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At a Glance - In the Real World

Scientific Notation

Earth-shattering fact: scientists use math. Some mathematicians even use science. Too bad more of them don't use soap. Ooh, ice burn.

We're not only talking about white-haired dudes and ladies in lab coats. All scientists, from astronomers to chemists to alientologists, play with numbers that are so big or so small that they had to to think of a way to write them without their hands getting sore. Their hands are already sore enough from having to sign so many autographs for young fans.

The solution? Scientific notation, which is shorthand for writing really big or really small numbers. This is the notation used by movie producers when writing out the amount on one of Johnny Depp's paychecks.

It's called scientific notation because it is useful to all different types of scientist—astronomers use it to measure the size of the universe (big) and chemists use it measure atoms (small)—but even non-scientists such as yourself are allowed to use it. What do you say to the nice scientists? Thank you, scientists.

If you're rusty on decimal multiplication, it would be a good idea to review...oh, about now.

Sample Problem

What is 3,400,000 in scientific notation?

Instead of writing 3,400,000, we could write 3.4 × 106. Using scientific notation in this instance may not be saving us a ton of work, but wait until the numbers become incredibly huge. Hopefully, they'll still remember all the little numbers.

For example, 34,000,000,000,000 is a ridiculously big number. If it wore pants, it would need to shop at a Big and Tall store. It has twelve—count them, twelve—zeros. In order to avoid writing all those zeros, we abbreviate this insanely big number as 3.4 × 1013. If we start with the number 3.4 and move the decimal point 13 places to the right, we find 34,000,000,000,000. If we moved the decimal any more to the right, it would need to change its zip code.

Sample Problem

When numbers become incredibly tiny, negative exponents have a chance to make themselves useful.

We know that 0.000003 is the same thing as . If we start with 3 and move the decimal point 6 places to the left, we'll have 0.000003. By the way, this decimal point is starting to become a little annoyed with all of the back and forth. In fact, it's starting to develop severe motion sickness.

Since we can write  as 10-6,here's how we write 0.000003:

0.000003 = 3 × 10-6

A number like 0.0000000000123 can also be written as 1.23 × 10-11. If we start with the number 1.23 and move the decimal point 11 places to the left, we find 0.0000000000123.

To recap, a number is written in scientific notation if it's in this form:

(number with one value before the decimal place) × 10some integer

Here are a couple more examples:

-24 = -2.4 × 101

3800 = 3.8 × 103

Our first number (before the multiplication sign) always needs to have only one digit before the decimal point. So something like 44 × 105 isn't legit scientific notation. Switch that thing to 4.4 × 106.

Sometimes other symbols besides "×" are used for scientific notation. You may see 123,000 written as 1.23 × 105 or even 1.23 E5. Remember that time you sat on your calculator and that weird number with the "E" appeared? Mystery explained.

When Should Scientific Notation Be Used?

When your teacher or a problem tells you to use scientific notation, it's probably a good idea to follow instructions. If you start questioning authority for no reason, you'll be headed down a slippery slope of juvenile delinquency. You don't want to be sentenced to 1.02 × 103 detentions.

When it's up to you, go with whichever kind of notation makes your hand the least tired. You don't need to use scientific notation for things like "3 + 4 = 7," but you might want to for things like "3,000,000,000,000 + 4,000,000,000,000 = 7,000,000,000,000." Hm, someone must be adding up the scores for Kevin Durant's last three games.

Calculations with Scientific Notation

To do calculations with numbers written in scientific notation without having to turn the numbers back into regular numbers, we use the rules of exponents. A rule of exponents is like a rule of thumb, but with more digits. Ooh, double meaning!

Sample Problem

Find (4 × 103) + (8 × 103). Write the answer in scientific notation.

First we rewrite, using the distributive law, to find (4 + 8) × 103. Then we simplify to 12 × 103. This isn't quite in scientific notation, since the number 12 has two values before the decimal point, so we adjust to get 1.2 × 104 as our final answer. We're making that poor decimal point dizzy again.

Sample Problem

Find (4 × 10-3) ÷ (2 × 102). Write the answer in scientific notation.

Division by 2 × 102 is the same as multiplication by , or . Therefore:

(4 × 10-3) ÷ (2 × 102) = (4 × 10-3)(12 × 10-2)

And we can rearrange things a bit to solve:

This answer is already in scientific notation, so we're done. Now we have a bit of free time to catch up on all our DVR'd episodes of Cake Boss.

Example 1

Find (5.4 × 107) + (3.23 × 109).

Example 2

Find (3 × 104)(5 × 102). Write the answer in scientific notation.

Exercise 1

Write 1,234,000,000,000 in scientific notation.

Exercise 2

Write 5,000,000,000,000,000,000,000,000 in scientific notation.

Exercise 3

Write 45 in scientific notation.

Exercise 4

Write 0.001 in scientific notation.

Exercise 5

Write 0.0000000000989 in scientific notation.

Exercise 6

Write 1.5302 × 103 as a regular number.

Exercise 7

Write 3.22 × 109 as a regular number.

Exercise 8

Write 1.1 × 100 as a regular number.

Exercise 9

Write 3.4 × 10-2 as a regular number.

Exercise 10

Write 2.002 × 10-5 as a regular number.

Exercise 11

Simplify (3 × 107)(4.2 × 109). Write answers in scientific notation.

Exercise 12

Simplify (1.24 × 104) + (2.5 × 105). Write answers in scientific notation.

Exercise 13

Simplify (4 × 1018) – (6 × 1018). Write answers in scientific notation.

Exercise 14

Simplify (4.2 × 1031) ÷ (2.1 × 1019). Write answers in scientific notation.

Exercise 15

Simplify (4.23 × 1099) + (1.1 × 1097). Write answers in scientific notation.

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