$10^6 = 1000000$, which has $6$ zeros before decimal. Then why does $10^{-6} = 0.000001$, has only $5$ zeros after decimal?

Question

I just can't get my head around this. Now I know that $10^{-n}$ = $\frac{1}{10^n}$. But I just can't understand the logic. My question may sound stupid to most people here but I am a beginner at math and I would be grateful if someone can provide a simple explanation.

Answer 1

Think about it in terms of patterns:

If you multiply by $10$ (increasing the exponent by $1$), you will expect to add one zero in front of the number.

So, $1\times10=10$

$10\times10=100$

$100\times10=1000$

$1000\times10=10000$

$10000\times10=100000$

$100000\times10=1000000$

Starting with $1$ (which is $10^0$), We have multiplied by $10$ a total of $6$ times to get to $1000000=10^6$, which has $6$ zeros. Ok, so what is the pattern here? Multiplying by $10$ $0$ times gives $1$. Multiplying once gives $10$. Twice gives $100$. And so on. Well, it seems like the exponent equals the number of zeros, right? But wait! What about dividing by $10$? What if we repeatedly divide by $10$ (equivalent to decreasing the exponent by $1$)? Well, let's see:

$1\div10=0.1$

$0.1\div10=0.01$

$0.01\div10=0.001$

$0.001\div10=0.0001$

$0.0001\div10=0.00001$

$0.00001\div10=0.000001$

Ok so we have divided by $10$ a total of $6$ times, and now we see the pattern emerging here as well. We divide by $10$ zero times and we get $1$. Divide once and we get $0.1$, which has zero $0$'s after the decimal place. Divide twice and we get $0.01$, which has one $0$ after the decimal place. Divide three times and we get $0.001$, which has two $0$'s after the decimal place. Notice that how many times we divide by $10$ is one more than the amount of zeros after the decimal place. And also realize that this pattern will hold, because every time we divide by $10$, another zero will appear after the decimal place. Now continue this to dividing by $10$ a total of $6$ times (equivalent to $10^{-6}$) and we can infer that it will have $5$ zeros after the decimal place!

Answer 2

@JyrkiLahtonen basically nails it, but as a quip. I'll expand on that:

The decimal place separates the negative exponents from the positive exponents. Let's write out those exponents:

$$... 4, 3, 2, 1, 0, -1, -2, -3, -4 ...$$

You get to make one cut (with your decimal point) and split them into two groups. But where does $0$ go? Well, we decided to make the cut between $0$ and $-1$, which puts $0$ with the positives. So the units column, which corresponds to $10^0$, goes to the left of the decimal point. And that's your extra zero.

This naturally prompts the question "what if I made the cut exactly at $0$?" (as @Aaalol_dude hints). Well, you would have irritated the typesetters, but if you could have convinced them to step away from their single line of characters, you could have $1\underset{\bullet}{0}$ and $\underset{\bullet}{0}1$ as "ten" and "one tenth" respectively, and your question would disappear.

Answer 3

A little acrobatics that does not answer your question directly.

$10^1=10;$ $10^{-1}=1/10^1=0.1;$

$10^1×10^{-1}=10^0=1=$ $10×0.1$

Left hand side uses power rule, right hand side: recall that by multiplying with 10 the decimal point moves one digit to the right.

$10^6=1000000;$ $10^{~6}=1/10^6=0.000001;$

$10^6×10^{-6}=10^0=1=$ $=1000000×0.000001$

Use the above rules. Last line: Multiplying by 1000000 move the decimal point 6 digits to the right in $0.000001$.