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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: \(3.65 \times 10^{-8}\)

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: \(2.39 \times 10^6 \)

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 \(4.53 \times 10^{-4}\) 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 \(6.21 \times 10^4 + 1.33 \times 10^4\)

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

\( \begin{array}{l} 6.21 \times 10^4 + 1.33 \times 10^4  &=& (6.21 + 1.33) \times 10^4 \\ &=& 7.54 \times 10^4\\ \end{array} \) 

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 \(8.6 \times 10^7 \div 2.0 \times 10^4\).

Solution: We are dividing, so the numbers are divided and the exponents are subtracted:

\( \begin{array}{l} 8.6 \times 10^7 \div 2.0 \times 10^4 &=&(\frac{8.6}{2.0}) \times 10^{(7-4)} \\ &=& 4.3 \times 10^3 \end{array} \) 

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: