dividing small number over a larger number concept

Hannibal lecter · 2022-03-09 09:02:35

What is the concept of dividing small number over a larger one (ex: 1/6)
1/6=0.16666666666
is 1 divided into 6 pieces and become 0.16666666666
and these pieces called units of 1 or something like that?
so 0.16666666666 * 6 = 1
what is the concept of this division and why number one cut into pieces and become exactly 0.16666666666
how to measure that or represent it on Number line!
also when calculation 0.16666666666 * 6 it's 0.99999999996 not 1 !!!
and is there a resources and math books to illustrate this but not YouTube channel please it's only give the ways to calculate it not explantion.

and I have example of a second problem it's 6/15 as following in the picture ..
my question is who gives as the rights or standards to put zero down? (u will see what I mean from the picture)
and my second question is who gives as the right and standards to place a decimal point (you will see that also from the picture)

Jai Ganesh · 2022-03-09 14:00:14

Hi,

6/15 (0.4) is a terminating decimal whereas 1/6 is a recurring decimal.

See the links below:

Decimals, Factions and Percentages.

Ordering Decimals.

Bob · 2022-03-09 22:14:58

I'm old enough to have used mechanical calculators to do arithmetic. To do a multiplication you had to do repeated addition (easy example 5 x 4 = 4 + 4 + 4 + 4 + 4) And to do division you had to do repeated subtraction.

example. If the calculation was 48 divided by 6 you would set up 48 in the register, then set up 6 as the amount to subtract, and then wind the handle backwards so the display showed 42, then 36 then 30 and so on until zero was showing. A side register would show how many times you had turned the handle and it would be showing 8 in the case.

Even after the invention of electronic calculators the machines were really useful for teaching what division actually is.

If the calculation was 1 divided by 8 the procedure was as follows:

Set up 1 in the register, and 8 on the dials. Turn the handle back once. A bell would ring to indicate that you had taken away more than you had to start with (8 from 1). you would turn the handle back once to restore the register and then move the carriage one place so the subtraction was now not -8 but rather -0.8. Turn the handle once and the machine does 1 - 0.8 = 0.2. Do it again and the bells rings. Too much! Turn back once and move the carriage. Now you are subtracting 0.08 each time. Turn once 0.2 - 0.08 = 0.12. Again 0.12 - 0.08 = 0.04. Again and the bell rings as you've taken away too much. Move the carriage so now you are subtracting 0.008 each time. Turn once, 0.04 - 0.008 = 0.032. Again, 0.032 - 0.008 = 0.024 . Again 0.024 - 0.008 = 0.016. Again 0.016 - 0.008 = 0.008. Again 0.008 - 0.008 = 0. As zero has been reached the calculation is over. The handle turning register is showing 0.125 because it registered 0.1 when the carriage was moved the first time, another 0.02 when the carriage was moved again, and finally 0.005 for the final turns.

I hope you were able to follow all that. It would obviously be easier if you has one of these machines, but you can do it on paper like this:

1-
0.8
0.2-
0.08
0.012

etc

Our number system is based on powers of ten so its easy to do say 3 x 40 by doing 3 x 4 and adding a zero to shift the place value up one place.

When you set out a division you are actually doing a repeated subtraction.

example. 6 divided by 15.

First you try 6 - 15. Cannot do that as you're trying to take away too much. So 'shift the carriage' and try to take away 1.5s. You can do that exactly 4 times; 6 - 1.5 - 1.5 - 1.5 - 1.5 = 0. So we record that in the answer space as not 4 subtracts but 0.4 subtracts because we shifted the carriage one place.

example. 1 divided by 6

Cannot do 1-6, so 'shift the carriage' and do 1 - 0.6 = 0.4. You can do that once but then there's not enough left to do it again.
So 'shift the carriage' and do 0.4 - 0.06 - 0.06 - 0.06 - 0.06 - 0.06 - 0.06 = 0.04. Once again we have to stop as there's not enough left to subtract any more 0.06s. Move the carriage to subtract 0.006 each time. Once again you can do this 6 times and then you have to 'shift the carriage' again. You can see that this will go on for ever which is why the answer space contains recurring digits.

It's more likely such a calculation will recur rather than terminate. Why? 10 = 2 x 5. 100 = 2 x 2 x 5 x 5. 1000 = 2^3 x 5^3

If you divide by a number that only has 2s and 5s as factors then it will terminate. That's why 1/8 terminates. But when you divide by 3 or 7 or 11 etc the calculation will never terminate (because no matter how many times you 'shift the carriage' the top number will not be an exact factor of the bottom number)

6/15 looks like it should be recurring as the bottom has 3 as a factor, but actually 6/15 is not yet fully cancelled down. If you divide top and bottom by 3, you get 2/5 and now you can see it ought to terminate as the divisor is 5.

1/7 is interesting. When you divide by 7, if it doesn't divide exactly then there are 6 possible remainders. As you carry out the division, all 6 remainders occur. After that there can only be a repeated remainder so the division recurs with a cycle of 6 digits.
1/7 = 0.142857142857142857........

From this idea you can move on to realise that every fraction conversion to decimal must either terminate or recur.

Bob

Hannibal lecter · 2022-03-13 10:09:43

Hannibal lecter wrote:

Bob wrote:
++++++++++++++++++++++++++++++++++++
I'm old enough to have used mechanical calculators to do arithmetic.
can you please tell me or send me a photo of that mechanical calculators and where to do you have a similar links to buy it from amazon or ebay
And to do division you had to do repeated subtraction.
++++++++++++++++++++++++++++++++++++
I couldn't understand how division is a repeated subtraction
++++++++++++
example. If the calculation was 48 divided by 6 you would set up 48 in the register, then set up 6 as the amount to subtract, and then wind the handle backwards so the display showed 42, then 36 then 30 and so on until zero was showing. A side register would show how many times you had turned the handle and it would be showing 8 in the case.
++++++++++++
but 48 divided by 6 is 8 directly how could it displayed 42 at first from where did we get this number even in papers couldn't find it
++++++++++++
If the calculation was 1 divided by 8 the procedure was as follows:
Now you are subtracting 0.08 each time.
1-
0.8
0.2-
0.08
0.012
++++++++++++
the number is 0.08?? is very ambiguous to me, from where we get that number even in papers I tried a lot I couldn't figure it
you said " subtracting 0.08 each time "
and thanks you so much you understand what I want it's really my problem I can't understand division process

Last edited by Hannibal lecter (2022-03-13 10:12:10)

Mathegocart · 2022-03-13 16:48:29

Hannibal lecter wrote:

Hannibal lecter wrote:
Bob wrote:
++++++++++++++++++++++++++++++++++++
I'm old enough to have used mechanical calculators to do arithmetic.
can you please tell me or send me a photo of that mechanical calculators and where to do you have a similar links to buy it from amazon or ebay
And to do division you had to do repeated subtraction.
++++++++++++++++++++++++++++++++++++
I couldn't understand how division is a repeated subtraction
++++++++++++
example. If the calculation was 48 divided by 6 you would set up 48 in the register, then set up 6 as the amount to subtract, and then wind the handle backwards so the display showed 42, then 36 then 30 and so on until zero was showing. A side register would show how many times you had turned the handle and it would be showing 8 in the case.
++++++++++++
but 48 divided by 6 is 8 directly how could it displayed 42 at first from where did we get this number even in papers couldn't find it
++++++++++++
If the calculation was 1 divided by 8 the procedure was as follows:
Now you are subtracting 0.08 each time.
1-
0.8
0.2-
0.08
0.012
++++++++++++
the number is 0.08?? is very ambiguous to me, from where we get that number even in papers I tried a lot I couldn't figure it
you said " subtracting 0.08 each time "
and thanks you so much you understand what I want it's really my problem I can't understand division process

Hey Hannibal lecter,

As for the mechanical calculators, this site provides an eloquent and comprehensive overview of their functions and history. There are also some captivating pictures that the site owner has graciously provided.

They're available on eBay(this one appears to be going for $200 including shipping costs where I live.) Can't seem to find any on Amazon though - I'm afraid eBay's your best bet here.

Last edited by Mathegocart (2022-03-13 16:50:49)

Bob · 2022-03-14 00:20:52

https://www.youtube.com/watch?v=aDN4s8ElxqE

Bob

Hannibal lecter · 2022-03-16 07:47:23

First you try 6 - 15. Cannot do that as you're trying to take away too much. So 'shift the carriage' and try to take away 1.5s. You can do that exactly 4 times; 6 - 1.5 - 1.5 - 1.5 - 1.5 = 0. So we record that in the answer space as not 4 subtracts but 0.4 subtracts because we shifted the carriage one place.

bob why we can't do repeated subtraction
like 8 ÷ 4
8 - 4 = 4 (1 time)
4 - 4 = 0 (1 time)
then the result is 2

why in 5 ÷ 16 you said (Cannot do that as you're trying to take away too much)
what you mean by that can you please explain further to me

and after that you said :-

, so 'shift the carriage' and do 1 - 0.6 = 0.4. You can do that once but then there's not enough left to do it again.

what you mean by "there is not enough left"

can you please explain that further for a beginner

Hannibal lecter · 2022-03-16 13:20:48

So we record that in the answer space as not 4 subtracts but 0.4 subtracts because we shifted the carriage one place.

I mean we made 15 become 1.5 the number is changed who gives the right to do such a thing is that legal in math and clearly understandable? is there explaintion for that ..
I mean we chose a different number it's 1.5
I can't imagine that is there interactive example like 6 apples iver 16 people

Bob · 2022-03-16 21:04:06

Suppose a group of 15 friends have 6 dollars that they want to share out equally. How to do this without doing a division?

Let's start with an easier share out.

Suppose it was $120 between 15.

120 - 15 = 105 105 - 15 = 90 90 - 15 = 75 75 - 15 = 60
60 - 15 = 45 45 - 15 = 30 30 - 15 = 15 15 - 15 = 0

I was able to subtract 15, eight times so give each person $8.

Check. $8 x 15 = $120.

So a way to do this is "Keep subtracting 15 until you reach zero. Count how many subtractions you have done."

So let's try that with $6.

We hit a problem straight away because I don't have enough to do the subtraction even once.

'Move the carriage' in this example means change $6 into 60 dimes. I've moved the decimal point one place.

I can take 15s away now.

60 - 15 = 45 45 - 15 = 30 30 - 15 = 15 15 - 15 = 0

So I've done the subtraction 4 times. Does that mean each friend gets $4 ? Well obviously not! So what do they get?

Because I 'moved the carriage' I was subtracting dimes not dollars. So does it work if I give each friend 4 dimes?

4 dimes x 15 = 60 dimes = $6. Yes that works. So 6 / 15 = 0.4

The underlying math that allows you do do this is called 'place value'.

Here's a simple sum that shows it in action. Hope it displays properly for you.

240 / 6 ??

_4_
6 | 240

6 into 24 goes 4 times. But the answer isn't 4, because the place value of that 4 is 4 tens not 4 units. To complete the calculation you have to fill the units space with a 0, so the answer is 40.

Suppose the question was 2.4 / 6 ??

You can set out the same sum but you need to put in a decimal point in the right place.

_.4_
6 | 2.40

To make sure that you always get the point in the right place set out your calculations neatly keeping the points lined up vertically.

Because our number system goes in tens you can turn any sum into a simpler one if you're happy to use the point to correct it afterwards.

Let's say you want to do 1 / 3 as a decimal.

Let's try 100 / 3 first.

100 - 3 = 97 97 - 3 = 94 Oh dear! This will take ages. Let's speed it up by taking 30 at a time.

100 - 30 = 70 70 - 30 = 40 40 - 30 = 10

Now I haven't got enough left to take another 30 so I'll switch to taking 3s.

10 - 3 = 7 7 - 3 = 4 4 - 3 = 1

Now I haven't got enough to take any more 3s so I'll switch to taking away 0.3 each time.

1 - 0.3 = 0.7 0.7 - 0.3 = 0.4 0.4 - 0.3 = 0.1

This is going to go on forever because I always get a bit left over.
If I was setting this out as a division the answer space would have this:

33.333333

If I want to do 1 / 3, it's exactly the same except the decimal point shifts two places left.

1 / 3 = 0.33333333

Bob

Hannibal lecter · 2022-03-17 03:05:56

Bob wrote:

Let's say you want to do 1 / 3 as a decimal.
Let's try 100 / 3 first.
100 - 3 = 97 97 - 3 = 94 Oh dear! This will take ages. Let's speed it up by taking 30 at a time.
100 - 30 = 70 70 - 30 = 40 40 - 30 = 10
Now I haven't got enough left to take another 30 so I'll switch to taking 3s.
10 - 3 = 7 7 - 3 = 4 4 - 3 = 1
Now I haven't got enough to take any more 3s so I'll switch to taking away 0.3 each time.
1 - 0.3 = 0.7 0.7 - 0.3 = 0.4 0.4 - 0.3 = 0.1
This is going to go on forever because I always get a bit left over.
If I was setting this out as a division the answer space would have this:
33.333333
If I want to do 1 / 3, it's exactly the same except the decimal point shifts two places left.
1 / 3 = 0.33333333
Bob

I understand everything now completely but that last segment of your post it's a little not clear to me
because you take 100 - 3 until you reached 40 - 30 = 10 then stopped (first time)
then you take 10 - 3 = 7 until you reached 4 - 3 = 1 then stopped (second time)
finally you take 1 - 0.3 = 0.7 until you reached 0.4 - 0.3 = 0.1 then stopped (third time)
so the number would be 0.3? right? it's 3 times right?? how it's 33.333333 or 0.33333333 it should be 0.3 right

Bob · 2022-03-17 03:23:52

Multiplication is repeated addition eg. 4 x 5 = 4 + 4 + 4 + 4 + 4

Division is the opposite, so repeated subtraction. You have to count how many times the divisor is subtracted.

100 - 30 = 70 70 - 30 = 40 40 - 30 = 10

The divisor is 3. We are taking away 30 at a time (ten 3s) so we've taken away 10 + 10 + 10 3s = 30 3s.

30 is the first part of the answer (correct word is the quotient.)

Now I haven't got enough left to take another 30 so I'll switch to taking 3s.

10 - 3 = 7 7 - 3 = 4 4 - 3 = 1

I've taken away another three 3s, so add that to get the new quotient 30 + 3 = 33

Now I haven't got enough to take any more 3s so I'll switch to taking away 0.3 each time.

1 - 0.3 = 0.7 0.7 - 0.3 = 0.4 0.4 - 0.3 = 0.1

This time I'm only taking away one tenth of a 3 each time and I've done that three times so add another 0.1 + 0.1 + 0.1 to the quotient to make the new quotient 33.3

This keeps happening (take 0.03 each time next) so I can guess ahead that the final quotient is 33.333333 for ever.

I'm thinking of creating an example that shows that repeated subtraction, short division, and long division are all the same thing. Would you like to see it?

Bob

Hannibal lecter · 2023-06-28 23:34:09

1/7 is interesting. When you divide by 7, if it doesn't divide exactly then there are 6 possible remainders. As you carry out the division, all 6 remainders occur. After that there can only be a repeated remainder so the division recurs with a cycle of 6 digits.
1/7 = 0.142857142857142857........

From this idea you can move on to realise that every fraction conversion to decimal must either terminate or recur.

How to find 0.142857 using pen and papers
I keep subtracting 1−0.7−0.07−0.07−0.07−0.07−0.007−0.007−0.0007−0.007−0.0007−0.00007−0.0007−0.0007−0.00007−0.0007−0.00007−0.00007−0.00007−0.00007−0.00007−0.00007−0.00007−0.00007−0.00007−0.00007−0.00007

Until the calculator exceed the maximum number
And I didn't get any of these values 0.142857

I know it's non terminated decimal but I can't even get the first numbers while doing reapeted subtracting as you tought me

Also when I do 1/4 I get 1−0.4−0.4−0.04−0.04
−0.04−0.04−0.04=0
But I don't get the results which is 0.25

Also with 3/4=0.75
Is 3−0.4−0.4−0.4−0.4−0.4
−0.4−0.4−0.04−0.04−0.04
−0.04−0.04=0
But I didn't get the value 0.75

Last edited by Hannibal lecter (2023-06-28 23:55:08)

Bob · 2023-06-29 00:44:58

Further explanation.

Also when I do 1/4 I get 1−0.4−0.4−0.04−0.04
−0.04−0.04−0.04=0

First try to subtract 4. Cannot do this. So subtract 0.4 (twice). So that's not 2 subtractions of 4 but rather 2 subtractions of 0.4. Because we've shifted one place right in terms of units, tenths, hundredths etc we record this as 0.2.

Cannot take off any more 0.4 so take off 0.04. You've managed to do this 5 times so record this as 0.05.

Put that together with 0.2 and we get the answer 1/4 = 0.2 + 0.05 = 0.25

Bob

Math Is Fun Forum

#1 2022-03-09 09:02:35

dividing small number over a larger number concept

#2 2022-03-09 14:00:14

Re: dividing small number over a larger number concept

#3 2022-03-09 22:14:58

Re: dividing small number over a larger number concept

#4 2022-03-13 10:09:43

Re: dividing small number over a larger number concept

#5 2022-03-13 16:48:29

Re: dividing small number over a larger number concept

#6 2022-03-14 00:20:52

Re: dividing small number over a larger number concept

#7 2022-03-16 07:47:23

Re: dividing small number over a larger number concept

#8 2022-03-16 13:20:48

Re: dividing small number over a larger number concept

#9 2022-03-16 21:04:06

Re: dividing small number over a larger number concept

#10 2022-03-17 03:05:56

Re: dividing small number over a larger number concept

#11 2022-03-17 03:23:52

Re: dividing small number over a larger number concept

#12 2023-06-28 23:34:09

Re: dividing small number over a larger number concept

#13 2023-06-29 00:44:58

Re: dividing small number over a larger number concept

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