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json | {"to":"rjjb76519","cards":["C3-331-0ENHGKDVM8"],"app":"splinterlands/0.7.139","n":"fABjU3rBW7"} |
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json | {"type":"quest","quest_id":"70c89e76c74e3707229f80d4574f6867d90fc717","app":"splinterlands/0.7.190","n":"PJ0qqvCfS2"} |
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json | {"items":["cca12f09747d64a10c5c827d2bb153a6b9c1b6dc-0"],"currency":"DEC","days":1,"app":"splinterlands/0.7.139","n":"PQ8k8ds1Ad"} |
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1. | required_auths | |
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json | {"to":"chaseparker","cards":["C3-336-HPC2WTZ61C","C3-336-J48TZVV728","C3-336-I9R14C6PCW","C3-336-ILOV7DWVI8","C3-336-JCPQHO7GYO","C3-336-IT9U4I14G0","C3-336-HGVDLAX1WG","C3-336-GXPDAAYMR4","C3-336-H90W4WET6O","C3-336-I3VWAVCH40","C3-337-NOURKXKP4W","C3-337-O9FS24ZIFK","C3-337-OMRCWBXTCG","C3-337-NV70SG8ZHS","C3-337-ORW376ZMS0"]} |
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json | {"type":"quest","quest_id":"a7952810929ca5dc3fba83be2e8f8fe175b80357","app":"sl-mobile/0.7.139","sm_id":"oxJyRnQyBw"} |
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json | {"token":"DEC","qty":18.5,"to":"kent0327","memo":"kent0327","app":"splinterlands/0.7.139","n":"3YB3Tjh8fO"} |
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json | {"to":"tuanchimto","qty":11,"token":"DEC","type":"withdraw","memo":"tuanchimto","app":"splinterlands/0.7.139","n":"44EDjKvgHL"} |
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json | {"items":["fc941720b7000100902747d04e6446c95770ebae-97"],"days":2,"currency":"DEC","player":"xoi300","market":"peakmonsters","app":"peakmonsters"} |
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json | {"to":"jk6618","qty":11.441,"token":"DEC","type":"withdraw","memo":"jk6618","app":"splinterlands/0.7.187","n":"WYRADtAcW7"} |
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operations | 0. | 0. | comment |
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1. | parent_author | "" |
---|
parent_permlink | hive-163521 |
---|
author | dkmathstats |
---|
permlink | approximating-pi-squared-over-6-with-python-programming |
---|
title | "Approximating Pi-Squared Over 6 With Python Programming" |
---|
body | "Hi there. In this math & Python post, I cover one way of approximating the value of pi in Python programming. It is based on this .pdf link [here](https://d32ogoqmya1dw8.cloudfront.net/files/matlab_computation2016/activities/estimating_pi_lab.pdf). That .pdf link mentions MATLAB but I'll use Python here.
This post is a bit technical with the mathematics.
Pictures are either from Quicklatex.com for math text or screenshots.
<center><img src="https://cdn.pixabay.com/photo/2016/06/13/07/58/pi-1453835_960_720.jpg" /></center>
<center><a href="https://cdn.pixabay.com/photo/2016/06/13/07/58/pi-1453835_960_720.jpg">Pixabay Image Source</a></center>
## Topics
---
* For Loop As Sigma Sum Notation Or Product Notation
* Approximating Pi-Squared Divided By 6
* Approximating Pi
## For Loop As Sigma Sum Notation Or Product Notation
---
You can relate the for loop from computer programming with the Sigma sum notation from mathematics. In mathematics, the sum notation is as follows:
<center>https://quicklatex.com/cache3/25/ql_e668a0b45dfacce9d539d5f803658825_l3.png</center>
The for loop equivalent of the above in Python is something like:
```{python}
total = 0
for k in range(1, n, 1):
total = total + k
```
Instead of adding, there is also one for multiplying numbers to obtain a product. It is like this:
<center>https://quicklatex.com/cache3/bb/ql_13fec97b16b1a17a8d2404db3ef651bb_l3.png</center>
In Python, expression the product notation above would be something like:
```{python}
product = 0
for k in range(1, n, 1):
product = product * k
```
## Approximating Pi-Squared Divided By 6
---
Here is the infinite series which is equal to pi-squared divided by 6.
<center>https://quicklatex.com/cache3/5e/ql_fad48c1e2d2dc238fb3357d4c7872c5e_l3.png</center>
I create a function in Python that outputs the number of terms in the series given a maximum error (tolerance). This maximum error is based on the difference of the approximation to pi-squared over 6 and pi-squared over 6. A lower maximum error does give a better approximation but it does take longer. A balance between speed versus precision.
<center>`Error = Approximation - pi-squared over 6`</center>
<center>`Loop stops just when error is less than acceptable error (tolerance)`.</center>
As a reference pi to 10 decimal places is about `3.1415926536`. Pi-squared over 6 is about 1.64493 to 5 decimal places. (Screenshot from Desmos.)
![pi_reference_desmos.PNG](https://images.hive.blog/DQmT4yqHBTdqDkjRRXQTKgBxsXFWBGSPvr8udiAYuXBUMa8/pi_reference_desmos.PNG)
**Python Code**
```{python}
import math
# Approximate pi-squared over 6 given tolerance (acceptable error from approx to actual value)
# Number of terms given
def approx_pisq_tolerance(tolerance = 0.01):
# Initialize total/sum, error, k (number of terms)
total = 0
error = 1
k = 1
# Keep looping until error is less than acceptable tolerance error:
while error > tolerance:
total += (k**(-2))
error = abs(total - ((math.pi)**2 / 6))
k += 1
print('Given a tolerance of ' + str(tolerance) + ' the number of terms needed is ' + str(k - 1) + '.')
return(total, error)
```
In the function I have set the default tolerance error amount to be one hundredth. The user can change this if he or she wants to add more precision with the approximation to pi-squared over 6. Given a tolerance error, the function would output the number of terms needed in the series to obtain an error just less than the submitted tolerance error.
I first initalize the total/sum to zero and the error to 1. The error is set to 1 such that the while loop starts.
This while loop operates as long as the error is greater than the max error accepted. In each loop in the while loop I obtain the value that corresponds to `k` and add that to the running total/sum. Then I increment k by 1 just like with sum notation.
Note that in the print statement I use `k-1` as the `k-1` term is not evaluated.
**Function Call**
Doing a function call of tolerance of 0.1 gives this output. Screenshot from my jupyterNotebook and Desmos screenshot to verify.
![functionCall01.PNG](https://images.hive.blog/DQmfRix3idJzk97WWAUQpSqLudZvDitbXkzBVC3vvUB4VGV/functionCall01.PNG)
![pi_reference_desmos_02.PNG](https://images.hive.blog/DQmVs8HbBGiQeqUSNirmsnWF53RCqDMELgca1qcKLtPRHLM/pi_reference_desmos_02.PNG)
## Approximating Just Pi
---
With rearrangement of the formula and code adjustments, the value of pi can be approximated in Python.
<center>https://quicklatex.com/cache3/1a/ql_45945f5d4c1b739f125ede2c4f93f41a_l3.png</center>
**Python Code**
```{python}
def approx_pi_tolerance(tolerance = 0.01):
# Initialize total/sum, error, k (number of terms)
total = 0
error = 1
k = 1
# Keep looping until error is less than acceptable tolerance error:
while error > tolerance:
total += 6 * (k**(-2))
error = abs(math.sqrt(total) - math.pi)
k += 1
print('Given a tolerance of ' + str(tolerance) + ' the number of terms needed is ' + str(k - 1) + '.')
return(math.sqrt(total), error)
```
The code is not much different in terms of format compared to approximating pi-squared over 6. In this code, it is important to take the square root when computing the error.
**Function Calls**
Here are the function calls along with a Desmos screenshot as a check.
![functionCall02.PNG](https://images.hive.blog/DQmTZLPBgHZ68qTdWcvvmgZY9A6Wb6zprehYm8cZoqzZrYH/functionCall02.PNG)
![pi_reference_desmos_03.PNG](https://images.hive.blog/DQmcCnNa2NHwLhVqtEBgwik6N3nvdGPBuzywwRPh5NTpJ7c/pi_reference_desmos_03.PNG)
<center>Thank you for reading.</center>
Posted with [STEMGeeks](https://stemgeeks.net)" |
---|
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id | sm_start_quest |
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id | cbm__building__restore_condition |
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json | "{"name":"building","action":"restore condition","payload":{"identifier":{"building_name":"Water Pump"},"user":"friendlymoose#41170","restored_condition":{"quantity":98.12,"unit":"percent"},"building_condition":{"quantity":100.0,"unit":"percent"},"cost":{"quantity":0.02592,"currency":"CBM"},"timestamp":1647197491805}}" |
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id | sm_claim_reward |
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json | {"type":"quest","quest_id":"57a7f52239cc735326f2e368fc5d9384c8cd2308","app":"splinterlands/0.7.139","n":"3ju4b4bl6u"} |
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id | sm_claim_reward |
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json | {"type":"quest","quest_id":"7101efa7cd792eb2e181f9563c4355f1fc40ed08","app":"splinterlands/0.7.139","n":"w3km0x34xd"} |
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1. | required_auths | |
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id | sm_market_renew_rental |
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json | {"items":["149da89c23bc085bf9a5015d1e6ff6c6310fb751-32"],"days":2,"currency":"DEC","player":"soc439","market":"peakmonsters","app":"peakmonsters"} |
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id | sm_market_rent |
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json | {"items":["bb484c0acaa67df903dac2fbdd01b9071cc0c138-77"],"days":2,"currency":"DEC","limit_price":0.6,"player":"subaxis","market":"peakmonsters","app":"peakmonsters"} |
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required_posting_auths | |
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id | sm_submit_team |
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json | "{"summoner": "starter-437-nuVyQ", "monsters": ["starter-174-C2NwH", "C3-337-ZJTVNJAJ34", "starter-385-A64JL", "C3-338-6KUI72UHWG", "starter-172-SIdCP", "starter-169-5xvnm"], "trx_id": "981247802f447cca4b023fe3758d9dd48591ef61", "app": "splinterlands/0.7.139", "secret": "nktWk5BtVT", "team_hash": "1f352076f10d35674db63bca73a9d940"}" |
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extensions | [] |
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required_posting_auths | |
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id | sm_claim_reward |
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json | {"type":"quest","quest_id":"84990432bd52684cb02e7dceb6ee0d60fa41994e","app":"splinterlands/0.7.139","n":"6w3BKK2mXz"} |
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extensions | [] |
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1. | required_auths | |
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required_posting_auths | [] |
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id | ssc-mainnet-hive |
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json | {"contractName":"market","contractAction":"sell","contractPayload":{"symbol":"DEC","quantity":"352727.574","price":"0.00360914"}} |
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extensions | [] |
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required_posting_auths | |
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id | sm_start_quest |
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json | {"type":"daily","app":"splinterlands/0.7.190","n":"VBtBAhJpVr"} |
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extensions | [] |
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1. | required_auths | |
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required_posting_auths | [] |
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id | sm_market_renew_rental |
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json | {"items":["a31e99953230c617c146c5bb1ed122232a885527-0"],"days":2,"currency":"DEC","player":"haigame5629","market":"peakmonsters","app":"peakmonsters"} |
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extensions | [] |
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expiration | 2022-03-13 19:19:06 |
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operations | 0. | 0. | custom_json |
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1. | required_auths | [] |
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required_posting_auths | |
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id | sm_undelegate_cards |
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json | {"cards":["C3-225-7H0GDRXCSW","C3-234-Z4KYTX2Y1C","C3-240-RU7JL90FB4","C3-243-Q8YC2USCTC","C3-253-7RWLVRXLE8","C3-256-OW7QK28RU8"]} |
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