Solving Division: Understanding Zero as a Placeholder

[roger]

HI

please can someone help me with this quick question on division :

If I have to do 3 divided by 4
____
4 |3


4 doesn't go into 3 so I put a zero on top .

When it gets to 4 into 2(from 30-4x7), it doesn't go, so why don't I put a zero on top but instead put a zero beside the 2 to make it 20 ?


thankyou



roger

 

[dav2008]

I have no idea what you're doing.

If you're doing 3 divided by 4 using long division:

http://home.comcast.net/~iberiaforums/division.GIF

 

[arildno]

roger:
Do you want to perform 3:4 or 4:3?

 

[roger]

roger:
Do you want to perform 3:4 or 4:3?

DEAR ARILDNO,

I AM TRYING TO DO 3 DIVIDED BY 4 LONG DIVISION

BUT WHY DONT I PUT A ZERO ON TOP FOR 4 INTO 2 ?

ROGER

 

[dav2008]

DEAR ARILDNO,

I AM TRYING TO DO 3 DIVIDED BY 4 LONG DIVISION

BUT WHY DONT I PUT A ZERO ON TOP FOR 4 INTO 2 ?

ROGER

Look at my link.

You never put a 0 on top.

What you do in the first step is put a decimal point on top.

 

[roger]

Look at my link.

You never put a 0 on top.

What you do in the first step is put a decimal point on top.

just looking at the link...

when you do 7x4 =28

and then 30 minus 28 = 2.

from here

4 doesn't go into 2 so why don't I put a zero on top ?

This is what I want to know ...


thanx


roger

 

[arildno]

Now, since to write down the long-division algoritm is a bit difficult (and besides, we do it differently in Norway from the US), I will show you the RATIONALE behind the long division technique instead; long division is simply a condensed version of what I'll present.

1. What is meant by "division" in this case?
Ordinarily, "division" is meant to be that process that rewrites a number, given as a FRACTION, into the equivalent DECIMAL REPRESENTATION of that number (which, as it happens, is a particular TYPE of fractional representation).
2. Let's look at the number given by fraction 3/4

We want to write 3/4 in its decimal representation, that is to find digits $$a_{i}$$ between 0 and 9, so that we have:
$$\frac{3}{4}=0.a_{1}a_{2}a_{3}...$$
Where the notation $$0.a_{1}a_{2}a_{3}...$$ MEANS:
$$0.a_{1}a_{2}a_{3}...\equiv\frac{a_{1}}{10}+\frac{a_{2}}{100}+\frac{a_{3}}{1000}++$$

3. Let's start!
a) We note that 3<4, so our first step is to multiply 3/4 with an appropriate representation of the number "1":
$$\frac{3}{4}=1*\frac{3}{4}=\frac{10}{10}*\frac{3}{4}=\frac{1}{10}*(\frac{30}{4})$$
Henceforth, we will work with the expression included in the parenthesis.
b)
We note that 30>4, and the greatest multiple of 4 which is less than 30, is 4*7=28.
Hence, we write 30=4*7+2
We therefore have the equality:
$$\frac{30}{4}=\frac{4*7+2}{4}$$
c)We now use the fact that we can split up a sum in the numerator into a sum of fractions:
$$\frac{4*7+2}{4}=\frac{4*7}{4}+\frac{2}{4}=7+\frac{2}{4}$$
The last step follows since 4 is a common factor in both the numerator and denominator in the first fraction.
d) Hence we have shown:
$$\frac{3}{4}=\frac{1}{10}*(7+\frac{2}{4})$$
This can be rewritten as:
$$\frac{3}{4}=\frac{7}{10}+\frac{1}{10}(\frac{2}{4})=0.7+\frac{1}{10}(\frac{2}{4})$$
e) We will now work with the parenthesized 2/4.
Since 2<4, we multiply 2/4 by an appropriate version of 1:
$$\frac{2}{4}=1*\frac{2}{4}=\frac{10}{10}*\frac{2}{4}=\frac{1}{10}*(\frac{20}{4})$$
f) We note that 20=5*4, so we have:
$$\frac{2}{4}=\frac{1}{10}*(\frac{5*4}{4})=\frac{1}{10}*(5)=\frac{5}{10}$$
g) We now look back at the equation in d):
$$\frac{3}{4}=0.7+\frac{1}{10}(\frac{2}{4})$$
With the result from f), we have:
$$\frac{3}{4}=0.7+\frac{1}{10}(\frac{5}{10})=0.7+\frac{5}{100}=0.7+0.05=0.75$$

And that's our result..

 

[HallsofIvy]

I'm confused about your question "WHY DONT I PUT A ZERO ON TOP FOR 4 INTO 2 ?" because I certainly WOULD put a 0 on top!

4 goes into 3 0 times so I would put a 0 before the decimal point and continue:

__0.
4) 3.0

Now, 4 divides into 30 (ignore the decimal point now- we've already taken care of it) 7 times: 4*7= 28

__0.7
4) 3.00
_28_
20
and now 4 divides into 20 5 times, evenly

__0.75_
4) 3.000
_2 8__
20
20

Of course, you don't HAVE to put the "0 on top" because 0, after all, means NOTHING! 0.75 is the same as .75 although I think you will find that "0.75" is preferred in "formal" writing.

 

[roger]

I'm confused about your question "WHY DONT I PUT A ZERO ON TOP FOR 4 INTO 2 ?" because I certainly WOULD put a 0 on top!

4 goes into 3 0 times so I would put a 0 before the decimal point and continue:

__0.
4) 3.0

Now, 4 divides into 30 (ignore the decimal point now- we've already taken care of it) 7 times: 4*7= 28

__0.7
4) 3.00
_28_
20
and now 4 divides into 20 5 times, evenly
this is where my concern was...
I thought it was 4 divides into 2 NOT 20.
So is that to say because the decimal point has been put down, it is really the calculation of 30 divided by 4 ?
Because if it had been 2 instead of 20, my question was if 4 does not go into 2, then why do I not put a zero on top to indicate this .And then I would have brought down a zero to make it 20 and proceed from there...
If I ever have to bring down a zero, where does it come from ? __0.75_
4) 3.000
_2 8__
20
20

Of course, you don't HAVE to put the "0 on top" because 0, after all, means NOTHING! 0.75 is the same as .75 although I think you will find that "0.75" is preferred in "formal" writing

 

[HallsofIvy]

"Because if it had been 2 instead of 20, my question was if 4 does not go into 2, then why do I not put a zero on top to indicate this "

And my answer was you certainly can "put a zero on top to indicate this."

You don't HAVE to since ".75" is commonly understood as "0.75" although, in my opinion, 0.75 is better.

 

[roger]

"Because if it had been 2 instead of 20, my question was if 4 does not go into 2, then why do I not put a zero on top to indicate this "

And my answer was you certainly can "put a zero on top to indicate this."

You don't HAVE to since ".75" is commonly understood as "0.75" although, in my opinion, 0.75 is better.

It still doesn't answer my query because if I had done it my way, I would have got an answer of 0.705

This is wrong it should be 0.75 but I'm referring to the zero after the 7 not before the decimal point ?


Please help me to understand this


Roger

 

[faust9]

It still doesn't answer my query because if I had done it my way, I would have got an answer of 0.705

This is wrong it should be 0.75 but I'm referring to the zero after the 7 not before the decimal point ?


Please help me to understand this


Roger

Ok, I don't understand why you WANT to put the zero on the top?

Lets walk through an easy example the way you want to do it and the correct way:

116/4

___2
4)116

2*4=8

11-8=3

Now 3/4=0

___20
4)116

bring down the 6

36/4=9

___209
4)116


Does that look like the correct answer? No. The reason you don't put the 0 on the top is that you don't redivide the remainder from one step to the next.

Good luck.

 

[dav2008]

It still doesn't answer my query because if I had done it my way, I would have got an answer of 0.705

This is wrong it should be 0.75 but I'm referring to the zero after the 7 not before the decimal point ?


Please help me to understand this


Roger

Your question is like asking "why don't you add a zero when you divide 30/5 and get 6, not 60"

You just don't. That's how it's done.