Question

I’m trying to understand what is happening in the python code below. I take the square root of 2 and divide its decimals by 1. Doing this for 5 times keeps giving the same value but the 6th and 7th time I get different values.

Why does the output change at the 6th time when the input value is the same as in the previous 5 calculations?

import math as M
frct = M.sqrt(2)

# 1
frct = 1 / (frct - int(frct))
print frct # 2.41421356237

# 2
frct = 1 / (frct - int(frct))
print frct # 2.41421356237

# 3
frct = 1 / (frct - int(frct))
print frct # 2.41421356237

# 4
frct = 1 / (frct - int(frct))
print frct # 2.41421356237

# 5
frct = 1 / (frct - int(frct))
print frct # 2.41421356237

# 6
frct = 1 / (frct - int(frct))
print frct # 2.41421356238

# 7
frct = 1 / (frct - int(frct))
print frct # 2.41421356235

Answer 1

Short version, Python is rounding the output 🙂

import math as M
frct = M.sqrt(2)

for i in range(7):
    frct = 1 / (frct - int(frct))
    print 'Attempt %d: %.20f' % (i, frct)

Long version, floating points don’t store the real (no pun intended) value, they store an exponent and mantissa. See this wikipedia page for more info: http://en.wikipedia.org/wiki/Floating_point

Basically, a floating point number is stored like this:

Significant digits × base^exponent

If you want a more precise version in Python, try the decimal module:

import decimal
context = decimal.Context(prec=100)
frct = context.sqrt(decimal.Decimal(2))

print 'Original square root:', frct

for i in range(7):
    frct = context.divide(1, frct - int(frct))
    print 'Attempt %d: %s' % (i, frct)

Output:

Original square root: 1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
Attempt 0: 2.414213562373095048801688724266222622763067167798368627068136427003657772608039155697953022512189319
Attempt 1: 2.414213562373095048801688723683379910288448158038030882339615025168647691299718507620657724911891709
Attempt 2: 2.414213562373095048801688727180436185136162216600057354932063779738350752352175486771948426117071942
Attempt 3: 2.414213562373095048801688706780941248524496874988236407630784335182989956231878308913506955872772859
Attempt 4: 2.414213562373095048801688825680854593346774866097140494471009059332623720827093783193465943198777227
Attempt 5: 2.414213562373095048801688132680869461024772261055702548455940065126184103929661474210576202848416747
Attempt 6: 2.414213562373095048801692171780866910134509900201052604731129303452089934643341550673727041448985316