7 Scientific notation

7.1 Going up: powers of ten for large numbers

It is estimated that the total volume of water stored on the Earth is 1 460 000 000 km³.

When dealing with large numbers like one thousand four hundred and sixty million (1 460 000 000), it becomes tedious to write out the number in words or to keep writing out all of the zeros. Worse still, it is very easy to lose some of the zeros or add extra ones by mistake. Fortunately, we can refer to large numbers without having to write out all of the zeros. The powers of ten notation is less prone to errors and tedium because it removes the zeros. We will introduce the powers of ten notation with some numbers more manageable than 1 460 000 000, though.

One thousand is ten times ten times ten:

We can use powers notation (introduced in Section 1.2) to write

When expressing 100 and 1 000 in powers of ten, there are no great savings on writing zeros, but what about one million (1 000 000)? One million is the product of multiplying together six tens:

so it is written as Now you begin to see the benefit of the powers of ten notation.

One thousand is often written not just as but as Spoken aloud, this would be expressed, ‘one times ten to the power three’ or just ‘one times ten to the three’. Likewise one million is either or simply Now we can give two alternative explanations that may help you to get to grips with powers of ten. First, the power of ten shows how many times 1 has been multiplied by 10. Taking as an example, 1 000 is seen to be Second, the power of ten shows how many places the decimal point has to move to the right to give the actual number. If we write 1 as 1.0 to remind ourselves where the decimal point is, then one move to the right would turn 1.0 into 10.0, a second move would give 100.0 and a third move would give 1 000.0, that is, one thousand.

You do not have to recall both of these ways of understanding powers of ten; just use the one that suits you best, or develop your own way of fixing the idea in your armoury of mathematical techniques.

Let’s go back to the total amount of water on the Earth. Using the powers of ten notation, 1 460 000 000 could be written as A significant saving on zeros! The complete number would be spoken as ‘one point four six times ten to the power 9’ or just ‘one point four six times ten to the nine’. The power of 9 tells us how many times 1.46 has been multiplied by 10 to give the final number of 1 460 000 000. It is nine times. That is, our number is comprised of:

To see clearly that this expression is still one thousand four hundred and sixty million, it helps to begin with 1.46 and work our way to the number we want by multiplying each time by ten:

If we carry on doing this, we end up with:

Alternatively, you can think of each increase by one in the power of ten as moving the decimal point one place to the right. That is, if you multiply 1.46 by 10 the decimal point moves one place to the right, giving 14.6.

Likewise, to multiply 1.46 by one thousand, the decimal point moves three places to the right, giving 1 460.0. In the powers of ten notation, this is written

There is a convention called scientific notation that is used when writing a number with a power of ten. Scientific notation requires the number accompanying the power of ten to be less than 10 but equal to or greater than 1. Let’s take the example of one million. It could be correctly expressed as , and so on, or even as , but only the first of these obeys the convention of scientific notation and this is the one that should be used. As a second example, it is quite correct mathematically to write 85 000 as , or , but correct scientific notation would demand