Scientific Notation

Introduction

Scientific notation is a useful way to deal with numbers that are extremely large or extremely small. Scientific notation is written as a product of some number between 1 and 10 and an integer power of 10. Scientific notation is of the form:

(Decimal between 1 and 10) x 10 ^{integer exponent}

CONVERTING TO SCIENTIFIC NOTATION

Place the decimal between the 1^st and 2^nd digit or to right of the 1^st non-zero number.
Count the number of places the decimal was moved to reach the position in step 1.
The number you got in step 2 becomes the integer exponent (negative if the decimal was moved to the right).

Example:

Express 0.0000000365 in scientific notation.

Solution:

Step 1: The decimal is placed between the 3 and 6.

Step 2: Since the decimal is moved 8 places to the right to reach this position, the exponent is -8.

Answer: 3.65 x 10 ^-⁸

Example:

Express 2390000 in scientific notation.

Solution:

Step 1: The decimal is placed between the 2 and 3.

Step 2: Since the decimal is moved 6 places to the left to reach this position, the exponent is 6.

Answer: 2.39 x 10⁶

CONVERTING TO STANDARD NOTATION

Move the decimal the number of places indicated by the exponent.

If the exponent is positive, the decimal moves to the right.
If the exponent is negative, the decimal moves to the left.

Often, zeros will be needed as placeholders in these situations.

Example: Express 4.53 x 10^-4 in standard notation.

Solution:

The decimal moves 4 positions to the left (since the exponent is - 4).

Answer: 0.000453

ADDING AND SUBTRACTING IN SCIENTIFIC NOTATION

Numbers must be written with the same exponent. Move the decimal place if needed to accomplish this (add 1 to the exponent for each time the decimal is moved to the left; subtract 1 from the exponent each time the decimal is moved to the right).
Once the numbers have the same exponent, simply add or subtract them (the exponent stays the same).

Example: Find 6.21 x 10⁴ + 1.33 x 10⁴

Solution: Since the numbers have the same exponent when written in scientific notation, we can simply add them to obtain:

6.21 x 10⁴ + 1.33 x 10⁴

= (6.21 + 1.33) x 10⁴

= 7.54 x 10⁴