One Upping the Babylonian Method on 254

The ‘#BabylonianMethod' to get the #squareroot of any positive number.

We must set an error for the final result. Say, smaller than 0.001. In other words we will try to find the square root value with at least 2 correct decimal places.

• Step 1: 
•  Divide the number (254) by 2 to get the first guess for the square root .
•  First guess = 254/2 = 127.
• Step 2:
•  Divide 254 by the previous result. d = 254/127 = 2.
•  Average this value (d) with that of step 1: (2 + 127)/2 = 64.5 (new guess).
•  Error = new guess - previous value = 127 - 64.5 = 62.5.
•  62.5 > 0.001. As error > accuracy, we repeat this step again.
• Step 3:
•  Divide 254 by the previous result. d = 254/64.5 = 3.9379844961.
•  Average this value (d) with that of step 2: (3.9379844961 + 64.5)/2 = 34.2189922481 (new guess).
•  Error = new guess - previous value = 64.5 - 34.2189922481 = 30.2810077519.
•  30.2810077519 > 0.001. As error > accuracy, we repeat this step again.
• Step 4:
•  Divide 254 by the previous result. d = 254/34.2189922481 = 7.4227785014.
•  Average this value (d) with that of step 3: (7.4227785014 + 34.2189922481)/2 = 20.8208853748 (new guess).
•  Error = new guess - previous value = 34.2189922481 - 20.8208853748 = 13.3981068733.
•  13.3981068733 > 0.001. As error > accuracy, we repeat this step again.
• Step 5:
•  Divide 254 by the previous result. d = 254/20.8208853748 = 12.1992890998.
•  Average this value (d) with that of step 4: (12.1992890998 + 20.8208853748)/2 = 16.5100872373 (new guess).
•  Error = new guess - previous value = 20.8208853748 - 16.5100872373 = 4.3107981375.
•  4.3107981375 > 0.001. As error > accuracy, we repeat this step again.
• Step 6:
•  Divide 254 by the previous result. d = 254/16.5100872373 = 15.3845340942.
•  Average this value (d) with that of step 5: (15.3845340942 + 16.5100872373)/2 = 15.9473106658 (new guess).
•  Error = new guess - previous value = 16.5100872373 - 15.9473106658 = 0.5627765715.
•  0.5627765715 > 0.001. As error > accuracy, we repeat this step again.
• Step 7:
•  Divide 254 by the previous result. d = 254/15.9473106658 = 15.9274504224.
•  Average this value (d) with that of step 6: (15.9274504224 + 15.9473106658)/2 = 15.9373805441 (new guess).
•  Error = new guess - previous value = 15.9473106658 - 15.9373805441 = 0.0099301217.
•  0.0099301217 > 0.001. As error > accuracy, we repeat this step again.
• Step 8:
•  Divide 254 by the previous result. d = 254/15.9373805441 = 15.9373743569.
•  Average this value (d) with that of step 7: (15.9373743569 + 15.9373805441)/2 = 15.9373774505 (new guess).
•  Error = new guess - previous value = 15.9373805441 - 15.9373774505 = 0.0000030936.
•  0.0000030936 <= 0.001. As error <= accuracy, we stop the iterations and use 15.9373774505 as the square root.
• So, we can say that the square root of 254 is 15.93737 with an error smaller than 0.001 (in fact the error is 0.0000030936). this means that the first 5 decimal places are correct. Just to compare, the returned value by using the javascript function 'Math.sqrt(254)' is 15.937377450509228.
  Note: There are other ways to calculate square roots. This is only one of them.

So I was curious about finding simpler way to convert Metric to Imperial and back again and found this page above about the Babylonian Method. So Google says (as everywhere else) that 1 in = 25.4 mm. Too inelegant. And Metric! :)

I tried the old "Please Excuse My Dear Aunt Sally” to sq. root 254 and that’s when I discovered the ancient art of the Babylonians below. Too old and long. Its 8 steps. Simpler needed.

I then wanted to find out about the conversion of millimeters to inches and saw a site with a unique way of finding the sq. root of things such as 254 in this case. So I tried converting the Imperial 1 inch into a decimal (0.1) to see what would happen then by solving the equation knowing that 2.54 in = 0.1 mm. So then 0.1 in = 27.432. (i.e. 1.08)  What? Now they’re BOTH decimals. How’d they pull that off?

I ended up seeing how 0.0197 inches = 0.50038 mm.The  0.5 mm rounded is still very accurate only off 3 thousandths and easy to work w/. A nice constant along w/ the 0.2 in. So we can conclude that 1.0 mm = 0.4 in. rounded out. So we have a 50% and a 20% constant. Helpful but unworkable still.

So according to the Metric Rebels (think Star Wars) they see us a just another decimal. Metric and Imperial literally don’t add up rationally. That’s gut push to. my theory. But I pressed on, obsessed to see if its still there. In this flick we’re the Imperial Stormtroopers. :)

So I noticed that the results are about 10% different with that first conversion but it gets worse as u go. It doesn’t equalize and level off. The difference keeps growing about 4% everytime u convert fro metric to imperial and back. But we learn that the metric system has a ‘constant' of sorts. An easy decimal we’re all used to on both sides of the measurement divide. 5%. So now I had a ‘constant.’ Sorta.

5% of as an example of lets say, 25 mm’s as an ex. = 1.25 mms or 5% of 25.4 mm = 1.27 mm's. 25 is great to use as an ex. bcz the decimals increase exactly* 0.25 for 1%, .50 for 2%, etc. (*its actually 0.254!) 1.27 x .95 should be 100% or 1 inch but no its 1.0541. 95% of 25.4 is 24.13 while 5% of 25.4 is 24.13. 24.13 +  1.27 = 25.4. 

That 1.27 (25%) isn't insignificant bcz we kinda ’solved’ the equation w/ a decimal to convert from decimals. So if we pick as an example a number like 33 inches and convert using the 1.27 number we get 41.91 mm’s or 1.65 inches. Not wonderful math but different. :) But we press on.  

So lets simplify again. Let’s convert back to metric again w/ the 41.91 mm’s to get the total 100% total..  41.91 mm = 1.65 inches. If 95% of 41.91 mm = 39.8145 mm and 5% of 41.91 = 0.0825 mm. Then how does the total found in the equation 0.0825 mm + 39.8145 mm = 39.897 mm happen? :) So now we’ve seen it ADD 2.013% too! Alien numbers!

29.16 inches = 1 centimeter = 2.916 mm’s. So knowing that we can ANOTHER 10% off from 27. Its getting fuzzier as we go! So let’s erase the 33% example and we get 317.5524. (962.28 x .33 = 317.5524.) If we convert the 10% back out we get 31.75524

Now lets’s take that and simplify again. 31.75524 mm's = 1 inch. Now we’re off another 4.75524. Its getting worse and worse.  Now we’re up to 20% off! or 1.20783096 inches. 

Review: We made the conversion ‘universal' by solving for two ‘constants.’ 254 & 10. That’s more than enuf to do some teeny weeny algebraic equations on the fly. If our problem is what is 4 inches in metric? The equation is: 4 in x 2.54 mm = 1.08 inches. Notice that even without the long tail of number beyond 2.54 the equation solves 1 only 1 decimal point. That’s not enuf for certainty. i.e. if u think about it the Metric System was designed (I didn’t claim planned!) to make inches into decimals themselves! Its like an alien movie where the foreigner/aliens impregnate and lay eggs and it looks like  either 2 eggs got into inch or is that 8 mean 8 eggs. Hmmm. (I actually try to figure these out and u’d be surprised how many simple things just weren’t studied bcz everyone said it was impossible. I wasted a lot of my life believing that ’smarter’ ppl than me made the rules and I should follow them. I’ve learned to chuck all the rules of man out. God is my only ruler. :)

Convinced yet? Of what?! That the Metric System is like an alien brood trying to convert his all to metric and decimals. They’re already defined us as 1.08 inches converted from inches into metric and back to inches. And it gets worse each time we try to ’solve’ it. My advice is throw away all metric rulers and don’t buy foreign cars. ;) 

And I thot I could ‘beat’ the Babylonians! hah!

SO let’s review since I packed a lot in.

The ‘Constants;’ 10 Imperial and 254 Metric

Metric converted to Imperial’s simplified decimal is 2.54 mm = 1 inch.

BUT, Imperial converted to Metric is 1.08, NOT 1 inch. Hmmmm…They planned this I’m sure! Its their plot to take over the numbers. Have some Pi on me.