Decimal Place and crossing the boder

Question

I always was a little confused by this notion but never thought to investigate it. In school and as I grew older people in this world (mathematics) would just say " that's the way it is " as in other subjects i would try to understand and accepted it so and moved on with my younger life to trash it away on other things. BUT, this one thing never left the back of my head. In the decimal world it is said: 1,234.567 1 (thousandths) 2 (hundredths) 3 (tenths) 4 (singles) And once we cross the border (decimal) we say it like this 5 (tenths) 6 (hundredths) 7 (thousandths)

WHY!

shouldn't the 5 be singles. I understand we are cutting the whole number(s) from the left side up into even smaller amounts on the right side, but shouldn't it still exactly reflect it's other side. Instead it skips the (singles) and goes straight to (tenths). what happen to the " what you do to one side you must do to the other " rule in mathematics.

p.s. I apologize for the incorrect tag. I don't know what it's called, that's what the site gave me.

Answer 1

You're noticing a pattern that holds when you have an even number of items that are arranged in a mirror pattern:

$$A B C D | D C B A$$

Here, you can put a dividing line like a mirror, separating the figure into a left half and a right half. Also, every item appears twice: once on the left side, and once on the right.

But notice what happens when you have an odd number of items arranged in a mirror pattern :

$$A B C D \hat{E} D C B A$$

In the odd case, there is a "middle" item which falls between the left and right sides. As a result, you can't place a dividing line neatly between the left and right sides. Instead, the "middle" item marks the mirror point.

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I think you see decimal numbers as an even-like case, with the decimal point $\LARGE .$ dividing the left side from the right side. As such, you expect each item to appear twice, once on the left and once on the right.

But in fact, decimal numbers are an odd-like case (!). The units (or "singles") position marks the midway point in the way that $\hat{E}$ does in the diagram above —the decimal $\Large .$ is not part of the picture and should be ignored for now.

Look at it this way, when you ignore the decimal point:

$$\ldots\text{thousands}\quad \text{hundreds}\quad \text{tens}\quad \hat{\text{units}}\quad\text{tenths}\quad\text{hundredths}\quad\text{thousandths}\ldots$$

Do you see the mirror symmetry now? Because it is odd-like symmetry, there is no place to put a dividing mirror at the halfway point.

So where can we put the decimal point? We choose to put the decimal point on the righthand side of the units position. This has the feature of putting all of the whole number positions to the left of the decimal point, and all the fractional number positions to the right of the decimal point. The decimal point therefore breaks the symmetry — but the notational advantages are worth it.

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But why does the decimal have odd-type symmetry in the first place, instead of even-type symmetry? Why couldn't it be something like:

$$\ldots\text{hundreds}\quad \text{tens}\quad \text{units}\quad| \quad{\color{blue}{\text{uniths}}}\quad\text{tenths}\quad\text{hundredths}\ldots$$

One answer is that we are listing all the integer powers of 10 in order, and so the odd-type symmetry of the decimal system comes from the odd-type symmetry of the integers:

$$\ldots 10^{3},10^2, 10^1, 10^0, 10^{-1}, 10^{-2}, 10^{-3},\ldots$$

where here $10^0$ is the midpoint of the odd symmetry. So now the question becomes: why do the integers have odd-type symmetry like this:

$$\ldots +3 \quad +2 \quad +1 \quad 0 \quad -1 \quad -2 \quad -3 \ldots$$

One answer is that our number system does not have two different numbers called, say, $0^+$ and $0^-$. If we did, we would have even-type symmetry exactly the way we'd like:

$$\ldots 10^{+2}\quad 10^{+1}\quad 10^{0^+} \quad {\color{blue}{10^{0^-}}} \quad 10^{-1}\quad 10^{-2}\ldots$$

And the reason why our number system does not have two different numbers $0^+$ and $0^-$ is because then you'd have to tell me how to do arithmetic such as $0^+ + 0^-$—and it's hard to give a satisfying definition. For example,

• If $0^+ + 0^-$ is just $0^+$, then $0^+$ isn't "really" like zero. We know that zero should obey a law like $0 + a = a$ for any number $a$.
• $0^+ + 0^-$ is just $0^-$, then $0^-$ isn't "really" like zero. We know that $0^-$ should obey a law like $a + 0 = a$ for any number $a$.
• If $0^+ + 0^-$ is some new number $0^\star$, then neither $0^+$ nor $0^-$ is is like 0.

And now it seems we've run out of options, but it's good to keep thinking.

Answer 2

It has to do with powers of 10. The coefficient of $10^0$ is singles, for $10^1$ or $10^{-1}$ we have tens or tenths, for $10^2$ and $10^{-2}$ we have hundreds and hundredths, and so on. Since +0=-0, we have only one singles.

Answer 3

If I give you the choice between $1$ pie and $0.1$ pie, which do you prefer? Clearly the place after the decimal point does not represent singles. We only need to allocate one place to singles. For historical reasons, that is placed in front of the decimal point. It is similar to the number line, where we only need one zero, but we need $\pm 1, \pm 2, $ etc.

Answer 4

Nice question. The names for the digits in a decimal representation: (..., thousands, hundreds, tens, units, tenths, hundredths, ...) can be thought of as what we multiply the value of the corresponding digit in evaluating our number. Since "decimal" corresponds to base ten, taking a number: $$\begin{align}123.45 &=1*(100)+2*(10)+3*(1)+4*\left(\frac{1}{10}\right)+5*\left(\frac{1}{100}\right)\\ &= 1*10^2+2*10^1+3*10^0+4*10^{-1}+5*10^{-2}\end{align}$$

Note that our "singles" or "units" place is the position where $10^k$ has an exponent of $0$. To the left and right the symmetry is more obvious between "tens" and "tenths", "hundreds" and "hundredths" and so on.

Writing it out this way should make it clear that the symmetry is around the units place instead of the decimal point, which just separates the integer part (where the exponent $k \ge 0$ in $10^k$) and the fractional part (where it's smaller than zero).

Answer 5

" In the decimal world it is said: 1,234.567 1 (thousandths) 2 (hundredths) 3 (tenths) 4 (singles) And once we cross the border (decimal) we say it like this 5 (tenths) 6 (hundredths) 7 (thousandths)"

We don't say "1 (thousandths) 2 (hundredths) 3 (tenths) 4 (singles)"! We say "1 (thousand) 2 (hundreds) 3 (tens) 4 (singles)"

The units are units. The figures to the left of the unit are multiples: tens, hundreds, thousands, etc. The figures to the right of the units are the fractions the tenths, the hundredths, the thousandths. et.c

If we were consistent we'd write the number as

123 4 567

Where the singles/units would have an exhalted place as the units values--- the figures to the left would be understood to be groups of tens and groups of tens of tens (and tens of tens of tens) and those to the right are the fractional tenths and tenths of tenths.

However such would be impractical. We usually just write integers as 1234 where it's understood that the last time is units and only the further terms are groups of tens.

We only introduce the decimal point "." when we need to. So that we use it at all indicates "as soon as we cross it we are into the fractional area".

THink of the units zone as a comfort zone between the "fractions" on the right of "." but before the "deep" powers once we really get into the left of the units place.