Alternate Method for Adding Time

Date: 02/09/2001 at 19:08:06
From: Siri
Subject: Siri Factor use in time conversion formulae

To: Dr. Math,

I have found a factor to be used in a formula to make calculations 
easy (it is like math made easy). My school and others have been using 
the method. It has been published in the school magazine and has aired 
on TV stations. My dad sent it in my name to be trademarked and named 
after me (the Siri Factor). The trademark is pending. If you are 
interested in publishing it, I can send the details of the Siri Factor
and Formula so that whole world will know about it. Please let me know, 
so that I can ask my dad to send to you.

Date: 02/12/2001 at 17:55:13
From: Doctor TWE
Subject: Re: Siri Factor use in time conversion formulae

Hi Siri and Janardhan - thanks for writing to Dr. Math.

I am a colleague of Dr. Ian, and I saw your initial posting this 
morning. Ian shared the message copied below with me, and I told him 
that I'd respond to you.

From: Janardhan
To: Dr. Ian
Subject: Siri Factor
Date: Mon, 12 Feb 2001 11:56:50 -0800

SIRI FACTOR (40)

The Siri Factor is a constant number, 40. This Factor is named for my 
first name, Siri. I am 11 years old and I invented this factor to use 
in the formula described below.

The Siri Factor (40) can be used to make mathematical calculations 
involving the time units hours, minutes and seconds easier. 
Calculations involving time units can be done easily by using the 
Siri Factor in a decimal-like format.

Formula:

   A +/- B = C (minutes or seconds in A&B > 60) +/- Siri Factor (40)

Addition:

   A + B (minutes/seconds or hours/minutes) = C + Siri Factor (40)

(If the total number of minutes in A and B exceed 60)

Subtraction:

   A - B (minutes/seconds or hours/minutes) = C - Siri Factor (40)

(If the minutes in B exceed the minutes in A)

Example:

Suppose A = 3 hours, 40 minutes and B = 1 hour, 50 minutes.

Adding A to B:

     3:40 + 1:50 = 4:90 + Siri Factor (:40)

                 = 5:30 (5 hours, 30 minutes)

Subtracting B from A:

     3:40 - 1:50 = 1:90 - Siri Factor (:40)

                 = 1:50 (1 hour, 50 minutes)


IMPORTANT FACTS:

1. The Siri Factor, 40, is a constant.

2. This Factor provides a simpler way of calculating time units when 
   compared to conventional ways that involve additional steps.

3. Time units of hours and minutes in the above examples can be 
   substituted with minutes and seconds.

4. This factor can also be used in multiple additions and subtractions 
   of time units, by adding or subtracting the Siri Factor as 
   necessary.

5. The Siri Factor can also be used in the calculation of 
   multiplication and division of time units.


Siri, 11 years old, 6th Grader

---------------------------------

First, let me congratulate you, Siri, on a very clever mathematical 
insight. Most 11-year-olds would not come up with ways to make a 
tricky task like this easier. Well done!

The idea, however, is not new - although applying it to sexagesimal 
(hours-minutes-seconds or degrees-minutes-seconds) calculations *may* 
be new. A similar technique, using what is commonly called a "decimal 
adjust" or "fudge factor," is used when calculators and computers add 
or subtract decimal numbers directly. Let me explain how your Siri 
Factor works mathematically, and then I'll explain how this relates to 
the computer's "fudge factor."

For reference, let's assume we are dealing with hours and minutes, 
although the same logic holds for minutes and seconds. When you change 
a time value from, for example, 5:37 (h:mm) to 5.37 (decimal), you are 
changing the minutes from 1/60ths of an hour to 1/100th of an hour. 
Now let's add 5:37 and 2:48...

     "Traditional" way     With Siri Factor
     -----------------     ----------------
        5:37                  5.37
      + 2:48                + 2.48
       ------                ------

In both cases, we first add the units digit of the minutes. 7+8 = 15, 
so we record 5 and carry the 1. There is no difference here because 
both systems require 10 for a carry here. So we have:

     "Traditional" way     With Siri Factor
     -----------------     ----------------
          1                     1
        5:37                  5.37
      + 2:48                + 2.48
       ------                ------
           5                     5

Next, we add the tens digits:

     "Traditional" way     With Siri Factor
     -----------------     ----------------
          1                     1
        5:37                  5.37
      + 2:48                + 2.48
       ------                ------
         :85                   .85

Then we add the hours:

     "Traditional" way     With Siri Factor
     -----------------     ----------------
          1                     1
        5:37                  5.37
      + 2:48                + 2.48
       ------                ------
        7:85                  7.85

In both cases, we have to examine our results. With the traditional 
method, we see that we have 85 minutes, and we realize that we can't 
go above 60. So we subtract 60 minutes and add 1 hour. With the Siri 
method, we also realize that we added more than 60 minutes, so we add 
the Siri Factor, .40

     "Traditional" way     With Siri Factor
     -----------------     ----------------
          1                     1
        5:37                  5.37
      + 2:48                + 2.48
       ------                ------
        7:85                  7.85
      -  :60                +  .40
       ------                ------
        7:25                  8.25
      + 1:00
       ------
        8:25

In both cases, we end up with 8:25. We can shorten the "traditional" 
method a bit by making the correction immediately as we add the 
minutes. When we add the tens digits of the minutes, we get 1+3+4 = 8. 
But 6 tens of minutes (i.e. 60 minutes) make an hour, so we make 6 of 
our 8 a carry and record the remaining 8-6 = 2.

     Modified traditional way
     ------------------------
        1 1     <- the carry into the hours represents 6 groups of
        5:37        10 minutes
      + 2:48
       ------
         :25         

Now we add the hours (with the carry from the tens of minutes) and we 
have:

     Modified traditional way
     ------------------------
        1 1
        5:37
      + 2:48
       ------
        8:25         

The difference between these methods is in the carry from the tens of 
minutes to hours. In the modified traditional way (and essentially in 
the traditional way - though more indirectly), we carry a group of 6 
tens to make an hour. In the Siri method, we carry the normal 10 tens 
of the decimal system. However, because the result (.85 in the 
example) is not valid, we add the Siri factor of .4. This makes the 
group of 6 tens that should have been carried into a group of 10 tens 
that are carried.

Subtraction works similarly, reducing a borrow of 10 tens into the 
required borrow of 6 tens. The Siri Factor takes away the extra 4.


As I mentioned earlier, calculators and computers use a similar 
technique when they add decimal numbers directly. Calculators and 
computers traditionally work in binary, or base 2, because they use 
tiny electronic "on/off switches" to represent the 1's and 0's. 
However, when dealing with larger numbers, it is easier to think of 
them as working in hexadecimal, or base 16. If you are not familiar 
with other bases - and binary and hexadecimal in particular - you 
may want to check out our "Number Bases" FAQ at:

    http://mathforum.org/dr.math/faq/faq.bases.html   

Suppose a calculator is used to add 38 and 77 without converting them 
to hexadecimal. It will start by adding the units digits, 8+7 = 15. 
But in hexadecimal, 15 is a valid digit and is represented by an F. So 
the calculator will have:

       38
     + 77
      ----
        F

Next, the calculator will add the "tens" digits (actually, they are 
sixteens digits in hexadecimal), 3+7 = 10. Again, 10 is a valid digit 
in hexadecimal and is represented by a A. So the calculator now has:

       38
     + 77
      ----
       AF

Of course, AF is not a valid decimal answer (just as 7:85 was not a 
valid time answer), so the calculator adds a "fudge factor" of 6. 
First, it adds 6 to the units (because F is an invalid units digit). 
F+6 = 15 in hexadecimal (15+6 = 21):

       38
     + 77
      ----
       1    <- carry from the units when adding the fudge factor
       AF
     +  6
      ----
       B5

Next, it adds 6 to the tens (because B is an invalid tens digit). 
B+6 = 11 in hexadecimal (11+6 = 17):

       38
     + 77
      ----
       1
       AF
     + 66
      ----
      115

And thus it gets the correct answer.

Why was the fudge factor 6? Because that makes the 10 ones that we 
should have carried over to the next digit into a group of 16 ones 
that the calculator does carry over in hexadecimal. 

When is the fudge factor of 6 used? When either [a] a digit is greater 
than 9 (and thus invalid in decimal), or [b] a carry is generated from 
that digit. In case [a], the fudge factor is needed to make a group of 
10 that should be carried a group of 16 that will be carried. In case 
[b], the fudge factor is needed to "give back" the extra 6 that were 
carried (the calculator carried 16, when it should have only carried 
10).

Looking at how calculators handle decimal numbers, I would not be 
surprised if those calculators that have the capability to add and 
subtract sexagesimal (DMS) values don't already use the Siri Factor - 
but I'm not sure. I haven't explored how they handle DMS calculations. 
If I discover this, I'll write you back.

One final thought on the Siri Factor: Like the fudge factor above, you 
have to be careful to be sure to use it properly. Consider the 
following two additions:

       1:55        2:05
     + 2:50      + 2:00
      ------      ------

Using the Siri method:

       1
       1.55        2.05
     + 2.50      + 2.00
      ------      ------
       4.05        4.05

Just looking at the results, both look valid and we can't tell whether 
they need a Siri Factor added on. The first one needs a Siri Factor 
(because of the carry into the hours), but the latter does not. So to 
use the Siri Factor properly, you have to examine what you're doing - 
and I'm not sure that it's *significantly* simpler that just doing it 
the old fashioned way.

Again, congratulations on your work - you definitely have the attitude 
to become a renowned mathematician!

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/   

Date: 05/14/2007 at 13:06:16 
From: Jack
Subject: The Siri Factor

Siri's method has been known to surveyors, as the following web page 
indicates:

    http://www.fer3.com/arc/m2.aspx?y=200302&i=009016

Date: 05/14/2007 at 21:28:43
From: Doctor Rick
Subject: Re: The Siri Factor

Thanks, Jack, that's interesting.  Even if the method has been known
before Siri discovered it, it's still pretty neat that an 11-year
old would come up with that insight.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/