Scientific Notation
INTRO
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) × 10 Integer Exponent
CONVERTING TO SCIENTIFIC NOTATION
• decimal placed between 1st and 2nd digit or to right of 1st non-zero number
• count number of places the decimal was moved to reach position in step 1
• the number you got in step 2 becomes the integer exponent (negative if decimal
was moved to right)
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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.
−8
Answer: 3.65 × 10
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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 × 10
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6
CONVERTING TO STANDARD NOTATION
•
move decimal the number of places given by the exponent
o if exponent is positive, decimal moves to right
o if exponent is negative, decimal moves to left
Often zeros will be needed as place holders in these situations.
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Example: Express 4.53 × 10 −4 in standard notation.
Solution: The decimal moves 4 positions to the left (since the exponent is - 4)
Answer: 0.000453
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ADDING AND SUBTRACTING IN SCIENTIFIC NOTATION
•
•
numbers must be written with same exponent
o move decimal place if needed to accomplish this (add 1 to exponent
for each time decimal is moved to left, subtract 1 from exponent
each time decimal moved to right)
once numbers have same exponent, simply add or subtract them (exponent
stays the same)
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Example: Find
6.21×10 4 + 1.33 ×10 4
Solution: Since the numbers have the same exponent when written in scientific notation,
we can simply add them to obtain:
6.21× 10 4 + 1.33 × 10 4
= (6.21 + 1.33) × 10 4
= 7.54 × 10 4
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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
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8.6 × 10 7
Example: Find
.
2.0 × 10 4
Solution: We are dividing, so the numbers are divided, and the exponents are subtracted:
8.6 × 10 7
2.0 ×10 4
8.6
=
× 10 ( 7−4 )
2.0
= 4.3 × 10 3
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For a detailed explanation of some more difficult examples, check out the mini-clips!