Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2010-08-25 13:59:30

irrational
Member
Registered: 2010-08-25
Posts: 6

### Explain why 0.3 = 0.30 = 0.300 etc.

Hello all. Together with a private maths tutor, I am trying to teach a bit of basic maths to a 55 y.o. who had never attended school (yet who is rather bright). The pupil cannot understand why 0.3 = 0.30 = 0.300 etc. and both the tutor and myself are at a loss to explain it in clear terms. I have researched the internet and did not find a clear explanation either. Any idea?

Offline

## #2 2010-08-25 14:11:00

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Hi irrational;

There are two interpretations of that number. Since in the positional number system to the base 10, .3 really is 3 / 10 and .33 is 3  / 10 + 3 / 100 and .3304 is really 3 / 10 + 3 / 100 + 0 / 1000 + 4 / 10000. when you have:

Keeping in mind that 0 / 10 = 0 and 0 / 100 = 0 and 0 / 1000 = 0 etc.

You can see that those trailing zeros are not changing the value of that decimal at all. You are adding nothing to .3 = 3 / 10 many times.

For the sake of completeness there is another definition that applies for computers or numerical analysis .3 does not equal .30 or .300 etc. When dealing with significant digits each one of those decimals implies a different accuracy or error.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #3 2010-08-25 14:23:47

irrational
Member
Registered: 2010-08-25
Posts: 6

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Thanks for the idea. If this is the simplest, then we'll have to go that way. I was hoping to find a more practical (less abstract) explanation.

Last edited by irrational (2010-08-25 14:24:50)

Offline

## #4 2010-08-25 14:33:22

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Hi irrational;

Try to remember that a decimal is a fraction and as such it is also a sum of fractions. The first definition will become more self evident as you play with it.

For example, .785003 is really

or

Feel free to ask as many questions as you need until it makes sense to you.

Try this:

http://www.mathsisfun.com/decimals.html

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #5 2010-08-25 14:41:18

irrational
Member
Registered: 2010-08-25
Posts: 6

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Hi bobbym and many thanks for the great help.

Now, I'll have to convert that into oranges, apples or (better) dollars. Hmmmmm.

Offline

## #6 2010-08-25 14:45:38

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Yes, turning it into an actual problem may help stimulate his synapses. If you can get him to understand that 3 / 10 and 30 / 100 and 300 / 1000 etc are all the same than you are there!

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #7 2010-08-25 22:48:23

bob bundy
Registered: 2010-06-20
Posts: 8,398

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

hi irrational,

Use diagrams whenever possible.  The attached image is not the answer to your question today but it shows how you can make a visual image for explaining something.  On the teaching pages of MIF there are loads of pictures that help to explain the concepts.  I also like to make my own for a specific lesson and my favourite software for this is called Sketchpad.

With regard to 0.3 = 0.30 I'd also use money as I'm sure he'd grasp that \$0.3 = \$0.30 (substitute currency of your choice).

See if your local school has an educational supplies catalogue they'll lend you because you might find lots of kit that would get you past sticky points.

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

## #8 2010-08-25 23:29:35

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

It's worth noting that those aren't always interchangeable.

If a question asks you to round 0.304 to 2 decimal places, then 0.30 is right but 0.3 isn't.

When taking measurements, the number of decimal places is important to show how accurate the measurement was.
When measuring a piece of string, you might use a ruler with cm graduations, and find its length to be 0.87m. This actually means the string is somewhere between 0.865 and 0.875m.

If you came back with a more accurate ruler, you might then find its length to be 0.870m, but this is different because the possible range of lengths is smaller.

But this probably doesn't matter in your context.

Why did the vector cross the road?
It wanted to be normal.

Offline

## #9 2010-08-25 23:36:44

bob bundy
Registered: 2010-06-20
Posts: 8,398

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

hi irrational

Here's a diagram to show that the decimal number system behaves the same under increasing magnifications ( x 10 each time).

Bob

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

Offline

## #10 2010-08-26 07:58:05

irrational
Member
Registered: 2010-08-25
Posts: 6

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Thanks for the ideas. I found this type of table (on ixl.com) which my pupil understood easily:. That image stands for 0.23.

Offline

## #11 2010-09-05 18:36:27

George,Y
Member
Registered: 2006-03-12
Posts: 1,332

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Actually, they are a little bit different.
The 0.30000 may not exist whilst 0.3 could in the real world when the entity can be divided into no more but 10 parts.

X'(y-Xβ)=0

Offline

## #12 2010-09-05 18:58:23

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Hi George;

Not following you on why .300000 does not exist in the real world and .3 does.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #13 2010-09-08 17:44:48

George,Y
Member
Registered: 2006-03-12
Posts: 1,332

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Hi bobbym, it involves the problem whether real matter is abitarily divisible. If not, there is some N that 1/10^M (M>N) of a particular piece does not make sense, like saying 1/100 of a water molecule.

X'(y-Xβ)=0

Offline

## #14 2010-09-08 17:51:38

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Explain why 0.3 = 0.30 = 0.300 etc.

Hi George;

Your thinking of physics, mathematics is a world of it's own. So much so that formalists believe math is just a game. So saying that .300000 doesn't exist because you can't divide an atom is like saying chess doesn't exist because there are no small horses.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline