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Steve teaches Decimals!

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Everything you need to know about decimals!

Decimals are mysterious things at first sight - but really they are a very natural way to describe a number.

When we have a whole number like 5382, we know that this is:

  • 5 thousands.
  • 3 hundreds.
  • 8 tens.
  • 2 ones.

Each digit is worth ten times less than the one to the left of it.

So why not have more digits that are yet smaller?   24.913 can be seen as:

  • 2 tens.
  • 4 ones.
  • 9 tenths.
  • 1 hundredth.
  • 3 thousandths.

We put a "decimal point" there so we know where the 'whole number' part ends and the 'less-than-one' part starts.   It's just a visual aid.   When we write large numbers, we often use a comma to make them easier to read.  We put a comma every three digits to make it easier on the eyes:  1,435,359.67 - and the decimal point is a similar thing.

(Interestingly, in some parts of the world, they use the dot and comma the other way around - so in Italy, you might see:  1.435.359,67 - others use a space to split up the digits and a comma as the 'decimal point':  1 435 359.76 ...surprisingly, the US and UK are somewhat in the minority here!)

So - what's the deal with fractions versus decimals?

As we saw in my blog about fractions (Steve teaches Fractions!) - a number like  100/160 is really just 100 divided by 160.  We can do that division for real and wind up with a decimal:  0.625...so:

Converting a fraction into a decimal:

Divide the top number by the bottom number.

Easy!   You can also convert "mixed numbers" into decimals by dividing out the fraction and just adding the whole number.

What about going the other way?

Converting a decimal into a fraction:

Write the decimal out without the decimal point - make that be the top of the fraction - then write out a '1' with that number of zeroes underneath:

So, if you have 0.423 then:

     423
------
1000

We used three digits after the decimal point - so we have a '1' and three zeroes under the fraction.

If you have zeroes after the decimal point, like 0.000423 - then do this:

     000423
---------
1000000

...then just drop the zeroes at the left of the top number

       423
---------
1000000

As always with fractions, it's worth trying to simplify them - so a number like 0.125 turns out like this:

       125    
------
1000

But we can obviously divide top and bottom by 5 to simplify:

       25    
-----
200

...and by 5 again:

       5    
----
40

...and AGAIN:

      1    
---
8

So 0.125 = 1/8.

You can also have decimals with whole numbers - so 6.125 is converted into a 'mixed number' 6 and 1/8.

Adding and Subtracting Decimals:

Line up the decimal points - add zeroes to the right of the shortest number to make everything line up - then just add them as usual.

Example:   12.345 plus 100.8...first line up the decimal points:

        12.345 
+ 100.8

Then add zeroes to make the decimal bits be the same length:

        12.345 
+ 100.800

...now you can just add them up - pretending that the decimal point isn't there:
        12.345 
+ 100.800
113 145

...and put the decimal point back into the result to get 113.145.

Multiplying Decimals:

Count the total number of digits after the decimal points - multiply the two numbers, ignoring the decimal point - then put it back.

So  1.23 x 4.5 has two digits after the decimal point in the first number - one after the second number - so a total of three digits.

Now multiply 123 by 45 - which is 5535.  Now we need to have three digits after the decimal point - so the answer is 5.535.

Dividing Decimals:

First add zeroes to one or other number so that they have the same number of digits after the decimal point.  Divide them as if the decimal point wasn't there.

So 6.35 / 1.4 can be divided as 6.35/1.40 - which is like 635/140 

 

Reader Forum: Steve teaches Decimals!

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From: Steve Baker  Date: 2017-10-24 11:25:01  

1. The problem lies in the structure of the website. In some sites, people post one long series of stream-of-consciousness "blog entries" to a single page that is their "blog". But others (like me) create separate pages for each topic and carry on conversations in each page separately. Hence each topic is a separate "blog". But this is a new word - the meaning is only slowly settling down. It's rash to start being too definite about it!

2. Yes - you're absolutely right. But I'm trying to write SHORT pieces that don't exceed the attention span of people who need this kind of remedial math! Precision is one of the things on my "To Do" list for future discussion. For the moment, I just want to get the basics across.

From: Archon Shiva  Date: 2017-10-24 08:39:01  

Two tiny corrections:

1. “my blog about fractions”: a blog (web log) is the log as a whole, so it should be “my blog post” or “my blog entry”. The word blog can be implied in either.

2. More technical: dividing and multiplying decimals works that way for pure numbers but if they are *measures*, you can’t create precision (add new decimal positions) by multiplying. 1.23 cm x 4.5 is 5.54 cm, not 5.535. 1.23cm means “somewhere between 1.225 and 1.235cm” However, 1.230cm x 4.5 would be 5.535cm - the zero is known information.