Mental Calculation - Basics ( Complements )

The unsung hero. A very large part of mental calculation is addition and subtraction.
Any weakness here is death. Alternately, Every tiny improvement here reaps benefits with a nice multiplier effect.

I often stumble when carrying, Having the idea that the complement of 7 is 3 and 47 is 53 at the forefront of your thoughts process removes a chain of reasoning that seems to take forever when you do it. I regularly find myself mentally carrying the one, reducing, adding the 10’s to the 1’s and then subtracting. This takes a bunch of time, is distracting from the larger calculation, and is error prone as heck. If I could choose to unwire something explicitly it would be how I learned to add and subtract at an early age. Observing the complement seems like a far more natural thing than long step addition and subtraction.

Maybe it is just the soroban practice turning my brain to mush but if you don’t have this nailed I expect that it is an opportunity for big gains at low cost.

It was this video that popped the idea into my head…

The missing video from this series is the subtraction with complements but if you think about additions in this fashion then drill subtraction in the same fashion then it should be reasonably obvious. We all know this but if thinking about it in the right way when calculating doesn’t come naturally then like me you may be wasting a bunch of time and energy. That and a bunch of practice to drill the technique. till I stop doing it the hard way.

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For people who don’t know what complements are and why they are so useful an example.
If you need to add 57 and 77 you have a double carry; there is a carry from the ones’ digit into the tens and from the tens in to the hundreds.
Because of the double carry, this addition is relatively difficult to do.

However; if I think, ok let’s start with 77. How much do I need to get to a hundred? => 23.
If I take 23 from 55, there is still 32 left to add. So my answer is 132.

If this is still unclear, let’s try this. Suppose you need to travel first to a town 77 miles away.
Then you go to another town 57 miles away.
What is your total distance driven?
To get to the answer, think of an in between town between town one and town two. This in between town is 100 miles from the start. So if we go from town one to the in between town, our total distance driven is 100 miles. How far is this from the first town that is 77 miles from the start? Well, 23 miles of course.
How much is left to be done from the in between town to town two? Since we have already driven 23 miles out of from a total of 55 miles, there is still 32 miles to drive.
That means the total distance is 132, because we need 100 miles tyo get to the in between town and then another 32 miles to get to town two. Now we have 100 + 32 miles = 132.

Sometimes the complement can be difficult to calculate. Suppose you cannot directly see that the complement of 77 is 23.
You can then ask yourself:
Start with 77 miles. How much to get to 80? => 3
Then how much from 80 to 100? => 20.
Total of the 2 steps is 3 + 20 = 23.

For 77 the complement is easy.
Try 8745, which is more challenging.

However add 55 to go from 8,745 to 8,800 => 55
Now add 200 to get to 9,000. => 200
And maybe add another 1000 to get to 1,000 -> 1,000

In general the whole idea is to go from a difficult (lots of carries) addition to a much easier subtraction.
In the case of 77 + 57, this becomes 77’s completent (23) subtracted from 55 (= 32). And don’t forget to add the 100 => 132.

Likewise subtractions with lots of carries become simple additions.
Example. 122 - 57. In the ‘borrowing method’ that most kids learn at school, you need to first borrow 10 from 20 in order to change the ‘2-7’ part of the ones digits to ‘12-7’.
Then don’t forget that the 2 in the tens digit is now a 1. This is usually done by putting a dot around the 2.
Again we look at ‘1-5’ and realise we need to borrow the 1 form the hundreds to get ‘11-5’.
Done.

Now the complements method, done with distances.
From 57 to get to a 100 is 43. from 100 to 122 is obviously 22.
43 +22 = 65.
Done.

Alternatively first ask youself how much to go from 57 to 60 => 3
Then from 60 to 120 => 60
And then from 120 we need to do the last 2 miles to get to 122.
3+60+2 = 65
Done.

Observe that in both the distance methods we never did a subtraction.
We only added the distance to get to the next big, round number. And whether that is 100 or 60 does not matter much.

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I like the explanation but I think it misses the point. Being able to think in complements removes the plus/minus. The speed is not in knowing what a complement is. The speed is in using it instead of adding or subtracting. Reading the number and recognizing its complement as you go.

Reading the long hand explanation of what it is I completely miss that you don’t need to calculate the complement. You read the complement. You are aware of the complement in the same way you are aware of the number you are looking at.

I see 37, I know => 63
I see 123, I know => 877

In my head, I don’t say “minus or plus”, I don’t distrust my calculation because I didn’t calculate.
I just look at the number from left to right and know my complement.

Now the next step is drilling the sweet bejeesus out of it till I simply use it when the opportunity appears. Stop subtracting or adding when you can use the complement more readily. Don’t add or subtract to get the complement.

All that adding and subtracting and transforming takes a million years in my brain. Accepting that a complement is a fact, not a calculation, removes a bunch of minor distractions… (that plus a serious bunch of practice) I have the idea but I need to wire it up so that it fires when I see it.

This seems incredibly trivial but not being able to be smarter means that I need to find better ways of thinking and beat them into my head with a rock.

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This is interesting!

If I see 77, I immediately see 23 also.
However; I think this comes with using mental calculation a lot.
In your post it looks like you are implying the complement comes automatically?

Does one not need a bit of training and understanding before one just reads the complement?

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I’d say some, but nothing crazy… it’s basically just the Nikhilam Sutra from Vedic Math: all from 9 and the last from 10.

What Bjoern said… It will take some practice but not a ton. I will tell you how much after I have done so as I spend too much time inverting and subtracting or adding when it isn’t necessary. Inefficient thinking.

Doesn’t your soroban practice teach you with every digit you do what the complement to 9 is? The last digit is then simply to 10 to close out the number.

You should see the complement to 10 with squares all the time. Numbers ending in 1 or 9 will end in 1, numbers ending 2 or 8 will end in 4, etc.

With a soroban you are not counting. While it does use complements extensively and is what gave rise to my “aha” moment. What makes the soroban so fast is that working the beads tends to take no mathematical thought at all. Visual patterns and physical manipulations let you (someone other than me) move faster than the thinking part of your brain. Even with anzan, if I think about the numbers I am doing it wrong.

But when I am using numbers the complement makes total sense and reading it left to right when reading the regular number is an obvious improvement. If we are using the vedic analogy then the same can be said for using any base.