Scientific notation

Scientific Notation is a way to express very large or very small numbers by the number’s most significant digits combined with an exponential expression of base 10. This notation usually includes a whole number with one or two decimals after it: 2.38. This will then be followed by an exponential expression of:  × 10^a , where a is going to either be a positive or negative number, generally with a magnitude greater than 3.

The exponent on the 10 shows if the number is a very large number or a very small number. If the exponent is positive, then the number being expressed is quite large. If the exponent is negative, then the number being expressed is very small.

The magnitude of the exponent then tells you how many places the original digits were moved by to get to the new expression. In other words, how many places were originally between the decimal point and the new currently visible digits of the number.

For example, 6.52  × 10⁸is another way of writing 652,000,000. The scientific notation version is raised to the 8th power because in our original number there were 8 places between the leading 6 and the decimal point. It’s important to note that this would be the same even if the number were 6.2  × 10⁸, but there would be 7 0s instead of 6, as now there is one fewer significant digit before the decimal point.

The inverse is true for when there is a negative exponent instead of a positive one. It’s still the number of places between the original decimal point and the first significant digit, but now it’s to the right of the decimal point instead of the left.

6.52  × 10⁻⁸ is another way of writing .0000000652. The simple conversion is the whole number will move to the right of the decimal point, and then zeroes are inserted. The number of zeros = |Exponent|− 1. In other words since the exponent here is -8, there are 7 zeroes added rather than 9 (8-1 rather than  − 8  − 1).

The last thing to note is that with positive scientific notation, sometimes there are two or three digits after the decimal point that “absorb” some of the zeroes places that are being multiplied by. With the negative exponents, there is generally just the single digit whole number to the right of the decimal, and so the number of zeroes will always be equal to that exponent minus one rule.