Is money discrete or continuous?

Discussion in 'Mathematics' started by deleted456, Sep 10, 2007.

1. deleted456

Just marking a homework and a pupil has put money down as discrete.

Thought (1) Money can only take certain values, i.e. whole pennies

Thought (2) Share price suddenly increases and I'm left with £3.38472 ... well it can take any value on the scale in theory, we just don't have the currency to use it in real life.

My instincts tell me it's continuous. Hmm....

2. deleted456

Just marking a homework and a pupil has put money down as discrete.

Thought (1) Money can only take certain values, i.e. whole pennies

Thought (2) Share price suddenly increases and I'm left with £3.38472 ... well it can take any value on the scale in theory, we just don't have the currency to use it in real life.

My instincts tell me it's continuous. Hmm....

3. Lee07

mmmmmmmmmmmmmmmmmmmm
I would have said to students that money is discrete cos "we can count it" however now i am not so sure - anyone else?????????????????

discrete

5. penchoNew commenter

I believe money is continuos, because you can have like you say lotsmore decimal places e.g. in currency coversions, share prices and interest in banks. Just because we round to 2 d.p for example does not mean it cannot be measured to a more accurate degree of accuracy.

A quick internet search desribes Money as continuous data too.
http://www.anticlue.net/archives/000788.htm

6. MathsHOD

Perhaps their is merit in differentiating between 'money' (the physical objects used to buy/sell items with) and 'value' (the theoretical cost of buying/selling an item)?

Could we then argue that 'money' (physical) is discrete but that 'value' (theoretical) is continuous?

If I buy 11 sweets for 3p (total) then this seems valid. The money is discrete but the value (of each sweet) is continuous.

7. salingers

This strikes me as one of those sterile debates that happen too frequently in statistics and give the subject a bad name. By focussing on whether money is discrete or continuous, we lose sight of when the distinction is important.

Does it really matter whether money is discrete or continuous?

It's like agonising over whether the median is in the (n+1)/2 or n/2 position in an ordered list of the data.

What do you want to do with your money data that makes you need to determine whether it is discrete or continuous?

8. penchoNew commenter

I personally think sometimes it is important to raise these questions and discuss them. If a pupl asks is Money discrete or continuous, I would perhaps like to give an answer instead of saying "It doesn't matter".

9. salingers

I'm not saying that it doesn't matter. I'm saying that you should at the same time be asking "does it matter?" Otherwise you are debating how many angels can dance on a pinhead.

10. MrJob

You can only trade in multiples of 1 pence, therefore the system is NOT continuous.

If you can pay 0.15 pence into my bank account I will agree that money is continuous and not discrete.

11. JSimpson

This is a very important point, and contrary to salingers is not a sterile debate - the difference affects how calculations are done which changes the answer.

Money is treated by both the banking and tax industries as continuous, NOT as discrete amounts in pence. As a result, calculations are done to a much higher degree of accuracy than 1 penny.

For an example of why this is important to calculations: Take a column of figures and add them up. Now convert them at a current exchange rate to (say) Euros. As exchange rates run to many decimal places, you will need to round figures. If you round each entry (assuming money is discrete) and add them up you will get one figure. If you add them up unrounded (and then round them if you like) you will get a different answer, which will be the same answer as if you had converted the original total.

Most banking and accounting systems will allow for the posting of fractional pence to and between internal accounts.

Cash transactions are discrete however, and limited to one pence as the smallest unit. This practical limitation of coinage is followed for most(but not all!) transactions between people and organisations, as people generally have the option to convert any balance to hard cash.

If you still think money is discrete, please let BT know. I am sure they will be happy to round up all your itemised call costs to the nearest (discrete) pence and then add them up when calculating your bill!

12. MrJob

I would put money in the same category as children.
Government figures used to say that the average family had 2.4 children based on their calculations.

For economic calculation purposes this is fine, but clearly if you count children the result will always be an integer. 2.4 children can only exist in a theoretical sense.

Children are also a discrete quantity.

13. salingers

The debate is sterile because it loses sight of why we have the terms discrete and continuous in the first place.

But since you insist . . . the value of money is never irrational (insert your own pun here) so it cannot be continuous.

There may be times when it is convenient to treat it as if it were continuous and there may be times when it is convenient to treat it as discrete. Statistics is about modelling; we make assumptions and simplifications and construct our models accordingly.

The sterility of the debate is the apparent underlying assumption that it is worth arguing about whether money is discrete or continuous in absolute terms. The argument should be: which model is the best model to use in a given situation.

14. PiranhaStar commenter

I recall a rather clever trick perpetrated by a bank employee some years ago. Whenever an amount of money was rounded to the nearest penny and the rounding was down, he arranged for the fraction of a penny to go into another account he owned (I am not sure what happened to the amounts rounded up; perhaps the bank always rounded down). He managed to make a decent amount before he got found out.

Coming back to the OP, I think it is usally best to consider money as discrete; not just because it is important from the statistical point of view (it doesn't make any real difference) but because we are taught to round the results of money calculations to the nearest penny, and penalised if we don't. Share prices, exchange rates etc are quoted by convention to a fixed numbver of decimal places. However, when dealing with large amounts, it is far simpler to treat it as continuous.

15. coyoteNew commenter

Regardless of whether money is continuous, citing exchange rates, or otherwise quoting amounts in fractions of a penny is not a demonstration of this. As long as the unrounded amounts are rational numbers, that is.

16. valed

I'll bet anyone 2.038736 pence to a pound that there'll be no agreement on this........

17. penchoNew commenter

I'm probably being thick here but I don't get your comment Salingers

"But since you insist . . . the value of money is never irrational (insert your own pun here) so it cannot be continuous"

What has irrationality got to do with anything?

If Money was discrete then there must be two values for which there isn't a value between. I don't think there is.

Filling up at the petrol atation this morning, Diesel was 86.9p per litre (quoted on outside board).

I was given a receipt and interestingly it said I bought however many litres of diesel at 89.99p per litre. I have never noticed this before. I wonder how many decimal places the till uses.

I wonder if they multiply the number of litres by 89.99999...

Interestingly if I use 7 litres of petrol per day for the whole year that's 2555litres

2555 x 86.9 = £2220.295
2555 x 86.9999 = £2222.85

£2 saving - does it matter - probably not - but if 2 million people do the same per year, then thats £4million more to the petrol station.

18. penchoNew commenter

Mr Job
Why can you put money in same category of children. You can have 1, 2, 3. You can't have 3.6 children for example. Something like 2.4 children is an average - but the original data is discrete.

Now money might go 86p, 87p, but there is always a value that lies between any two values that you can choose (e.g. 86.9p for petrol) and then one petrol station charges (86.99p and so on). This is continuous.

19. salingers

Pencho asks, "What has irrationality got to do with anything?" and adds, "If Money was discrete then there must be two values for which there isn't a value between. I don't think there is."

At first I was going to say that the rationals are not continuous, so that if we preclude irrational values for money, we must conclude that money is not continuous.

Then I looked into it a bit more and the preponderance of the sources suggests that, countintuitively, the rationals are continuous, and for the reason Pencho describes.

So I may have to abandon my claim.

I'll replace it with this observation. Given that the rationals are continuous, it is possible to construct a continuous statistic from a discrete distribution.

All this being said, the question still remains: why does it matter whether money is continuous or not? What are you doing that you would do differently if money were continuous compared with if it were not?

20. MrJob

Each UK pound can be divided into one hundred equal parts. These discrete quantities are called pence. The money system defined by the Bank of England only allows you to TRADE in multiples of 1 pence. Therefore 39.3p cannot exist in this money system.

When you see petrol advertised it is a rate and not a quantity of money. i.e. 86.7p/litre. If you buy one litre of petrol the garage cannot charge you 86.7p beacause this quantity does not exist in the money system defined by the bank. They can only charge 87p or 86p and it is only possible for you to to pay 87p or 86p.

The relevance of all this is that discrete amounts are easier for most people to handle and it speeds things up in the real world. While the lack of accuracy can upset some mathematicians it can make life much more practical for the general public. If money were continuous you would need an infinite number of coins - the money system is discrete rather than continuous for practical reasons.