TI-84 Plus CE Python (and TI Basic): Rounding vs. Truncating
Rounding vs Truncating: Introduction
The program ROUNDING and the Python script rounding.py takes two numbers and estimates one of four arithmetic operations (+, - , ×, ÷).
Rounding: rounds a real number to the nearest integer. The traditional 5/4 rules are applied. In the case of the integers: for all numbers with fractional parts of 0.5 and above (to 0.9999...), the integer is rounded up.
Truncating: the fractional part of the number is dropped.
For example:
round(14.2) -> 14
round(14.5) -> 15
round(14.7) -> 15
trunc(14.2) -> 14
trunc(14.5) -> 14
trunc(14.7) -> 14
This results are shown in text form (Python program) or stored in a matrix:
[ [ x and y are rounded individually before the arithmetic operation, percent error ]
[ x and y are truncated individually before the arithmetic operation, percent error ]
[ result is rounded after the arithmetic operation, percent error ]
[ result is truncated after the arithmetic operation, percent error ] ]
The rounding percent program allows the user to compare rounding methods.
TI-84 Plus CE Python: Python script rounding.py
from math import *
# rounding comparison
# 2021-12-21 EWS
# rounding
def rd(x):
return int(x+0.5)
# truncate
def tr(x):
return int(x)
# percent change
def pc(x,y):
return (y-x)/x*100
print("integer rounding arithmetic")
pirnt("x [+,-,*,/] y")
x=eval(input("x>0, x? "))
y=eval(input("y>0, y? "))
print("1 + 2 - 3 * 4 / ")
op=float(input("Operation? "))
# test, force error
if op<1 or op>4:
print("not a valid operation")
1/0
# operations
if op==1:
a=rd(x)+rd(y)
b=tr(x)+tr(y)
c=rd(x+y)
d=tr(x+y)
z=x+y
if op==2:
a=rd(x)-rd(y)
b=tr(x)-tr(y)
c=rd(x-y)
d=tr(x-y)
z=x-y
if op==3:
a=rd(x)*rd(y)
b=tr(x)*tr(y)
c=rd(x*y)
d=tr(x*y)
z=x*y
if op==4:
a=rd(x)/rd(y)
b=tr(x)/tr(y)
c=rd(x/y)
d=tr(x/y)
z=x/y
print("results")
print("rounding individual")
print(str(a)+" ,"+str(pc(z,a))+"%")
print("truncate individual")
print(str(b)+" ,"+str(pc(z,b))+"%")
print("rounding result")
print(str(c)+" ,"+str(pc(z,c))+"%")
print("truncate result")
print(str(d)+" ,"+str(pc(z,d))+"%")
TI-84 Plus CE Python (TI-Basic): ROUNDING
"2021-12-21 EWS"
Disp "INTEGER ROUNDING ARITHMETIC","X [+,-,*,/] Y"
Input "X>0, X? ",X
Input "Y>0, Y? ",Y
iPart(X+.5)→N
iPart(X)→S
iPart(Y+.5)→J
iPart(Y)→T
Menu("OPERATION?","X+Y",1,"X-Y",2,"X*Y",3,"X/Y",4)
Lbl 1
N+J→A
S+T→B
X+Y→Z
Goto 5
Lbl 2
N-J→A
S-T→B
X-Y→Z
Goto 5
Lbl 3
N*J→A
S*T→B
X*Y→Z
Goto 5
Lbl 4
N/J→A
S/T→B
X/Y→Z
Goto 5
Lbl 5
iPart(Z+.5)→C
iPart(Z)→D
ClrHome
Disp "ROUND IND","TRUNC INC","ROUND RESULT","TURNC RESULT"
Wait 0.5
Pause [[A,(A-Z)/Z*100][B,(B-Z)/Z*100][C,(C-Z)/Z*100][D,(D-Z)/Z*100]
Examples
17.36+56.19
Rounding Individual: 73, -0.7477906186%
Truncate Individual: 73, -0.7477906186%
Rounding Result: 74, 0.611828688%
Truncate Result: 73, -0.7477906186%
82.8 - 17.9
Rounding Individual: 65, 0.1540832049%
Truncate Individual: 65, 0.1540832049%
Rounding Result: 65, 0.1540832049%
Truncate Result: 64, -1.386748844%
9.28 * 10.26
Rounding Individual: 90, -5.474894132%
Truncate Individual: 90, -5.474894132%
Rounding Result: 95, -0.2234993614%
Truncate Result: 95, -0.2234993614%
46.5 / 1.5
Rounding Individual: 23.5, -24.19354839%
Truncate Individual: 46, 48.38709677%
Rounding Result: 31, 0%
Truncate Result: 31, 0%
It just goes to show, rounding or truncating numbers before calculation may give results that aren't as accurate as one would like. The following paper, "Rounding versus truncation estimates in difference calculations" by Leonard Van Wyk (see source) argues that sometime truncation is more accurate than rounding in subtraction problems.
Source
Van Wyk, Leonard "Rounding versus truncation estimates in difference calculations" The Mathematical Gazette, Volume 103, Issue 557, July 2019. pp 285 -292
Eddie
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