Scientific Notation
Copyright, 1993 Dr. F.W. Boyle, Jr.

Converting numbers into scientific notation at first appears complex but is in reality simply being able to count spaces and learning the sign based on the direction the decimal point is moved.

```   Scientific numbers all have several things in common.

1) They represent relatively large or small real numbers.

2) The value in front of the decimal point is required to be
between 1 and 9.

3) The sign of the exponent (power) of the 10 (ten) is for the
purpose of converting the number from scientific notation back
to the original number when necessary, i.e. adding or
subtracting two numbers.  The direction to move the decimal
is to the right, adding zeros (0) as needed, when the sign
is positive or absent and left, placing zeros (0) in front as
needed when the sign is negative.

4) These numbers are always expressed as:

a.bc x 10Y

where a, b, c are numbers and Y is a number representing
the number of zeros (decimal places) that needed to be
added to the left (-Y) or to the right (+Y or Y) to
return the number to its more common form.

An easy way to remember how to expand scientific notation
is to look at the sign of the power.  If the sign is
negative (-Y), then write the number portion to the right
and add zeros IN FRONT of the numbers.  Then count from
old decimal position toward the left to locate the new
decimal point position.  If the sign is positive (+Y or
Y), write the number to the left and add zeros AFTER the
number.  Then count from the old decimal position toward
the right to find the new decimal position.

5) a number which fits the format as shown in No. 4 is
already in scientific notation and does not need
transformation.

6) the exponent sign is negative if the decimal point position
is moved to the right when going into scientific notation.
The exponent sign is positive is the decimal point position
is moved to the left going into scientific notation.  These
are the reverse of the moves discussed in No. 4 so be
careful not to confuse which way the number is being
transformed.
```
To figure the transformation from a real number into scientific notation, you must look carefully at the number being transformed.

For example:

1234567

Even though there is no decimal present, standard mathematical rules state the decimal is assumed to be at the right end of the number.

1234567.

Since scientific notation requires the number in front of the decimal to be between 1 and 9, the decimal must be moved to the left. Let's see how many places the decimal must move.

```
One (1) place  123456.7
Two (2) places 12345.67
Three (3) places 1234.567
Four (4) places 123.4567
Five (5) places 12.34567
Six (6) places 1.234567

```
Now the number in the position to the left of the decimal has a value between 1 and 9. Since the decimal moved to the left, the sign of the number for the power of 10 (ten) will be positive. The decimal was moved six places to the left so the value of the power will be +6.

Rewrite the number as a power of 10 (ten) using the new number and adding the x 10 portion as is shown below.

1.234567 x 10

Next simply add the power as an exponent to the 10 (ten).

1.234567 x 10+6

or simply

1.234567 x 106

For another example:

0.000001234567

Since scientific notation requires the number in front of the decimal to be between 1 and 9, the decimal must be moved to the right. Let's see how many places the decimal must move.

```  One (1) place  00.00001234567
Two (2) places 000.0001234567
Three (3) places 0000.001234567
Four (4) places 00000.01234567
Five (5) places 000000.1234567
Six (6) places 0000001.234567
```
Now the number in the position to the left of the decimal has a value between 1 and 9. Since the decimal moved to the right, the sign of the number for the power of 10 (ten) will be negative. The decimal was moved six places to the right so the value of the power will be -6.

Rewrite the number as a power of 10 (ten) using the new number and adding the x 10 portion as is shown below.

1.234567 x 10

Next simply add the power as an exponent to the 10 (ten).

1.234567 x 10-6

When converting to scientific notation, KEEP ALL the original digits. Digits are not dropped in scientific notation. (See the paper on Significant figures to learn about dropping digits and rounding.)

Try the following numbers for practice in changing numbers into the scientific notation format.

```       A) 987.88    B) 0.00203     C) 1.005       D) 15432
E) 0.0003    F) 111.1       G) 0.00000001  H) 2.135
```