Scientific notation

​ ​If you haven't seen the video about orders of magnitude yet, watch it now, because you will need the material from it. The distance between the earth and the sun, is 150 billion meters. This can be expressed using orders of magnitude. In fact you can use different powers with base ten to write the same thing. It can be written as 1.5 times 10 to the power of 11 or 15 times 10 to the power of 10 or 150 times 10 to the power of 9.

The first factor is called the co-efficient and the other is an exponential expression with the base ten. The first case is special, it is called, "scientific notation." ​ ​Scientific notation is a special type of order of magnitude where the co-efficient is larger than or equal to one but smaller than ten. Scientific notation is often used in a natural sciences and is also known as the standard form. This is especially practical if you dictate, calculate, or compare numbers. ​ ​Here are two large numbers: the masses of the Sun and of the Earth in kilograms. If you want to know how many times the Sun's mass is larger than the Earth's mass divide the Sun's mass by the Earth's.

Just as you can write a multiplication of fractions on a common fraction bar, you can also split it apart like this. Now you have two divisions. The first one you can calculate using a calculator or a pen and paper - the result is approximately one third. ​ ​To simplify the second fraction you use the rules for dividing exponential expressions with the same base. Here you take the numerator's power minus the denominator's. 30 minus 24 is six.

This is an order of magnitude, but it is not a scientific notation, because in scientific notation the coefficient must be greater than one but less than ten. If you now think about the exponential expression as a repeated multiplication, it's easy to see how we can turn this into scientific notation. Use one of the ten factors to make the coefficient ten times larger. And to revert the remaining tens back to a power again. So, it is now ten to the power of five instead of ten to the power of six.

The power part of the exponent is now worth ten times less, and the coefficient is worth ten times more. The product remains unchanged and here is your scientific notation. If you want to write this in a decimal form, you move the decimal comma in the coefficient as many places as the power says, five in this case. One, two, three, four, five. The mass of the sun is 333,000 times more than the mass of the earth.

Scientific notation is also used for very small numbers. An Ebola virus is approximately eight times ten to the power of minus eight meters in diameter. Do you remember how to count orders of magnitude with negative powers? You move the decimal comma as many places to the left as the power says. One, two, three, four, five, six, seven, eight.

The virus is 800 millions of a meter in diameter. Scientific notation. An order of magnitude with the coefficient one, or greater than one and less than ten. This is useful when you compare very large or very small numbers