Scientific and Engineering Notation

09/22/2012

A really simple way to do Scientific and Engineering Notation,

that always gives you the right answer

(Don't even think about "moving decimal points")

A lot of people struggle with Scientific Notation and Engineering Notation.

Sometimes this is because they see what appears to be a lot of involved math.

Other times, it just looks extra complicated.

I would like to illustrate a method that is so easy to learn and use, that I have been able to teach it to a group of 5th grade boys. These young fellows (we were using slide-rules with this method) could handle any number you gave them, for multiplication or division, and always get the right answer, and have the decimal point in the right place. There is a catch however ... It seems that a lot of adults see this method as a less-sophisticated way of doing math, and as a result they tend to shy away from it. All I can say is that if it always works, and is so terribly easy to learn... what could be wrong with doing it this way?

Like to see just how easy this is? OK, here we go....

First, consider when given any number in its basic form, is shown as such, with a x 10⁰ notation attached to it. here's an example: Given a number like 2.53, we would now express that number as the same 2.53, but with x10⁰ attached... i.e. 2.53x10⁰.

Everyone knows that our actual worth can be determined not by just what we have in our pocket, or how much we either have in the bank (or our bank deficit), but it is the combination of the two. We can move what funds we have in our pocket to the bank, or move funds from the bank (either savings or borrow) to our pocket, without changing our worth. In our method here, we simply maintain that type of relationship as follows:

Example #1

Consider the number 2.53 as a single digit of "2", with some numbers behind the decimal that we temporarily ignore. The 10⁰ we consider as like a "bank account" with a balance of "0" digits. Let's think of that single digit as a coin of some sort, as a single coin, and an initial bank balance of "0". There's a very important concept here that we must keep in mind, in that the number of coins on the left side of the "x" and the number of coins either saved in the bank, or owed to the bank, will always balance out.

Now, if we wanted to have more than one coin in our pocket, we could borrow a couple of coins from the bank. The end result would look like 253.0x10^-2, where the 253 is like having 3 coins now, but the 10^-2 shows that we owe the bank 2 coins. We actually just simply moved some digits from the bank into our pocket (the left side), which created the -2 deficit (x10^-2), and gave us 2 more coins making our single digit 2.53 change to 3 digits as 253.

The method here is that by disregarding what the digits are, and by only thinking of them as just simple digits, we can simply move them around "by count".

Borrowing 2 more digits would increase our current 253.00x10^-2 to become 25300.0x10^-4. Notice how the actual value of those digits is ignored.

We could take the number 0.0012x10⁰ as having less than 1 digit and we could borrow 3 digits from the bank, which would result in 1.2x10^-3. Note again that we have not changed the actual total value of that number.

Example #2

Take the number 45.67x10^-3 (which could represent milliamps). If we wanted to express this as x10^-6 (microamps) we would simply borrow 3 digits from the bank. The result would be 45,670x10^-6 (45,670 microamps).

Example #3

The number 782.3watts would be expressed as 782.3x10⁰ watts. Since "Kilo" represents 1,000, this could be converted to kilowatts by taking 3 digits from the left side and putting them in the bank. The result would be expressed as 0.7823x10³ watts, which is the same as 0.7823 kilowatts.

The significant difference between Scientific Notation and Engineering Notation is that Scientific Notation is usually expressed as a single digit number ahead of the decimal point, with the necessary x10ⁿ following. Engineering Notation always has the numeric power of 10 expressed in 3's, such as 10^-6 10^-3 10⁰ 10⁺³ 10⁺⁶.

The general format for those Engineering Notation descriptions is shown best in table form: