Scientific Notation (Year 10)

In science, we deal with numbers that are incredibly large or incredibly small. Scientific notation gives us a neat way to write them.

A number in scientific notation looks like this: a times 10 to the power of n. a is always between 1 and 10, so the decimal is after the first digit. The power n tells us how many places to move the decimal point.

Let’s convert 3.2 times 10 to the 5 to an ordinary number. Take the 3.2 and write it out. The power on the 10 is 5, so we move the decimal 5 places to the right. That gives us 320,000.

Now the other way. Write 4,700,000 in scientific notation. We need to write it in the form a times 10 to the n. a comes from the first two digits of our number — 4 and 7. And a needs to be between 1 and 10, so we put the decimal between them: 4.7. Now let’s think about what power of 10 we need. We count: one, two, three, four, five, six. Now it matches, so we need 4.7 times 10 to the 6.

When the power is negative, we move the decimal to the left instead. Convert 5.1 times 10 to the minus 3 to an ordinary number. Move the decimal 3 places to the left, filling with zeros. That’s 0.0051.

Write 0.00082 in scientific notation. Again we need it in the form a times 10 to the n. a needs to be between 1 and 10, so that’s 8.2. Now what do we multiply 8.2 by to get the original value? We move the decimal one, two, three, four places left to match the original number. So our answer is 8.2 times 10 to the minus 4.

Here’s something useful: in scientific notation, the number of digits in the a part tells you the number of significant figures. That’s because scientific notation gets rid of all the ambiguous leading and trailing zeros.

How many significant figures does 6.020 times 10 to the 5 have? Count the digits: 6, 0, 2, 0. That’s four digits, so four significant figures. Every digit you write in the a part is significant.

Here’s a common one. Write 12 times 10 to the 4 in proper scientific notation. 12 isn’t between 1 and 10, so it’s not in scientific notation yet. Write 12 as 1.2 times 10. Then 1.2 times 10 times 10 to the 4 gives 1.2 times 10 to the 5.

Let’s multiply two numbers in scientific notation. Find 2.5 times 10 to the 6, multiplied by 3 times 10 to the minus 2. First, multiply the numbers: 2.5 times 3 is 7.5. Next multiply the powers of 10. By the index laws, add the powers: 6 plus minus 2 is 4. So the answer is 7.5 times 10 to the 4. And since 7.5 is already between 1 and 10, we’re done — it’s already in scientific notation.