Powers of 10 (2024)

The decimal system, which is based on the number 10, is the number system usedmost in the world. Other number systems you are familiar with are binarynumbers, which are based on zero and 1 and are used in computers, and time,which is divided (mostly) into units that are multiples of 60 - there are60 seconds in a minute and 60 minutes in an hour, for example. Since thedecimal system and powers of 10 are so important in science, I'll talk a bitabout them here.

• 1.1 What is 10x?
• 1.2 Multiplying and dividing powers of 10
• 1.3 What about ``logs''?
• 1.4 Doing all this on your calculator

1.1 What is 10^x?

Don't panic about the phrase ``powers of 10'' - you are already used to usingthem, even if you aren't aware of it. Quick - what's 10 × 10? 100, ofcourse. And what's 10 × 1/10? It's 1. These are simple examples to showyou that you already know how this works.

Powers of ten are written like10^x,where x is whatever power I'm talking about and is called the``exponent''.10<sup>x</sup> means ``10 × 10, xtimes.''10^-x means ``1/10 × 1/10, x times.''In math, not words, it looks like

Here are a couple examples to make it a bit clearer:

Table1lists numbers in both common notation and math notation. Some powers of tenare used frequently in science, and have special ``prefixes'', which are listedas well. Also, some powers of ten have been given names, some of which are nodoubt familiar to you; I've put the names in the table.

If you look at Table1,you may notice something useful: the exponent for each entry in the table isequal to the number of zeros in the corresponding number written ``normally''.That makes it easy to remember what a given power of ten is when written in theusual way. You may also be able to see why scientists and engineers prefer towrite things like 10²⁰,rather than write out a 1 followed by 20 zeros (100 000 000 000 000 000 000)-- it's shorter. We will talk more about scientific shorthand later, whenwe talk about scientific notation.

The ``prefix'' in the fifth column inTable1is used as a shorthand when talking about numbers of things. For example,computer memory is usually measured in Megabytes, or millions of bytes, andcomputer hard disk storage is now often measured in Gigabytes, or billions ofbytes. Also, you might get a prescription for a cold medicine which is, say,20 milligrams of some drug. That's a shorthand way of saying 20 thousandthsof a gram.

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1.2 Multiplying and dividing powers of 10

Maybe you're not convinced that it's useful to be able to write numbers inpowers of ten. Well, powers of ten are helpful when doing math, as well.Let's say I ask you ``what is 10 times 1,000?'' Not a bigdeal - it's just 10,000. But what if I asked you ``what is one trillion timesone quadrillion?''

It turns out that multiplication of really big numbers is easy with powers often. All you have to do is add up the exponents, and you're done. Let's usethe example I just gave you. What is one trillion times one quadrillion?First, using Table1, you can see onetrillion is 10¹²,and one quadrillion is10¹⁵.So the answer is10²⁷.which is a really big number - and you can see that almost immediately, withoutneeding a calculator or a piece of paper to do it longhand. Here are some moreexamples:

Division works similarly, except that you subtract the exponents.What is one trillion divided by one quadrillion? Well, it is10¹² ÷ 10¹⁵,so the answer is 10^-3, or one thousandth. Hereare some more examples:

Again, while this may not seem useful for small numbers, imagine dividingone trillion trillion trillion, which is10³⁶,by one thousand million billion, which is10¹⁸,longhand. It would take you a while. (By the way, the answer is10¹⁸.)

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1.3 What about ``logs''?

You may be wondering if there's an opposite to powers of 10; somethinglike how division is the opposite of multiplication or subtraction is theopposite of addition. It turns out that there is just such a thing:logarithms, or ``logs''. It used to be that logs were really important formultiplyingand dividing, and were used all the time in doing arithmetic using slide rules.Now that hand calculators are commonplace, the use of logarithmsfor basic calculations is fading away, but it can still be worthwhile.And while using logarithms for simple arithmetic is uncommon, there are otheruses for logarithms in many areas of science, and we may see some of these uses later in the class.

The logarithm (log) of a number written as a power of ten is easy to calculate.It's just the exponent. So, for example, the log of10³ is 3. The log of10^-2 is just -2. And swapping that around isalso easy: if I tell you the log of some number is 6, all you have to do is saythe number is10⁶, or one million. Some more examples:

Where logs are really useful is in doing multiplication and division ofnumbers. You know from the previous section that to multiply numbers whichare powers of ten, you add the exponents, and to divide such numbers, yousubtract the exponents. Well, since the log of a number which is a power of10 is just the exponent for that power, to do multiplication, you just add thelogs. To do division, just subtract the logs. An example:

You could have seen the answer from Section 1.2,but you can see how easy it is to do this using logs. And so you can probablysee why, back before powerful computers, logarithms made complicatedcomputations simpler. As I said, few people do multiplication this wayanymore, but it can be done.

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1.4 Doing all this on your calculator

If you have a basic scientific calculator, you will have a button on itsomewhere that looks like ``10<sup>x</sup>''.That's the button to make a power of 10. You just enter a number, say 5, andhit the ``10^x'' button, and the numberyou get back will be 10⁵, or 100 000. In somecases, if you enter a big number, such as 50, and hit the10^x button, you won't see a 1 followed50 zeros, but something which looks like ``1 E 50'' or ``1 EE 50''. Don'tpanic - that's just your calculator's version of scientific notation,which we will talk about in the next section.

Also on scientific calculators, you will find a button for doing logs - it'susually written ``log'', and is often very near the button for doing 10<sup>x</sup>.You just punch in a number, say 1 000 000, and hit the ``log'' button, and you will get the log of a million, which is 6.

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