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To whom it may concern,Im trying to explain an old method…

<p>To whom it may concern...
<p>To whom it may concern, </p><p>I'm trying to explain an old method of addition that is referred to in a text (unpublished) writen by someone who is no longer with us - therefore I can't ask them to clarify. The text states "...they could take a column of figures about five figures wide and about ten figures tall and add them up in a peculiar way, some kind of a crisscross addition and they would arrive with almost an immediate answer." </p><p>The author himself didn't know the technique well, but saw it done and stated roughly what he recalled the person (who did it) saying "Nine added to something gives you itself, so all you do is go down the column and find all the combinations which make nine and forget those, and add the remainder and you get the total." If this example throws you off, forget it. I don’t know the accuracy of it. However, it seems there is an old method to quickly add a column of numbers that seems to be a crisscross method and that's what I'm looking for. </p><p>Any help I can get in understanding this method, I would greatly appreciate!</p>
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Answered in 1 hour by:
7/5/2010
Ryan
Ryan, Engineer
Category: Math Homework
Satisfied Customers: 9,184
Experience: B.S. in Civil Engineering
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Hi,

Welcome! Thank you for using JustAnswer.

The procedure that you are referring about nines is called "casting out nines", and applies to calculating the digit sum of a number, or a column of digits.

In calculating the digit sum, you keep repeating the summing process until you get a single number as an answer.

For instance, if you were calculating the digit sum of 359, you have:

3 + 5 + 9 = 17

Sine 17 is a two digit number, repeat the process with 17:

1 + 7 = 8

Now that you have a single digit as an answer, you're done. Note that the result is the same number you would have reached if you had neglected the nine in the first sum.

Calculating the digit sum can be used to check your addition of a column of multi-digit numbers, but doesn't help you get the answer. It only tells you if your answer it wrong. The digit sum of 8 and 9, for instance is 8, but if you were adding 8 and 9 in a column of numbers, you want the digit in that column to be 7. That is why this trick doesn't help with the addition itself, but only as a check.

There is a process of addition where you start from the left and use "complements". A complement is the difference between 10 and a number. So the complement of 7 is 3, the complement of 8 is 2, etc.

The way this procedure works is that you never add beyond 10. If adding the next number in the column will cause your total to exceed ten, you subtract the complement of that number from your total, and make a note that you have counted a "ten". This keeps you from having to use the whole table of addition facts, and keeps the numbers that you hold in your head small and easy to work with.

An example will help illustrate this:

Add 7 + 5 + 8 + 6 + 9 + 7:

7 + 5 would exceed ten, so subtract the complement of 5 (which is 5), giving you 7 - 5 = 2, and note that you have 1 ten. (So at this point you have a total of 2, and one "ten". Put them together would give you 12, which is what 7 + 5 equals.)

Continue: 2 + 8 (2 is the current total, and 8 is the next digit to add)

Subtract the complement of 8 (it's 2) to get 0, and count another ten. Your running total is now zero, and you have two tens.

Add the 6:

0 + 6 = 6

Add the 9:

6 + 9 will go over ten, so subtract the complement of 9 (1) from the 6 to get 5, and add another ten.

Finally, add the seven. This will again push your total over 10, so subtract the complement and add another 10. (5 - 3 = 2, and four tens)

The end result is 42.

It is harder to describe than it is to do, but is does take practice.

I've probably thrown enough new info at you for now, but if you want to see how this works in a column of multi-digit numbers, let me know and I'll work up an example.

You can find more about this type of addition in the following book:

A Guide to High-Speed Mathematics, by XXXXX XXXXX, published by Nelson Doubleday, Inc in 1959.

You might also try researching the "Trachtenberg System of Speed Arithmetic".

I don't know if this is exactly the method that you are looking for, but it might be. I have a couple of books that might have more info, but I'll have to dig them out. If I find anything, I'll let you know with a follow-up answer.

Please feel free to ask if you have questions about this. I'll be happy to explain further if necessary.

Thanks,

Ryan
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Customer reply replied 8 years ago

Dear XXXXX,

 

I really appreciate your explaination and the quick response! I will need to read through this and digest it. I will let you know back either later tonight or tomorrow. Thank you again.

Customer reply replied 8 years ago

Dear XXXXX,

 

Thanks very much - I appreciate your taking the time to answer me with two possibilities.

 

As you mentioned, "casting out nines" isn't intended to give you an answer, but only to verify the result of a computation already done. So that's not it.

 

I tried the complement strategy you mentioned on a column of numbers 5 wide and 10 tall. It does indeed accurately add up, but it takes so much longer than normal addition, that I can't see how it could possibly be the super-fast "crisscross addition" I originally wrote you about.

 

The passage I wrote you about comes from a lecture and I'm charged with correctly defining all odd terms in the lecture; it's an internal organizational type of glossary for several such lectures. I have to find a positively accurate definition for this "crisscross addition." This was apparently a type of strategy in use in the late 1800s or early 1900s.

 

Have you any other ideas or persons or places I could write or some book I could look for such a thing?

 

Again, really appreciated -- thanks,

 

Nicole

Hi Nicole,

Yes, the process is certainly no time saver initially, but with practice is gets easier (or so I'm told.)

I will keep looking to see if I can find something that sounds more like the "crisscross" addition method.

Do you have a copy of the lecture? If so, could you possibly scan it into a document? I would be very interest in reading it. It might help give me a clue that I can use to find something for you.

Thanks,

Ryan
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Customer reply replied 8 years ago

Dear XXXXX,

 

Thank you for the offer to keep looking - I really do need it!

 

I have a hardcopy transcript of the lecture and it is on something else entirely. The author/speaker only mentioned this mathematical bit as an aside.

I typed and sent you most of what there was regarding this point. There was a little more before the part I sent to you. I'm including it below.

 

Well, I was quite interested in old McGuffey's Readers at one time to find out how adept at arithmetic somebody was expected to be in 1888. The problems which they were expected to solve were the problems of algebra. And they were expected to solve these with arithmetic. And what do you know? It was a great revelation to me that it was very possible to solve these algebraic problems with their "X's" and "Y's" and all that sort of thing by common, ordinary, garden-variety arithmetic. I've run into some old-timers who could take a column of figures about five figures wide and about ten figures...

 

This is where the text I already sent you picks up. I hope this extra bit sheds some light on the matter.

 

Good luck on your search, I'm excited to see what you might come up with.

 

Again, I appreciate your help!

 

Thanks, Nicole

 

 

Hi Nicole,

Just want to let you know that I haven't forgotten about your question, or given up on the search for some information for you. I haven't found anything yet other than procedures for adding numbers from left to right, and the Trachtenberg system that I mentioned before.

You said that you sent me a portion of the transcript, followed by the snippet about the McGuffey readers. I only got the last part. Perhaps there was something in the first part that I could use. Can you try sending it again?

Thanks,

Ryan
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Customer reply replied 8 years ago

 

Dear XXXXX,

 

Thank you for answering me, I wasn't sure where we stood.

 

I'm sorry about the bits of data being confusing. I've put it together, as it is in the transcript, and it is below. You now have the chunk of the lecture that pertains to this subject and the question at hand.

 

I would appreciate it if you could let me know if you are able to come up with anything else.

 

Thanks again!

Nicole

 

"Well, I was quite interested in old McGuffey's Readers at one time to find out how adept at arithmetic somebody was expected to be in 1888. The problems which they were expected to solve were the problems of algebra. And they were expected to solve these with arithmetic. And what do you know? It was a great revelation to me that it was very possible to solve these algebraic problems with their "X's" and "Y's" and all that sort of thing by common, ordinary, garden-variety arithmetic. I've run into some old-timers who could take a column of figures about five figures wide and about ten figures tall and add them up in a peculiar way, some kind of a crisscross addition and they would arrive with almost an immediate answer. And you say ‘how did they do that' . Well they say ‘Well it's very simple. Nine added to something gives you itself, so all you do is go down the column and find all the combinations which make nine and forget those, and you add the remainder and you get the total.' "

Nicole,

I've pretty much exhausted every source I can think of, and I still haven't found anything about "criss-cross addition". I don't think such a procedure actually existed. If it had, there should be some mention of it somewhere.

However, I do now know more about McGuffey Readers, abacus math, and Vedic math than I ever imagined I would. It's been an interesting research journey. Smile

Given that the speaker was making this comment as an aside, perhaps he mispoke.
There is a method called "criss-cross" multiplication.

The part about "nine added to something gives you itself" is simply not true where addition is concerned. It does work for the digit sum as we discussed, but not for column addition. If the speaker were to change "nine" to "ten", then the statement would at least be mathematically correct. But it still wouldn't be "criss-cross" anything.

My only suggestion at this point would be to seek out an older mathematics professor. I'm assuming that the older they are, the more likely they might be to have heard of something like this, if it even exists.

Thank you very much for the question. It was one of the most interesting ones I've seen, and you have been a pleasure to work with. Good luck.

Best wishes,

Ryan
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Customer reply replied 8 years ago

Dear XXXXX,

 

I understand everything that you've said and you may well be exactly correct about it being a "speako", as I like to call it.

 

You've been really helpful on this and I really appreciate all the trouble you went to in trying to help me find the answer!

 

Thank you so much for your help. I will do as you suggest, just in case there is something. Any tips you can give on finding an older math professor ?? (I'll do my best via Google)

 

All the best,

Nicole

 

Nicole,

Thank you very much for accepting my "answer". I truly wish I could have found something more helpful for you. Sigh...

Thank you for the positive feedback too. It is greatly appreciated.

"Speako" That's awesome, I love it!

As for finding a math professor, I would start at the nearest state college or university. Try looking up the school's website, find the math department, and send an email to the department chairperson. That way you don't have to pay (or fight) for parking! Ideally, what you would be looking for would be a "professor emeritus" (fancy name for retired). They hang around the school for the prestige, but are generally not involved in research or publishing papers, so they have more time for inquiries like yours. Plus it keeps them feeling important. Smile

Good luck in your search.

Ryan
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Customer reply replied 8 years ago

Dear XXXXX,

 

Thank you very much for your suggestion ! I'm going to give it a shot.

 

All the best,

Nicole

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