The scientific notation representation of numbers is a convenient way to write both huge and small numbers. The number of significant digits is always clear because all the digits in the number are significant.  This is a quick introduction to scientific notation.

A basic idea is that any number raised to the power ( exponent ) zero is equal to one ( 1 ).

1 = 1 x 10⁰ ; 1 = meter⁰ = 1 meter/ 1 meter

This is also a fundamental part of unit cancellation in unit conversion factors.

Exponential numbers have two pieces. The first is a number more than 1 and less than ten.  This is a multiplier or coefficient. The second part is ten raised to a power.

EXAMPLE 

3,000  is the same as 3  x 1000

1000 is the same as 10¹ x 10¹ x 10¹  or 10³ here the exponents are added when the numbers are multiplied

10¹ x 10¹ x 10¹  is the same as 10³ where the exponents are added when the numbers are multiplied

so 3000 is equal to 3. x 10³

The thousands part is based on this fact  10¹ x 10¹ x 10¹  or 10³ 

RULES FOR NUMBERS GREATER THAN "1"

 The exponential number is determined in two steps.  For numbers bigger than '1'  count the number of places the decimal point must be moved to the left to get a coefficient between 1 and 10.  This number equals the value for the exponent on ten.

EXAMPLE  Convert 4,000,000 to exponential notation

The coefficient is 4. The decimal must be moved 6 places.  The exponent on ten must be a +6.                     

The exponential notation number  is 4. x 10⁶ .

EXAMPLE  Convert 92,000 to exponential notation.

To separate the coefficient, 9.2, the decimal must be moved 4 places to the left.  The exponent on ten must be a +6.                      

The exponential notation number  is 4. x 10⁴ .

EXAMPLE     Convert 761,000,000,000 to exponential notation.

The coefficient is 7.61.   The decimal must be moved 11 places to the left.  The exponent on ten must be a +11. 

The exponential notation form is 7.61. x 10¹¹ .

Large numbers that are multiples of ten and their exponential equivalent are shown here.  The coefficient here is a 1.

100  = 10 x 10  = 10¹ x 10¹ = 10² decimal place moved 2 places to the left

1,000 = 10¹ x 10¹ x 10¹ = 10³    decimal place moved 3 places to the left

10,000 = 10¹ x 10¹ x 10¹ x 10¹ = 10⁴    decimal place moved 4 places to the left

100,000 = 10¹ x 10¹ x 10¹ x 10¹ x 10¹ x 10¹ = 10⁵   decimal place moved 5 places to the left

10,000,000 = 10⁷    decimal place moved 7 places to the left

EXAMPLE conversion to scientific / exponential notation

7,800,000,000,000  grams  =  7.8 x 10¹²  grams ;    

3,510,000 years  = 3.51 x 10⁶ years

RULES FOR NUMBERS LESS THAN "1"The exponential number is determined in two steps.  For numbers smaller than '1'  count the number of places the decimal point must be moved to the right to get a coefficient between 1 and 10.  This number equals the value for the exponent on ten. The coefficient is the rest of the digits that are a number more than one and less than ten.

EXAMPLES

Convert 0.000045 to exponential notation.     4.5 x 10^-5  decimal place moved 5 places to the right

0.001 = 10^-1 x 10^-1 x 10^-1 = 10^-3    decimal place moved 3 places to the right

0.000 000 000 000 000 000 000 008 2 = 8.2 x 10^-24    decimal place moved 24 places to the right