We know 1,000 equals 10 to the power of 3, and then 100 is 10 to the power of 2, and then 10 is 10 to the power of 1. We also know that 1 is 10 to the power 0. How do we use the powers of 10 to describe the place value of numbers less than 1? Well, if this decreasing pattern of the indices continues, then one tenth, or 1 on 10 to the power 1 equals 10 to the minus 1. One on a 100, or 1 on 10 to the power 2, equals 10 to the minus 2. And 1 on a 1,000, which is 1 on 10 to the power of 3 equals 10 to the minus 3 and so on. So powers of 10 with a negative index can be used when describing numbers between 0 and 1.
Now, how would 1 on a million be expressed as a power of 10? If you said 10 to the minus 6, then well done. Remember one million has six zeroes. So the power required is 6 and because the number required is 1 on 10 to the power of 6, we use a negative index. This technique of using negative indices is needed to express small numbers in scientific notation. Here’s an example so you can see how we approach converting a small number to scientific notation. Express 0.007 in scientific notation. Now the decimal point has to be moved 3 places to the right to follow the rules of scientific notation, which is move the decimal point so the answer is a number between 1 and 10. The number is now 7, which I like to write as 7.0. Now to show the number has the same value as before, we have to multiply 7.0 by a power of 10.
What power of 10 do we need to use? We moved three places, so the answer is minus three. If you’re a little confused watch and I’ll show you why it’s minus three. We’ll go through this a bit at a time to make sure you understand. Now from your work on decimals, you know that 0.007 as a fraction is 7 over 1,000. This can be written as 7 times 1 over 1,000. 1,000 is 10 to the power 3 and as we saw earlier, 1 over 10 to the power of 3 is 10 to the power of minus 3. So that’s why 0.007 is expressed as 7.0 times 10 to the minus 3 in scientific notation.
Can’t you see that there is a pattern here, just as there was for large numbers? For large numbers, we moved the decimal point to the left and the power of 10 used equaled the number of places the decimal point moved. For a number less than 1, we moved the decimal point to the right instead and the power of 10 used still equals the number of places the decimal point moves, but now it’s a negative power of 10. So if the decimal point is moved 4 positions to the right, you’ll be multiplying by 10 to the minus 4. For example, the decimal 0.0007 equals 7.0 times 10 to the minus 4. The point moved 4 places to the right, so we multiplied by 10 to the minus 4.