Tutorial: Converting to Scientific Notation

Converting to Scientific Notation

Most of the instructions on this website ask you to convert numbers to "scientific notation", and perform operations with the "coefficient" or "exponent". These terms are unfortunate because they sound complicated, but they're not. Converting to scientific notation just means shuffling the decimal point left or right, and the instructions intend for you to convert your number more or less instantly in your head. We certainly provide no slide rule scales for doing this.

There's a million textbooks and websites that describe scientific notation more elegantly than we do here, and cover important topics like significant figures. You should read a few. Be aware the terminology is not written in stone, and sometimes the "coefficient" is referred to with the even more obscure term "mantissa", and "exponent" can also be called the "power".

Scientific Notation is a method of writing an inconveniently large or small number as two smaller, more useful numbers: 1) A "coefficient", which is always between one and ten, and can be handled on slide rule scales, and 2) an integer power of ten, the "exponent", which can be easily manipulated in your head to determine final results.

Normally, a number in standard form and its equivalent representation in scientific notation are written like , but we have a lot of them to write in these tutorials so we're going to write the latter as 3.824E-8, the way it would show up on a cheap calculator. This will annoy the pedants but save a lot of time. Just remember that "E" means "exponent of ten", or "times ten to the power of". The coefficient comes before the letter E, and the exponent comes after.

To convert to scientific notation, shift the decimal point so that your number has a single nonzero digit to the left of the decimal, and then associate with it a power of ten representing the number of times you shifted the decimal left to achieve this.

1 ->1E0
3 -> 3E0
10 ->1E1
30 -> 3E1
300 -> 3E2
0.3 -> 3E-1 (shifted right 1, thus we shifted left "-1" times )
0.03 -> 3E-2
3243532 -> 3.243532E6
0.00003425 -> 3.425E-5
312.423E6 -> 3.12423E8 (left is stylistically wrong, but still the same number)

Let's explain a few of these.

3 -> 3E0: -let's write 3 as 3.0 ( to make the decimal point explicit ); -in 3.0 there's already exactly one non-zero digit to the left, so we shift the decimal zero times.; -thus we write 3E0 in scientific notation.

10 -> 1E1: -let's write 10 as 10.0 ( to make the decimal point explicit ); -There's more than one digit to the left of the decimal point.; -we shift left once and obtain 1.0. That's better.; -we write 1E1, which is our result, the letter E, and the number of times we shifted left.

300 -> 3E2: -let's write 300 as 300.0 ( to make the decimal point explicit ); -There's more than one digit to the left of the decimal point.; -we shift left once and obtain 30.0. Still no good.; -we shift left again and obtain 3.0. Excellent.; -we write 3E2, which is our result, and the number of times we shifted left.

Converting numbers in your head becomes routine after doing only a handful of them. Just move the decimal left or right, counting along, until it gets in the correct form. You also quickly learn to recognize the forms of the first -6 to 6 exponents, 31000 leaping to mind as 3.1E4, and 0.056 instantly recognizeable as 5.6E-2.

Numbers in scientific notation make a little more sense on the slide rule. The coefficient before the E can be found on the C or D scale, since it will always be between 1 and 10. The exponent after the E is just an integer, usually a small one, and the operations performed on it are simple integer arithmatic.