Study Guide

# Basic Operations - Scientific Notation

## Scientific Notation

A burst of energy created from two black holes smashing into each other 1.3 × 109 years ago was recently detected as gravitational waves. Uh, wait...exactly how long ago was that?

The number 1.3 × 109 is written in scientific notation, which is a way to write very large and very small numbers using exponents. Just like exponents, scientific notation was invented so that we don't need to waste a bunch of time writing out numbers like (our fingers just cramped typing all those zeros).

Instead, we can write this crazy long number as 2.5 × 1034.

### How It Works

Scientific notation has three parts to it: the coefficient, the base, and the exponent. The coefficient must be greater than 1 and less than 10 and contain all the significant (non-zero) digits in the number.

• 12.5 × 106 is not in proper scientific notation, since the coefficient is greater than 10.
• Neither is 0.125 × 107, since the coefficient is less than 1.

The base is always 10.

The exponent is the number of places the decimal was moved to obtain the coefficient.

### How to Convert from Scientific Notation to Standard Notation

To write a number in standard notation, we first look at the exponent. A positive exponent tells us how many place values to the right we need to move the decimal to get back to the original number.

When we write the number 2.5 × 106 as a regular number (in standard notation) we need to move the decimal in 2.5 to the right six places, filling all empty places with zeros.

2.5 × 106 =
2,500,000

Notice that the 106 part does not mean we add six zeros. It means we move the decimal point six places to the right.

We use positive exponents like this to write really large numbers.

A negative exponent tells us how many place values to the left we need to move the decimal to get back to the original number.

When we write the number 2.5 × 10-4 in standard notation, we move the decimal that's between the 2 and the 5 to the left by four places, filling in all empty places with zeros.

2.5 × 10-4 =
0.00025

We use negative exponents to write really small numbers.

### How to Convert from Standard Notation to Scientific Notation

To write large numbers in scientific notation, we move the decimal point from behind the ones place to the left until it only has one digit in front of it.

In the number 26,500,000, the decimal is to the right of the zero in the ones place, though it's invisible right now. To write this number in scientific notation, we abra-cadabra that decimal back into view and move it left across all the zeros until there's only one digit in front of it.

First, move the decimal behind the 2, so we get a number between 1 and 10. Note that it takes seven jumps to get there. Now drop all non-significant zeros. Multiply this by 10 to the power of 7, since the decimal was moved seven places. Bam, done.

To write small numbers in scientific notation, we move the decimal point from behind the ones place to the right until it only has one digit in front of it.

In the number 0.00006009, the decimal is to the left of the tenths place. To write this in scientific notation, we hop the decimal so that it has only one digit in front of it.

Like before, move the decimal until we get a number between 1 and 10. This time, it's five jumps to the right. Drop all of the non-significant zeros. So long, fellas. Since we moved the decimal right instead of left this time, the exponent will be negative. Multiply the coefficient by 10 to the power of -5, and we're done. So how long ago did those two black holes smash into each other?

1.3 × 109 = 1,300,000,000

Approximately one billion, three hundred million years ago. Glad we weren't there.

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