Question

Describe what the standard notation of a positive number will look like if the scientific notation of that number has a negative exponent?

Answer 1

By using exponents, we can reformat numbers. This can be helpful, in much the same way that it's helpful (that is, it's easier) to write "twelve trillion" rather than 12,000,000,000,000., or "thirty nanometers" rather than "0.00000003 meters".

For very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers).

The format for writing a number in scientific notation is fairly simple: (first digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an appropriate power).

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Scientific Notation on MathHelp.com

Scientific Notation

The conversion is fairly simple.

   Write 124 in scientific notation.

This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first convert the "124" to "1.24". This is not the same number as what they gave me, but (1.24)(100) = 124 is, and 100 = 102.

Then, in scientific notation, 124 is written as 1.24 × 102.

Actually, converting between "regular" notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places. To do the conversion for the previous example, I'd count the number of decimal places I'd moved the decimal point. Since I'd moved it two places, then I'd be dealing with a power of 2 on 10. But should it be a positive or a negative power of 2? Since the original number (124) was bigger than the converted form (1.24), then the power should be positive.

   Write in decimal notation: 3.6 × 1012

Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need to move the decimal point to the right, in order to make the number LARGER. Since the exponent on 10 is "12", I'll need to move the decimal point twelve places over.

First, I'll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:

3.6 _ _ _ _ _ _ _ _ _ _ _ .

Then I fill in the loops with zeroes:

3.600000000000.

In other words, the number is 3,600,000,000,000, or 3.6 trillion

Idiomatic note: "Trillion" means a thousand billion – that is, a thousand thousand million – in American parlance; the British-English term for the American "billion" would be "a milliard", so the American "trillion" (above) would be a British "thousand milliard".