a+1)+25

or even more simplified:

[a*(a+1)]&[25] (where “&” is the concatenation operator)

examples:
35^2=[3*4]&[25]=[12]&[25]=1225
75^2=[7*8]&[25]=[56]&[25]=5625
435^2=[43*44]&[25]=[1892]&[25]=189225

you may also prefer to calculate it as such:
35^2=[3^2+3]&[25]=[9+3]&[25]=[12]&[25]=1225
75^2=[7^2+7]&[25]=[49+7]&[25]=[56]&[25]=5625
435^2=[43^2+43]&[25]=[1849+43]&[25]=[1892]&[25]=189225

Now back to our square root:

we know sqrt(5329) is either 73 or 77 but we can quickly check that
75^2=[7*8]&[25]=[56]&[25]=5625>5329

so if 75 is too big, 77 is even bigger than the answer must be (correctly) 73.

Note that because a square of number ending with 5 always ends with 25 one doesn’t even need to calculate the whole thing and just have a look at 7*8 (7 times its successor).

Lets do another one quickly:

sqrt(107584)

32^2(=1024)<1075 (don’t need the ‘84’) <33^2(=1089)

32_

107584 ends with 4 so the rightmost digit is either 2 or 8

32^2+32=1024+32=1056<1075

too small so 322 is even smaller

so the answer is (correctly) 328.

As I already mentioned even-order roots from the form 4K need another step of elimination.

Higher numbers require more steps and non-perfect roots require algorithms from a different kind and approach but I am willing to write about those in the future.

He’s referring to this awesome submission.

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Submitted by Sav2718:

New business card sample. The back is an algorithm for calculating cube roots mentally and\or on paper (up to 6-digit numbers, higher numbers require extra steps and lookup tables) with examples.

That’s got to be the coolest business card I’ve ever seen.

There is a big difference between loving numbers and loving math.A mental calculator named Willem Bouman (set the world record for factoring 5 digit numbers) said he hates math and consider himself an "arithmetician". That's why I chose to focus on number theory and algorithms, so I could express my love for both math and numbers. From my experience with high level education (I studied math and physics in Tel Aviv university) not only you don't need to know arithmetic, but professors s*k at it.

Somehow, giving a girl tips on multiplication tables has turned into a discussion about the difference between mathematics and arithmetic. This is just one of the many reasons why I love running this blog. Everyone has so much to say!

Submission from sav2718:

Geometric art#2; “Discrete” by myself (“sav2718”, Gur Keren)

Artist Name: Gur Keren

Tumblr: http://sav2718.tumblr.com/

I do have synesthesia! I even got briefly mentioned in a final research project about synesthesia.

Wow! From Wikipedia:

Synesthesia is a neurological condition in which stimulation of one sensory or cognitive pathway leads to automatic, involuntary experiences in a second sensory or cognitive pathway. People who report such experiences are known as synesthetes. Recently, difficulties have been recognized in finding an adequate definition of synesthesia, as many different phenomena have been covered by this term and in many cases the term synesthesia (“union of senses”) seems to be a misnomer. A more accurate term for the phenomenon may be ideasthesia.
In one common form of synesthesia, known as grapheme (color synesthesia or color-graphemic synesthesia), letters or numbers are perceived as inherently colored, while in ordinal linguistic personification, numbers, days of the week and months of the year evoke personalities. In spatial-sequence, or number form synesthesia, numbers, months of the year, and/or days of the week elicit precise locations in space (for example, 1980 may be “farther away” than 1990), or may have a (three-dimensional) view of a year as a map (clockwise or counterclockwise). Yet another recently identified type, visual motion (sound synesthesia) involves hearing sounds in response to visual motion and flicker. Over 60 types of synesthesia have been reported, but only a fraction have been evaluated by scientific research. Even within one type, synesthetic perceptions vary in intensity and people vary in awareness of their synesthetic perceptions.

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The first part of a series of miniature drawings I’m working on.

Do you have to be good at arithmetic to obtain a phD in mathematics?

I have a bachelor’s in pure math, a masters in statistics, and am on my way to a PhD in the same—and I have dyscalculia. You do not need to be able to do arithmetic accurately to do higher maths. At all.-Anonymous

From my experience with high level education (I studied math and physics in Tel Aviv university) not only you don’t need to know arithmetic, but professors s*k at it.-Sav2718

I’ve had more than one professor who is horrible at arithmetic. You can get your calculator to do complicated arithmetic for you and it won’t lessen your understanding of what mathematics actually is at all.-Anonymous

Mathematician’s work isn’t just adding and multiplying of numbers. Mathematicians discover new things, so I want to say one important point: you need to know tables only to pass exams to uni.-Anonymous

In my experience of university math and physics, knowing (and understanding!) multiplication properties is far more useful than being able to multiply any two given numbers. (Except for labs, where you’re dealing with very precise values, I can only think of one class I’ve taken that uses actual numbers instead of variables.)-Endnotesandamperstands

There’s different levels of understanding. You can understand that math works, 5x5=25. You can understand how it works, 5 groups of 5 objects equals 25 object. But it’s very important, especially when looking for a degree in Mathematics, the why of math. That’s what makes math so beautiful, because understanding why something works, especially math, makes everything so much better.-The—-king

Submission from sav2718:

Geometric art#3; “Contradicting Modularities-Iteration” by myself (“sav2718”, Gur Keren)

Myself, solving some “mate in one” chess puzzles.

Maybe I’ll do a “mate in two” version…..

A fun fact- I have a personification type synesthesia to chess pieces.

Happy new year!

Submitted by Sav2718:

2012=2x2x503
2013=3x11x61
2014=2x19x53
2015=5x13x31

That’s four 3-almost-prime years in a row!

Thanks!