## MILESTONE 11: Decimal Numbers

The following lessons provide a basic understanding of decimal numbers.

MATH MILESTONE #B6: DECIMAL NUMBERS

A “division” or “ratio” notation is not the only notation possible to express fractions. We may use the place values to the right of ONE to account for fractions. Since the successive place values change by TENS, we call such numbers the DECIMAL NUMBERS.

The place values successively magnify by TENS as one moves to the left. And, the place values successively shrink by a TENS as one moves to the right. Thus, to the right of ONES we have, tenths, hundredths, thousandths, and so on. With these fractional place values we may express fractions to a desired accuracy.

The DECIMAL POINT is used in a decimal to separate the fractional portion. Thus, it appears  to the right of the place value of ONE in the number. It is this decimal notation that we use on calculators and computers.

Computation with decimals follows the same procedure as with the whole numbers. The only additional requirement is keeping the track of the position of the decimal point.

In short, the decimals are a natural extension of the existing whole number system to account for the fractions.

Here are some videos on decimal numbers from Khan Academy.

Subtracting decimals (old)

Dividing decimal

Converting fractions to decimals

.

• Chris Thompson  On July 11, 2012 at 9:17 PM

As a mathematician, do you feel that the following statement is TRUE or FALSE?

.999… = 1

Why or why not?

• vinaire  On July 11, 2012 at 9:22 PM

It is true. It can be demonstrated mathematically quite easily.

.

• Chris Thompson  On July 11, 2012 at 10:04 PM

• vinaire  On July 12, 2012 at 5:22 AM

Let
N = 0.9999… (1)
Then
10N = 9.9999… (2)
Subtract (1) from (2) to eliminate the recurring portion
10N – N = 9.9999… – 0.9999…
or, 9N = 9
or, N = 1
(3)
Comparing (1) and (3), we get
N = 0.9999… = 1

Q.E.D.

.

• Chris Thompson  On July 12, 2012 at 8:56 AM

Funny! So it is an exploitation of the semantics of the repeating decimal!

Thank you. I knew I was asking the right guy.

• vinaire  On July 12, 2012 at 9:22 AM

It is unknowableness of the infinity of repetition. 🙂

.

• Chris Thompson  On July 13, 2012 at 11:29 PM

That was helpful. Thank you. 🙂

Now I want to ask whether there is an important significance in formal math to what you’ve done. In other words, the infinite repetition in (1) is “subtracted” from (2) without knowing what precisely was subtracted. Regardless, this device has left the equation balanced with whole numbers remaining and a “proof” that two non-identical things are identical.

I don’t know if I can properly ask what I want to know. But regarding “formal mathematics,” how important is this proof, what does it demonstrate, and what is the significance of it? How were you introduced to it? In what context?

• vinaire  On July 14, 2012 at 5:00 AM

Here is the Wikipedia entry on this subject, which I have yet to study thoroughly.

0.999…

In my opinion, it boils down to the unknowable-ness of infinity, which may be approximated as

Infinity – 1 = infinity

It is the impossibility of defining infinity mathematically because, by definition,

infinity means “without limit”

I have written some essays on infinity that you may find on my blog.

.

• vinaire  On July 14, 2012 at 5:05 AM

You cannot know infinity with absoluteness. There is your unknowable again.

.