Scientific Notation

 Scientists have developed a shorter method to express very large numbers. This method is called scientific notation. Scientific Notation is based on powers of the base number 10.  The number 123,000,000,000 in scientific notation is written as 1.23 x  The number in scientific notation is written as 5.08 x  There is a significant advantage to writing very large or very small numbers this way – they take much less space!

 Weight of a rabbit: 1420 g  How many significant digits?  On first inspection, we would say 3 sig dig.  But, maybe the scale measures to the closest gram and we have 4 significant digits. How can we be sure? We can’t UNLESS …  We can take the ambiguity out by using scientific notation:  If the value is X 10 3, then we know that the fourth digit is significant

 For example, the number would be written 6.5 x  In this example the coefficient equals 6.5 (which meets the requirement that 1<y<10)  Since there are seven digits trailing the decimal between the 6 and 5 we must move the decimal point 7 places to the left:

 For example, the number would be written 9.87 x  In this example the coefficient equals 9.87 (which meets the requirement that 1<y<10)  Since there are seven digits preceding the decimal between the 9 and 87 we must move the decimal point 5 places to the right:

 An electron's mass is about kg. In scientific notation, this is written ×10 −31 kg.  The Earth's mass is about kg. In scientific notation, this is written ×10 24 kg.  The Earth's circumference is approximately m. In scientific notation, this is 4×10 7 m.  An inch is micrometers. In scientific notation, this is ×10 4 µm

 RULE #1: Standard Scientific Notation is a coefficient (y), with 1 ≤ y < 10 followed by a decimal and the remaining significant digits  y is multiplied by 10 raised to an exponent (where the exponent (b) is an integer).  y x 10 b :  y = coefficient or mantissa or significand  b = exponent or power  where 1 ≤ y < 10 and b = Z (integer)

 Converting a number in these cases means to either convert the number into scientific notation form, convert it back into decimal form or to change the exponent part of the equation.  None of these changes alter the actual number, only how it's expressed.

 RULE #2: When the decimal is moved to the left the exponent gets larger, but the overall value of the number stays the same. Each place the decimal moves changes the exponent by one. When the decimal is moved to the right the exponent gets smaller,  Example: 6000  = x 10 0 (Note: 10 0 = 1)  = x 10 1  = x 10 2  = x 10 3  All the previous numbers are equal, but only x 10 3 is in proper Scientific Notation.

 ▣ Decimal moves 6 places left▣ Decimal moves 4 places right ▣ Coefficient becomes 2.45▣ Coefficient becomes 4.72 ▣ exponent becomes (+) 6▣ exponent becomes -4  2.45 x x ) First, move the decimal point to make the coefficient’s (number's) value between 1 & 10. 2) If the decimal was moved to the left, increase the exponent (positive numbers will be produced). 3) If the decimal was moved to the right, decrease the exponent (negative numbers will be produced).

 x x ▣ Decimal moves 6 places left▣ Decimal moves 4 places right ▣ Coefficient becomes 2.45▣ Coefficient becomes 4.72 ▣ exponent becomes (+) 6▣ exponent becomes -4  ) When converting a number from scientific notation to decimal notation, first remove the x 10 b on the end 2) If the exponent (b) is positive, shift the decimal separator b digits to the right. You will have to place zeros for unfilled place values. See red zeros in the example. 3) If the exponent (b) is negative, shift the decimal separator b digits to the left. You will have to place zeros for unfilled place values. See red zeros in the example.

Convert Decimals to Scientific Notation 1) ) ) ) Convert Scientific Notation to Decimals 1) 3.39 × ) 8.05 × ) × ) × 10 -5

 RULE #3: To add/subtract in scientific notation, the exponents must first be the same.  Example:  (3.0 x 10 2 ) + (6.4 x 10 3 ); since 6.4 x 10 3 is equal to 64. x Now add.  (3.0 x 10 2 ) + (64. x 10 2 )  67.0 x 10 2 = 6.70 x 10 3 = 6.7 x 10 3

 RULE #4: To multiply, find the product of the numbers, then add the exponents.  Example:  (2.4 x 10 2 ) (5.5 x 10 –4 ) =  [2.4 x 5.5 = 13.2]exponents [ = -2] so  (2.4 x 10 2 ) (5.5 x 10 –4 ) =  13.2 x 10 –2 =  1.3 x 10 – 1

 RULE #5: To divide, find the quotient of the number and subtract the exponents.  Example:  (3.3 x 10 – 6 ) / (9.1 x 10 – 8 ) =  [3.3 / 9.1 =.36]; exponents [-6 – (-8) = 2], so:  (3.3 x 10 – 6 ) / (9.1 x 10 – 8 ) = .36 x 10 2 =  3.6 x 10 1

1) 4.90 × × ) 6.95 × × ) × × × ) × × 10 5

1) (9.2 × ) × (3.0 × ) 2) (3.5 × 10 6 ) / (5.0 × 10 2 ) 3) (4.18 × ) × (3.05 × ) 4) (7.15 × ) / (2.735 × ) 5) (3.0 × 10 7 ) × (4.0 × ) / (6.0 × 10 3 )

 Introduction (13:56)  G4rc6NI G4rc6NI  Just watch this one!  But if you need more help or more practice watch these Tyler DeWitt Videos (see next page)

 Practice with Scientific Notation (13:31)   Scientific Notation: Addition & Subtraction (7:12)   Scientific Notation: Multiplication & Division (5:31)   Scientific Notation & Significant Digits (7:58) 