**Swiss Micros DM41X and HP Prime: Rounding Numbers to One Significant Digit**

Introduction

This blog entry deals with rounding numbers to one significant digit, that is the first non-zero numbers. The general 5/4 rounding rules are used: round up if you are next digit is 5 or more.

Examples:

72.28 rounds to 70

78.28 rounds to 80

0.0283 rounds to 0.03

Rob Eastaway, author of Maths On the Back of an Envelope (see source below), introduces the concept of Zequals. Zequals is the practice of rounding numbers to one significant digit, with it's symbol similar to the symbol for the constellation Aquarius the Water Bearer ( ♒ ). The aim for rounding numbers to one significant digit is to aid in estimating arithmetic operations and make mental mathematics easier.

Is it always spot on?

The program RONE (Round to ONE Significant Digit) rounds positive, non-zero numbers to one significant digit. An algorithm to do so may look like this:

Number: x

a = log(x)

If a≥0, then a=int(a)

If a<0, then a=int(a)-1

n = int(x / (10^a) + 1/2) * 10^a

We can also use:

n = round(x / (10^a),0) * 10^a

I have included the HP Prime PPL code for RONE. RONEARITH for the HP Prime compares the actual versus rounded calculation for the four arithmetic functions.

**Swiss Micros DM41X Program: RONE**

01 LBL^T ZONE

02 CF 01

03 ENTER

04 LOG

05 X<0?

06 SF 01

07 INT

08 FS? 01

09 -

10 FS? 01

11 -

12 10^X

13 /

14 LASTX

15 X<>Y

16 2

17 1/X

18 +

19 INT

20 *

21 CF 01

22 RTN

23 END

Examples:

52.4564 XEQ RONE returns 50.0000

0.06834 XEQ RONE returns 0.0700

**HP Prime Program: RONE**

EXPORT RONE(n)

BEGIN

// Round to one significant digit

LOCAL a;

a:=LOG(n);

IF a<0 THEN

a:=IP(a)-1;

ELSE

a:=IP(a);

END;

RETURN ROUND(n/ALOG(a),0)*ALOG(a);

END;

**HP Prime Program: RONEARITH**

EXPORT RONEARITH()

BEGIN

// 2021-07-26 EWS

// Input form

LOCAL c,x,y,r,t,l;

l:={"+","-",CHAR(#D7h),CHAR(#F7h)};

INPUT({x,y,{c,l}},

"1 Sig Digit Arithmetic",

{"x: ","y: ","OP: "},

{"x>0","y>0","Arithmetic Operation"});

PRINT();

IF c==1 THEN

PRINT(STRING(x)+"+"+STRING(y));

r:=ZR(x)+ZR(y);

t:=x+y;

END;

IF c==2 THEN

PRINT(STRING(x)+"-"+STRING(y));

r:=ZR(x)-ZR(y);

t:=x-y;

END;

IF c==3 THEN

PRINT(STRING(x)+CHAR(#D7h)+STRING(y));

r:=ZR(x)*ZR(y);

t:=x*y;

END;

IF c==4 THEN

PRINT(STRING(x)+CHAR(#F7h)+STRING(y));

r:=ZR(x)/ZR(y);

t:=x/y;

END;

PRINT("Estimate: "+STRING(r));

PRINT("Actual: "+STRING(t));

END;

ZR(n)

BEGIN

// Round to one significant digit

LOCAL a;

a:=LOG(n);

IF a<0 THEN

a:=IP(a)-1;

ELSE

a:=IP(a);

END;

RETURN ROUND(n/ALOG(a),0)*ALOG(a);

END;

Source:

Eastaway, Rob. __Maths On the Back of an Envelope__ HarperCollins Publishers: Dublin, Ireland 3rd Ed. 2021. ISBN 978-0-00-844449-5

Eddie

All original content copyright, © 2011-2021. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

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