# Orders of magnitude and scientific notation - animation

In Physics it is often interesting to compare quantities without going into the detail of their exact value. In this way we can quickly get an idea of the size of one of the quantities compared to others. For instance, if two objects have masses of 2 kg and 123 kg respectively, the second one has approximately one hundred times more mass than the first one or, which is the same, the mass of the second one is two orders of magnitude greater than that of the first one.

In order to compare the order of magnitude of different quantities we should write them in scientific notation.

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##### Scientific notation

Scientific notation, also referred to as scientific form or exponential notation, it is a compact way of expressing large or small numbers using the powers of ten. A number expressed in scientific notation always has the form:

Where m is called the significand or mantissa and the exponent of 10 is called the order of magnitude. The significand is a real number whose absolute value is between 1 and 10. The exponent of 10 is the number of positions that the decimal separator must be shifted in order to convert a number expressed in scientific notation into the same number written in decimal notation. The decimal separator will be shifted to the right if the exponent is positive and to the left if it is negative.

The following table shows examples of numbers expressed in decimal and scientific notation:

Decimal
Scientific
Order of magnitude
200
2 102
2
237
2.37 102
2
-346.2
-3.462 102
2
0.200
2 10-1
-1
0.0237
2.37 10-2
-2
-0.00000376
-3.76 10-6
-6

The scientific notation allows an easy comparison of the orders of magnitude of two numbers by simply comparing the exponent of ten of both numbers. For example, the last number in the table is eight orders of magnitude smaller than the first one.

In Physics it is very frequent to use this notation, since there are many physical quantities that have very large or very small values and it is not practical to express them in decimal form. The charge of the electron expressed in units of the International System is – 1.60217662 × 10-19 C. Would you know how to express it in decimal notation?

In the following animation you can see the size of different objects with their corresponding orders of magnitude.

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