Scientific Notation

Do you recognize these numbers?

300,000,000 m/sec.
0.000 000 000 753 kg
.00000000000000000000000167
2750062
.001297

The numbers are …

the Speed of light!
the mass of a dust particle!
the mass of a proton!
the revolutions of a turntable!
the volume of a raindrop!

Scientists have developed a shorter method to express very large or very small 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 10¹¹.

The first number 1.23 is called the coefficient. It must be greater than or equal to 1 and less than 10.

The second number is called the base . It must always be 10 in scientific notation. The base number 10 is always written in exponent form. In the number 1.23 x 10¹¹ the number 11 is referred to as the exponent or power of ten.

To write a number in scientific notation:

Put the decimal after the first digit and drop the zeroes.
In the number 123,000,000,000 The coefficient will be 1.23
To find the exponent count the number of places from the decimal to the end of the number.
In 123,000,000,000 there are 11 places. Therefore we write123,000,000,000 as: 1.23 x 10¹¹

Exponents are often expressed using other notations. The number 123,000,000,000 can also be written as: 1.23E+11 or as 1.23 X 10¹¹

For small numbers we use a similar approach. Numbers less than 1 will have a negative exponent. A millionth of a second is: 0.000001 sec. or 1.0E-6 or 1.0 x 10^–6.

Convert from Scientific Notation to Real Number:
5.14 x 10⁵ = 514000.0

Scientific notation consists of a base (here 5.14) multiplied by 10 raised to an exponent (here 5). To convert to a real number, start with the base and multiply by 5 tens like this: 5.14 x 10 x 10 x 10 x 10 x 10 = 514000.0. Multiplying by tens is easy: one simply moves the decimal point in the base (5.14) 5 places to the right, adding extra zeroes as needed.

Convert from Real Number to Scientific Notation:
0.000345 = 3.45 x 10^-4

Here we wish to write the number 0.000345 as a base times 10 raised to an exponent. To convert to scientific notation, start by moving the decimal place in the number until you have a number between 1 and 10; here it is 3.45. The number of places that you had to move the decimal point is the exponent. Here, we had to move the decimal 4 places to the right, so the exponent is -4.