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Adding And Subtracting Integers (Whole Numbers)

Two Algorithms for Subtracting 173 from 342The Austrian physicist and philosopher, Ernst Mach (1838-1916) once claimed that “there is no problem in the whole of mathematics which cannot be solved by direct counting.”  This is an impressive claim, and there is a great deal of truth to it but, obviously some numbers are simply too big to count directly.  This is not a problem that can be ignored.  It means we must learn to manipulate numbers in more clever ways than simply counting (with) them.

Now, the manipulation of numbers is what we call arithmetic (from the Greek arithmos/αριθμός meaning number).  This is so fundamental a skill that the great German mathematician, Karl Frederich Gauss (1777-1855), observed, “Mathematics is the queen of sciences and arithmetic is the queen of mathematics.”  So, arithmetic … the art of counting and calculating … is very important indeed!

We need to ask the question, therefore, “Apart from simply counting on our fingers, how can we add and subtract quickly and efficiently?”

The contents of this page constitute my attempt to answer this question.

Learn to Add Quickly

I have developed an Excel workbook during the last few years.  It contains a number of addition and subtraction exercises that help my students master those skills.  I am sure that they can help you, too.

The prerequisites for the first exercise are not great. If you can recognise the ten basic numerals, know how to write numbers up to 20, and can add up on your fingers, you are ready to take on the challenge.

Now, in our normal numbering system, three and four always add to seven. In other words, 3 + 4 = 7. Although we all learn to add these numbers on our fingers it is important that, in time, we learn the answer.  In this way we can solve problems quickly without having to stop and think about the mechanics of adding.  Learning requires practice and that is what the workbook is for.  It randomly generates worksheets that will help you learn to add quickly.

Watch this video to see how to use the first spreadsheet and then download the sheet and start practising!

Record your time for the first sheet so that you can see how you improve.

Most students take 1:30 to 4:00 minutes to complete the sheet when they start (the average seems to be about 2:15). In time, you will find that you know the answers without counting on your fingers. This will happen gradually and may take from 1-2 days to a few weeks depending upon how hard you practise!  You will probably notice by this stage that you are saying the numbers in your head. Something like, "Three and four makes seven," or "Three and four, seven."

As you keep practising, you will start to write the answers without saying the names in your head. You will see the numerals '3' and '4' and know that the answer is '7.' When this happens, you will notice another big improvement in your speed. By this time, you should be taking less than 1:15 to complete the sheet (and possibly a lot less).

I have found that one or two students in every class of thirty will complete the sheet in less than one minute (after they have practised for a week or four). The best time that I have ever recorded for a student was 49 seconds. Time yourself and post your results in the comments section under my YouTube video or let me know directly. I would love to hear how you went!

Download the Excel Workbook here.

The worksheets help students develop basic addition/subtraction skills and rapid recall of addition/subtraction 'facts' through rapid drill work. A new and different worksheet is calculated every time the F9 key is pressed. This means that a different sheet can be printed for every child in a class or a different sheet can be printed every day to develop these skills. The sheets are designed so that most students will complete each one in under three minutes (and probably under one or two minutes).

The speadsheet is completely free and you are free to share it with friends but not to make money from it. Teachers may like to use it with their classes. Home educators might find it very useful to build solid addition skills in their children.

The Amazing 1089 Trick Demonstrated and Explained (Adding)

Many number patterns are entirely predictable.

If you rearrange any number to form a new number, and find the difference between the two numbers (i.e. subtract them), the answer will always be divisible by nine. This formed the basis of the coin trick that I shared on YouTube.

If you reverse a THREE-digit number and find the difference between the new number and the original one, the answer will always be divisible by eleven as well (and, therefore, by ninety-nine).

This REVERSE AND SUBTRACT pattern ensures that the answer conforms to a particular structure, a9b where the numbers a and b add up to nine, i.e., 099, 198, 297, 396, 495, 594, 693, 792, 891, or 990. Reversing this number (b9a) and adding it to the a9b will ALWAYS produce 1089 as I explain in the video. If you are trying this challenge out with someone, you can have a bit more fun by secreting your answer somewhere and producing it as 6801 ... which will produce a little bit of confusion, before you turn your answer upside down to reveal the true answer of 1089!

Remember that this trick ALWAYS WORKS provided two rules are adhered to (and your friend can calculate accurately!):

  1. The starting number must not be a palindrome (i.e. it must not read the same in reverse. If such a number is chosen, when it is reversed and subtracted, the difference will be found to be zero! Technically, this is still divisible by 99 (so the number theory pattern/rule still holds), but it is a redundant sort of result, so we exclude it by starting with a number that looks different when reversed.
  2. If, after subtracting, your friend ends up with 99, they must consider it as 099, i.e. as a three digit number, and reverse that to form 990 before adding.

Make these rules clear to your friend, and the trick should work every time.

Because I did not want the video to be unduly lengthy, I did not give a FULL explanation of the number theory principles involved. If you have any questions, please ask them in the comments below the video, and I will do my best to answer them for you.

Number Theory is a branch/aspect of mathematics in which we analyse how numbers work and what makes them behave the way they do. Number theorists are fascinated by prime and composite numbers, divisibility, number types and patterns. A whole world opens up ... and it is fascinating ... much moreso than you might think! I will be sharing more material relating to number theory as I am able.

How to Subtract (Large) Numbers Easily

This time I address a rather elementary skill, but one that many people struggle with and (in this age of electronic calculators) many people have forgotten as well.

I begin by sharing about my encounter with a VERY TALL taxi driver many years ago, and then explain how his joke about his height provides the key to understanding the structure of large numbers ... and how to subtract them. I explain and demonstrate the two most common subtraction methods (algorithms) and encourage you to practise both and then adopt the one that you prefer.

Now let me introduce you to the comic song-writer and pianist, Tom Lehrer. This US mathematician became popular the world over during the 1960s and 1970s (especially) for the satirical songs which he sang ... accomanied by what he described as his "88 stringed guitar" (his piano).

One of the songs that he wrote dealt with the "New Math" that was being introduced to American schools at that time. In it he SINGS how to subtract two numbers (and then adds an interesting 'twist'). I hope you enjoy the New Math (4:22)!

He is also famous for having put the entire Mendeleev chemical chart (Periodic Table) to music, although not in atomic mass order. You can enjoy his very entertaining song, called The Elements (1:25), as well.

And, if you are a fan of Elijah Woods, you might like to hear him sing it on the Graham Norton Show (on 12 November 2010) (1:43) ...

If you would like to listen to a larger sampling of his songs, here he is at Copenhagen (Denmark) in 1967, at about the height of his popularity (50:42) ...

It is unusual seeing a mathematician in this environment.

How to Subtract (Large) Numbers Easily (Some More Methods)

In my previous 'subtraction video' I explained and demonstrated the two most common methods/algorithms for subtraction that are taught in schools today.

As promised in that video, I now show you two more (unusual) methods (and one refinement). These will help you understand the mechanics of subtraction better. One of the methods utilises a concept used in computer design.

It is a long video because I not only demonstrate these methods, but explain why they work.

I hope you find this video informative and helpful.

Subtraction-fest 01 ~ Borrow Method (Common Today)

After having posted two lengthy videos in which I explained WHY different subtraction algorithms worked, some of you expressed interest in having shorter videos that simply DEMONSTRATED the use of the algorithms.  This is the first such video.

This (first) method/algorithm that I demonstrate is the one that is most often taught in schools around the world today. As we examine each column, if the numeral below is larger than the one above, then we "borrow" from the larger valued column(s) in the top number.

Subtraction-fest 02 ~ Carry Method (Easy)

This (second) method is the one that I was taught in Primary School. It seems to be used less often today.

As we inspect each column, if the number below is larger than the one above, then we add "ten" to the top number ... and add ten to the bottom number by adding a "1" in the next column to the left. By increasing both numbers by the same amount, the difference between them will remain the same. This is sometimes known as the "carry" method (or the "one up, one down" method).

Subtraction-fest 03 ~ Floating Digit Method (Change all Subtractions to Additions!)

This (third) method uses a concept that is used in designing computers. Since computer circuits can only add, in the ALU (Arithmetic Logic Unit), all subtractions are converted to additions using a 'complement' method.

Using our base ten number system, we also create a complementary number ... and it is so easy that you can learn to complete this step very quickly indeed! All that remains is to ADD the complementary number (and remove the 'floating digit').

This is not a method that I would teach to children as their primary method for subtracting numbers, but it is a rather 'fun' and ununsual method that I hope will appeal to you.

Subtraction-fest 04 ~ Floating Digit Variation Left to Right

This (fourth) subtraction method is a variation of the third one (which is used in designing computers).

Using some artful logic, we are able to convert all subtractions into a simple sum (addition) WITHOUT having to remove a floating point!

Again, this is not a method that I would teach to children as their primary method for subtracting numbers, but it is yet another ununsual subtraction method that I hope will find contains some merit.

Subtraction-fest 05 ~ Floating Digit Variation (Subtracting From Powers of Ten)

This (fifth) subtraction method is a variation on earlier methods, but deals with the special case when we subtract numbers from powers of ten ... such as 100, 1000 or 1 000 000.

Because the borrowing method (after borrowing from the initial "1") would leave a string of "9s" and a final "10" on the top line, we simply subtract all the leading digits from nine, and the final digit from ten. That is all there is to it!

Subtraction-fest 06 ~ Floating Digit Variation (Using Negative Markers)

This (sixth) subtraction method is also a variation on earlier methods. It shows how you can subtract large numbers in two steps if you are prepared to use negative numbers.

In the first step, the numbers in each column are subtracted with no borrowing at all. In the second step we resolve the problem of the "negative markers" by borrowing at that stage. This second step uses a combination of borrowing and the concepts that I shared in the floating digit subtractions.

Subtraction-fest 07 ~ Never Subtract Numbers Larger Than Ten!

This (seventh) subtraction method is related to some of the previous ones, but is quite unique. In this case you can subtract large numbers without ever having to subtract from a number larger than ten!

It was shown to me by a young student who was struggling with his mathematics during his first year of High School.  Some years earlier, he had not understood what his Primary School teacher had been trying to teach him about subtraction ... so he managed to work out his own subtraction algorithm!  Never underestimate the ingenuity of even some of the "poorer" students!

I have explained WHY this method works in the second (long) video of this series. In this short video I simply demonstrate the method for you.

Subtraction-fest 08 ~ Mental Subtraction By Counting

This video is the first of three in which I describe common mental subtraction methods.

The subtraction methods that we use on paper are, generally, too cumbersome for mental calculation. Also, we rarely subtract large numbers mentally ... so when subtacting smaller numbers in our head, we use techniques that don't require remembering lots of detail!

Because we count quite naturally by ones, tens and hundreds, the first method uses this simple skill to find the difference between numbers.

Subtraction-fest 09 ~ Mental Subtraction By Elevating Both Numbers

This video is the second of three in which I describe common mental subtraction methods.

If two numbers are a certain distance apart, then adding the same number to both of them will "shift them" by the same amount and their difference will remain the same. We utilise this knowledge in order to make the bottom number (the subtrahend) easier to subtract ... by rounding it upwards!

Subtraction-fest 10 ~ Mental Subtraction By Adding Differences

This video is the last of three in which I describe common mental subtraction methods.

In this case we actually find two differences and add them! By calculating how far one number is below, and the other number is above, some convenient reference number ... we then simply have to add those two differences.

Subtraction-fest 11 ~ Checking Your Subtractions By Adding

If you have had to complete a subtraction problem and need to check your answer to ensure that you "got it right," there are a few methods that you could use.

You could, of course, repeat what you had already done! I.e. repeat all the same steps on paper or on your calculator ... but you risk repeating an error if that is all you do. It would be sensible to check your paper calculation using a calculator, or your calculator result by working out the answer on paper, but I want to share two alternative strategies.

The first strategy is simply to perform the reverse operation (mathematicians would call this the inverse operation). This means that, to check a subtraction, we simply ADD the two smaller numbers to obtain the larger one! This is handy and also relatively quick since most of us are rather better and more speedy with our adding skills than we are with subtraction!

Subtraction-fest 12 ~ Checking Your Subtractions By Casting Out Nines

In the preamble to the last video I mentioned that I would share two strategies for checking your subtractions.  This is the second.

The method is known by the term "casting out nines." It is a wonderful skill that is based on number theory and you can get very fast at using it! I taught our daughter how to do this with the numbers on car number plates and, when she was quite young, she would calculate remainders as oncoming cars passed us while I was driving. I suppose that was a rate of one every second or two in heavy traffic.

Subtraction-fest 13 ~ Subtraction Terminology Demystified (Explained)

This is the last video in this "subtraction-fest" series.

So far, I have demonstrated a variety of ways that you may subtract numbers and how you can check your answers. This time I wanted to take a little time to explain the terminology that you may sometimes encounter in mathematics books.

The parts of a subtraction problem were named at a time when almost all scholars wrote and conversed in Latin. We have therefore inherited Latin names for these things.

minuend (that which is to be diminished/reduced in size) - (minus)
subtrahend (that which is taken away from underneath)
difference (the distance that the numbers are "apart")

The verb describing the action is subtract, so an appropriate instruction would be "Subtract 13 from 45."

The preposition used to name the sign is minus (less/diminished by), and the appropriate way to use the word is to read "13 minus 45." Although usage is changing today, in the past it was very poor form (and showed a poor education) to say "minus 13 from 45!" The correct usage was always "subtract 13 from 45."

I hope the series has been helpful for you. In time, I will also discuss addition, multiplication and division in this way.

I had the privilege of being one of Graeme’s students in 2005. As a creatively minded individual I struggled greatly with Mathematics and was falling behind in my HSC studies. Under his careful and creative tutelage I began to truly understand Mathematics for the first time in my life. Graeme patiently taught me to master mathematical concepts in ways I could understand. In the week intensive I spent with him he taught me the entire HSC course, and I went on to receive a Band 6. I cannot recommend him highly enough.

Roxanne G (student, 2005)

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