TI-84 Plus CE and TI-Nspire CX II: Square Root Trick - by @Mathsbook7474
Introduction
From the Instagram account @Mathsbook7474, it is stated that the square root of a number can be approximated by:
√x ≈ (x + y) ÷ (2 · √y)
where y is the number nearest to the root, preferably a perfect square. (see source for the link)
For example:
Approximate √52.
Let x = 52. Note that 49 is a perfect square near to 52. Since √49 = 7, let y = 49.
√52 ≈ (52 + 49) ÷ (2 · √49) = 7.214285714
Accuracy: 0.0031831631 (√52 = 7.211102551)
TI-84 Plus CE Program: SQAPPROX
The program SQAPPROX will calculate the approximation above, the actual root, and compare the results for the accuracy.
Listing:
Download the program here:
TI-Nspire CX II tns File: square root approximate.py
This document takes it a step further. The python program will create four lists that are used to display a table, graph the approximate answers, the actual answers, and the error.
Python program: spapprox.py
from math import *
from ti_system import *
# TI System command clear history
clear_history()
# intro and prompts
# characters use ctrl, [ book ]
print("√x≈(x+y)/(2*√y) @mathsbook7474")
print("For x:")
s0=float(input("start: "))
s1=float(input("stop: "))
# set up columns
col1=[]
col2=[]
col3=[]
col4=[]
for i in range(s0,s1+1):
col1.append(i)
w=sqrt(i)
col2.append(w)
a=trunc(w)
b=trunc(w)+1
c=fabs(a**2-i)
d=fabs(b**2-i)
if c<d:
s=(i+a**2)/(2*a)
else:
s=(i+b**2)/(2*b)
col3.append(s)
r=fabs(w-s)
col4.append(r)
# save columns to outside variables
# TI System module used
store_list("data",col1)
store_list("act",col2)
store_list("appr",col3)
store_list("error",col4)
print("data = x")
print("act = actual")
print("appr = approximate")
print("error = absolute error")
Download the file here:
Source:
"Square Root Trick" @mathsbook7474. Instagram account.
June 8, 2021. Retrieved June 21, 2021.
Eddie
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