# Number system

jackson6612
I'm trying to learn some math. I was studying about decimal number system.

The place values of base 10 are: ... 10000s 1000s 100s 10s 1s (decimal point) 1/10s 1/100s 1/1000s 1/10000s ...

In positional notation system every lower valued place is used to feed the higher one. Any place value left to the other is higher one.

999 means that there are 9 groups of 1s, 9 groups of 10s, and 9 groups of 100s (from right to left).

0.999 = 9/10 + 9/100 + 9/1000 = 999/1000.
0.9 = 9/10 = 90/100 = 900/1000

0.009 = 0/10 + 0/100 + 9/1000 = It means that there are 9 groups of 1/10000 things.

When 1000 is divided by 100 it can be said 1000 objects have been divided into 10 parts. What would be said when 1/100 is divided by 10? Please guide me on this with some example. Thanks.

jackson6612
Perhaps, it means this. Suppose, a pizza is divided into 100 slices and there are also 100 persons to eat those slices. Each person gets 1 slice which 1/100 of the pizza. One person goes out and he encounters some of his 9 friends as he was just going to take first bite of that slice. Now there are total 10 persons who need an equal share of that slice. It is divided into 10 smaller pieces of equal quantity. Now the original slice, 1/100 of the pizza, has been sub-divided into 10 pieces further pieces, i.e. 1/100 divided by 10 which is equivalent to 1/100 x 1/10 or 1/1000. Do I make sense?

jackson6612

sjb-2812
Sounds a reasonable interpretation to me.

jackson6612
Thanks, Sjb.

When the person went out he had one slice, 1/100 of the original pizza. Take that once as one whole piece, let's a mini pizza. That pizza was divided into 10 pieces, one piece being 1/10 of the mini. In the context of that mini pizza it would be 1/10 but taking that mini as a slice of the original pizza, it would be 1/1000 or (1/100 x 1/10), why? There aren't 1000 persons that one piece now equals 1/1000. That's where mathematics starts sounding weird. 10/1000 = 1/100 = one piece of the original pizza.

100/100 = 100 x (10/1000) = original pizza
for 1 slice, 1/100, there are ten 1/1000 pieces
for 10 slices, 10 x 1/100, there are hundred 1/1000 pieces
for 100 slices, 100 x 1/100, there are 1000 1/1000 pieces

So in this context 1/100 = 10/1000 means the original slice which was 1/100 of the pizza is equal to 10 new slices, where one slice is 1/1000 when viewed in context of the original pizza. Physically, not all the pizza was divided into 1000 pieces rather only one slice which was 1/100 was further divided into 10 pieces. But this is how mathematics work. In mathematics you have to manipulate expressions to find answers and solve real world problems at the expense of some nonsensical steps (you could call them unreal, abstract mathematical entities) while reaching the final 'sensible' answer.

Thank you for all the help.

epkid08
Why do say nonsensical?

jackson6612
Perhaps, it's a poor choice of words. But I think it's true sometimes (most of the time) mathematics uses steps, or make use of some ideas which aren't real or not too match close to reality. We have to make sense out such 'nonsense'.

Sometimes, one may get answer where a step involves 1/2 of a human being. But in reality a human being doesn't exist in a half piece. I hope you get my point.

Mentor
When 1000 is divided by 100 it can be said 1000 objects have been divided into 10 parts. What would be said when 1/100 is divided by 10? Please guide me on this with some example. Thanks.
When 1000 is divided by 100, the 1000 items are being divided into 100 parts, not 10 parts. Each part consists of 100 items.

Mentor
Thanks, Sjb.

When the person went out he had one slice, 1/100 of the original pizza. Take that once as one whole piece, let's a mini pizza. That pizza was divided into 10 pieces, one piece being 1/10 of the mini. In the context of that mini pizza it would be 1/10 but taking that mini as a slice of the original pizza, it would be 1/1000 or (1/100 x 1/10), why?
Because 1/10 of 1/100 = 1/1000. One tenth "of: something means 1/10 "times" that something.
There aren't 1000 persons that one piece now equals 1/1000. That's where mathematics starts sounding weird. 10/1000 = 1/100 = one piece of the original pizza.
The other 99 pieces of the original pizza went to the other 99 people. The one small piece (1/100 of the original pizza) is now being shared by 10 people, so each of these people gets 1/10 of the small piece, which is 1/1000 of the original pizza.

I don't understand what problem you are having with this.
100/100 = 100 x (10/1000) = original pizza
for 1 slice, 1/100, there are ten 1/1000 pieces
for 10 slices, 10 x 1/100, there are hundred 1/1000 pieces
for 100 slices, 100 x 1/100, there are 1000 1/1000 pieces

So in this context 1/100 = 10/1000 means the original slice which was 1/100 of the pizza is equal to 10 new slices, where one slice is 1/1000 when viewed in context of the original pizza. Physically, not all the pizza was divided into 1000 pieces rather only one slice which was 1/100 was further divided into 10 pieces. But this is how mathematics work. In mathematics you have to manipulate expressions to find answers and solve real world problems at the expense of some nonsensical steps (you could call them unreal, abstract mathematical entities) while reaching the final 'sensible' answer.

Thank you for all the help.

jackson6612
I don't understand what problem you are having with this.
\

Thanks a lot, Mark. I was just explaining it to myself. It would be nice of you if could comment a bit on that bold part too. Once again, thanks.

Mentor
Presumably you mean this:
jackson6612 said:
But this is how mathematics work. In mathematics you have to manipulate expressions to find answers and solve real world problems at the expense of some nonsensical steps (you could call them unreal, abstract mathematical entities) while reaching the final 'sensible' answer.
This doesn't make much sense to me. Can you think of some example problems you've seen that entail using nonsensical steps to arrive at a sensible answer? Preferably something other than your question about the pizza divided up into 109 pieces (99 pieces each 1/100 of the pizza + 10 pieces each 1/1000 of the pizza).

jackson6612
Mark: Thanks a lot for the comment. So you found my example about pizza being divided into 109 pieces nonsensical! :-) It was an extra, extra large pizza!

Perhaps, the use of the term 'nonsensical steps' is misleading. What I was unsuccessfully trying to say was that mathematics transform the entire real world into its own world where some of the steps or answers wouldn't make that much sense to us. Those steps aren't nonsensical (allow me to use this term again) in their own right, they seem that way from human point of view. Am I again speaking nonsense here?!

Best wishes
Jack

jackson6612
I'm trying to learn some math. I was studying about decimal number system.

The place values of base 10 are: ... 10000s 1000s 100s 10s 1s (decimal point) 1/10s 1/100s 1/1000s 1/10000s ...

In positional notation system every lower valued place is used to feed the higher one. Any place value left to the other is higher one.

999 means that there are 9 groups of 1s, 9 groups of 10s, and 9 groups of 100s (from right to left).

0.999 = 9/10 + 9/100 + 9/1000 = 999/1000.
0.9 = 9/10 = 90/100 = 900/1000

0.009 = 0/10 + 0/100 + 9/1000 = It means that there are 9 groups of 1/10000 things.

When 1000 is divided by 100 it can be said 1000 objects have been divided into 10 parts. What would be said when 1/100 is divided by 10? Please guide me on this with some example. Thanks.

Hi

Is it correct to say "9 groups of 1s"? It doesn't make any sense to me. To say that in '678' there are 6 groups of 100s makes sense.

In the quoted text I said in 0.009 there are 9 groups of 1/0000. I didn't say there 9 groups of 1/000. Why? Because the logic which holds true for the digits on the left side of decimal point (as I said there 6 groups of 100s in 678) doesn't hold true for the digits on the right side of the point. 1/1000 = 10 x 10/10000. That is, 1/1000 contains in itself 10 one-thousandth pieces of something. Correct. Please guide me.

Homework Helper
Hi

Is it correct to say "9 groups of 1s"?
Would it make sense to you to say that there were 9 groups, each containing 1 thing?

It doesn't make any sense to me. To say that in '678' there are 6 groups of 100s makes sense.
Why would it make more sense to say that "there are 6 groups of 100s" but not "8 groups of 1s"? I would be inclined to say "6 groups of 100 things" and "8 groups of 1 thing".

In the quoted text I said in 0.009 there are 9 groups of 1/0000. I didn't say there 9 groups of 1/000. Why?
Because you were wrong? "0.009" means 9/1000, not 9/10000.

Because the logic which holds true for the digits on the left side of decimal point (as I said there 6 groups of 100s in 678) doesn't hold true for the digits on the right side of the point. 1/1000 = 10 x 10/10000. That is, 1/1000 contains in itself 10 one-thousandth pieces of something. Correct. Please guide me.

jackson6612
Thanks a lot, Halls.

Would it make sense to you to say that there were 9 groups, each containing 1 thing?

No, it won't. Because the definition of 'group' implies that there are two or more things involved. So, guide me on this.

Why would it make more sense to say that "there are 6 groups of 100s" but not "8 groups of 1s"? I would be inclined to say "6 groups of 100 things" and "8 groups of 1 thing".

Okay, poor choice of words. I also intended to say 6 groups 100 things, and not of 100s. Sorry.

Because you were wrong? "0.009" means 9/1000, not 9/10000.

No, I'm not completely wrong. In the original post, I said:
0.009 = 0/10 + 0/100 + 9/1000 = It means that there are 9 groups of 1/10000 things.

It wouldn't make sense, at least to me, to say that there 9 groups of 1/1000 things. Then, how would extend that idea of grouping to the digits on the right side of decimal point?

1/1000 = 10 x 10/10000. That is, 1/1000 contains in itself 10 one-thousandth pieces/things of something. Correct? Please guide me.

Best wishes
Jackson

Mentor
Thanks a lot, Halls.
HallsofIvy said:
Would it make sense to you to say that there were 9 groups, each containing 1 thing?
No, it won't. Because the definition of 'group' implies that there are two or more things involved. So, guide me on this.
That's an overly narrow definition of this term. According to one of my dictionaries, a group is an assemblage of persons or objects. It doesn't specify that there must be two or more things in a group.

HallsofIvy said:
Because you were wrong? "0.009" means 9/1000, not 9/10000.
No, I'm not completely wrong. In the original post, I said:

0.009 = 0/10 + 0/100 + 9/1000 = It means that there are 9 groups of 1/10000 things.
No, you were wrong then and now. 0.009 means 9/1000. There are not "9 groups of 1/10000 things" whatever that means. There is a group with 9 things, each of which is 1/1000, to stay consistent with your phrasing. I would talk instead about how many tenths, hundredths, thousandths, etc., there are. In this case there are no tenths, no hundredths, and 9 one-thousandths. Period.
It wouldn't make sense, at least to me, to say that there 9 groups of 1/1000 things. Then, how would extend that idea of grouping to the digits on the right side of decimal point?

1/1000 = 10 x 10/10000.
No, not at all. On the left side you have 1/1000. On the right side you have 100/10000, which is the same as (equal to) 1/100.
That is, 1/1000 contains in itself 10 one-thousandth pieces/things of something.
Correct?
Again, no. 1/1000 could be divided up into 10 one-ten-thousandths. I.e., 1/1000 = 10 * 1/10000.

Think about it in terms of coins. A dime is 1/10 of a dollar, or $.10. A dime consists of 10 cents, or 10 hundredths of a dollar, so in terms of dollars as the unit, 1/10 = 10 * 1/100. jackson6612 That's an overly narrow definition of this term. According to one of my dictionaries, a group is an assemblage of persons or objects. It doesn't specify that there must be two or more things in a group. No, you were wrong then and now. 0.009 means 9/1000. There are not "9 groups of 1/10000 things" whatever that means. There is a group with 9 things, each of which is 1/1000, to stay consistent with your phrasing. I would talk instead about how many tenths, hundredths, thousandths, etc., there are. In this case there are no tenths, no hundredths, and 9 one-thousandths. Period. No, not at all. On the left side you have 1/1000. On the right side you have 100/10000, which is the same as (equal to) 1/100. Again, no. 1/1000 could be divided up into 10 one-ten-thousandths. I.e., 1/1000 = 10 * 1/10000. Think about it in terms of coins. A dime is 1/10 of a dollar, or$.10. A dime consists of 10 cents, or 10 hundredths of a dollar, so in terms of dollars as the unit, 1/10 = 10 * 1/100.

Thanks a lot, Mark.

Mark, I think now I'm also going to learn English from you too. But I would disagree on that definition of a group. Please remember that I'm an English learner. M-W nowhere says that a group could consist of one thing: http://www.merriam-webster.com/dictionary/group. Please correct me. Thanks.

'678' would mean that there are 6 groups with each containing 100 items, and 7 groups each with 10 items... I don't really get what I say for '0.009'. Should I say something like there 9 groups with each containing 1/1000 items. But the problem is '1/1000' is not even a one complete item, let alone items. Please guide me. Thanks.

Best regards
Jackson

Mentor
In my Merriam-Webster dictionary, definition 2a says "a number of individuals assembled together or having some unifying relationship"
2b says "an assemblage of objects regarded as a unit"

"a number of individuals" doesn't specify that there have to be at least two.
Also, I interpret "an assemblage of objects" to mean one or more.

For .009 it's fine to say 9 thousandths and be done with it.

With your first example, 678, as you point out, there are 6 groups of 100 each, 7 groups of 10 each, and 8 groups, each with 1 thing. Once you get on the other side of the decimal point, phrasing things in terms of groups is not very useful, I don't believe.

Maybe it's better to completely abandon the "groups" terminology and recognize that where a numeral is in a decimal number indicates which power of 10 it multiplies.

For example, 1205.37
Going from the left end to the right, we have thousands, hundreds, tens, ones, tenths, and finally hundredths.

Our decimal number system is a kind of shorthand for saying
1 x 1000 +
2 x 100 +
0 x 10 +
5 x 1 +
3 x 1/10 +
7 x 1/100

Or, in another way
1 x 103 + 2 x 102 + 0 x 101 + 5 x 100 + 3 x 10-1 + 7 x 10-2

Mensanator
I'm trying to learn some math. I was studying about decimal number system.

The place values of base 10 are: ... 10000s 1000s 100s 10s 1s (decimal point) 1/10s 1/100s 1/1000s 1/10000s ...

In positional notation system every lower valued place is used to feed the higher one. Any place value left to the other is higher one.

999 means that there are 9 groups of 1s, 9 groups of 10s, and 9 groups of 100s (from right to left).

0.999 = 9/10 + 9/100 + 9/1000 = 999/1000.
0.9 = 9/10 = 90/100 = 900/1000

0.009 = 0/10 + 0/100 + 9/1000 = It means that there are 9 groups of 1/10000 things.

When 1000 is divided by 100 it can be said 1000 objects have been divided into 10 parts. What would be said when 1/100 is divided by 10? Please guide me on this with some example. Thanks.

To divide rayionals, invert abd multiply:

1/100 / 10 = 1/100 * 1/10 = 1/1000