Scientific Notations

Scientific notation

 

Large numbers are common in scientific fields. In astronomy you travel with the light speed of approximately 300,000,000 meters per second. This is very fast.

 

In atom or particle physics the things get very, very small. There you can’t overlook the many 0 after the comma.

 

The invariant mass of an electron is approximately 9.109×10−31 kilograms or 5.489×10−4 atomic mass units. Can you write the 0 on a small sheet of paper? Probably not. So you can take the exponent writing.

 

The light speed in meters per second is:

 

[image]

 

Here it is easy to count the number of the 0 and write them as an exponent. After the 3 there are 8 zeros. So the exponent is 8 with the basis 10.

 

This procedure cannot be used, if the number begins with many digits greater then 0. In this case you must move the decimal point to the left until the first digit. The required steps are written as the exponent.

 

 

[image] Example 1

 

[image]

 

Move the decimal point to the left 3 and count the steps. There are 11 steps to reach the digit 3. These steps are the exponent 11.

 

 

Tiny numbers can also be written with an exponent. Here the exponent gets the minus sign. This indicates the steps after the decimal point. For instance 10-4 means the decimal point is located 4 steps left.

 

 

[image] Example 2

 

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Procedure: Move the decimal point 4 steps to the left. Here you fill up the leading zeros, until you reach the fourth step. The decimal point walks to the left.

 

10-4

 

Step 1: .1 (one step to the left.The decimal point is walking.)

Step 2: .01 (the next step to the left)

Step 3: .001 (the third step to the left)

Step 4: .0001 (the fourth step to the left.The decimal point has reached its target.)

 

Please complement one 0 before the decimal point. This zero doesn’t belong to the stepping.

 

Final act: 0.0001

 

Ready!

 

 

[image] Example 3

 

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There are 10 steps inclusive the digit 8. So the exponent is -10.

 

 

[image]  Exercises

 

1. Convert 51 810 to scientific notation.

 

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Move the decimal point to the left.

 

 

2. Convert 0.000 000 003 45 to scientific notation.

 

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Count the steps to the right, until you reach the digit 3 (inclusive).

 

 

3. Convert [image] into a number.

 

[image]

 

Move the decimal point 3 steps to the right and fill up with zeros.

 

 

4. Convert [image] to a number.

 

[image] = 0.007 13

 

Move the decimal point 3 steps to the left and fill up with zeros.

 

 

5. Convert 7 to scientific notation (a nice joke!)

 

[image]

 

You must know that base0 always is 1. If the exponent is 0, then the power always is 1.