Is "three ten-millionths of an inch" the same as "thirty millionths of an inch"?

Question

I was reading this article when I came across a fraction that was difficult for me to comprehend:

three ten-millionths of an inch

I thought to myself that wouldn't this be the equivalent of:

thirty millionths of an inch

However, the thought then occurred to me that they must be two completely different values. The problem is that with the numbers being so small, I'm having a difficult time scaling this up to a bigger number that would make more sense ($\frac{1}{3}$ for example).

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Are the two phrases interchangeable, or am I correct in thinking they have two different values? How can each be represented with a larger fraction to ease comprehension?

Answer 1

Don't confuse yourself.

Use simpler terms. "three quarters of an inch" means "$3\times \frac 14$ inch" or "$\frac 3{4}$ inch".

It does not mean "one three$\times$fourth" = "$\frac 1{3\times 4}=\frac 1{12}$ inch"

Like wise "three ten-millionths of an inch" means "$3 \times \frac 1{10,000,000} = \frac 3{10,000,000}$"

It does not mean $\frac 1{3\times 10,000,000} = \frac {1 }{30,000,000}$.

Answer 2

They are different values, three ten-millionth is $\frac{3}{10\times 1000000} = 3\cdot 10^{-7}$ and thirty-millionth is $\frac{1}{30\times 1000000}=3.\bar{333}\cdot 10^{-8}$.

Answer 3

Not the same.

Three ten-millionths would be (3/10 million) X 1 inch = 0.3 inches divided by 1 million. Thirty millionths of an inch would be 30 times an inch divided by 1 million. So they are different by two orders of magnitude.

I think 0.3 millionths of an inch would be the right description (not that it is any easier to comprehend...).

I think micro ($10^{-6}$), nano ($10^{-9}$), and pico ($10^{-12}$) come in really handy when dealing with small numbers. For example, 1 millionth of an inch can be called 1 micro-inch or 1 $\mu$i. Then, 0.3 millionths of an inch becomes $0.3\mu$i or 300 nano-inches.