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 1st and 2nd digit or to right of the 1st 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
MULTIPLYING AND DIVIDING IN SCIENTIFIC NOTATION
- When multiplying, the numbers are multiplied and the exponents are added.
- When dividing, the numbers are divided and the exponents are subtracted.
Example: Find 8.6 x 107/ 2.0 x 104.
Solution: We are dividing, so the numbers are divided and the exponents are subtracted:
8.6 x 107 / 2.0 x 104
= (8.6/2.0) x 10(7-4)
= 4.3 x 103
Scientific Notation Example 1:
Scientific Notation Example 2 - Converting to Standard Notation:
Scientific Notation Example 3 - Addition:
Scientific Notation Example 4 - Multiplication:
Scientific Notation Example 5: