Powers of ten and estimation techniques

Powers of ten and estimation techniques

We now consider numbers to the base ten and see how placing number in index form or standard form enables us to easily estimate quite complex arithmetic expressions.

The powers of ten are sometimes called `the technicians shorthand'. They enable very large and very small numbers to be expressed in simple terms. You may have wondered why, up to now in our fundamental study of numbers, we have not mentioned decimal numbers. Well the reason is simple, these are the numbers you are most familiar with, they may be rational, irrational or real numbers. Other numbers such as the positive and negative integers are a subset of real numbers. The exception are the complex numbers, these are not a subset of the real numbers and do not form part of our study in this course. Essentially then, decimal numbers may be expressed in index form, using the powers of ten. For example:

1,000,000                          = 1 X 10⁶

100,000                             = 1 X 10⁵

10,000                               = 1 x 10⁴

1000                                  = 1 X 10³

100                                     = 1 X 10²

10                                       = 1 X 10¹

0                                         = 0

1/10 = 0.1                          = 1 X 10^–1

1/100 = 0.01                     = 1 X 10^–2

1/1000 = 0.001                 = 1 X 10^–3

1/10,000 = 0.0001           = 1 X 10^–4

1/100,000 = 0.00001       = 1 X 10^–5

1/1,000,000 = 0.000001 = 1 X 10^–6

I am sure you are familiar with the above shorthand way of representing numbers. We show for example the number one million (1,000,000) as 1 X 10⁶ that is 1 multiplied by 10 six times. The exponent (index) of 10 is 6, thus the number is in exponent or exponential form, the exp button on your calculator!

Notice we multiply all the numbers represented in this manner by the number 1. This is because we are representing one million, one hundred thousand, one tenth, etc. When using your calculator, always input the multiplier (one in this case) then press the exp button and input the index (exponent) number, remembering to operate the +/– button if you are imputing a negative index.

When representing decimal numbers in index (exponent) form, the multiplier is always a numbers which is ³ 1.0 or <10, that is a number greater than or equal to     (³ 1.0) one or less than (< 10) ten.

So, for example, the decimal number 8762.0 is 8.762 X 10³ in index form. Notice with this number, greater than 1.0, we displace the decimal point three (3) places to the left, that is three powers of ten. Numbers rearranged in this way, using powers of ten, are said to be in index form or exponent form or standard form, dependent on the literature you read. What about the decimal number 0.000245? Well I hope you can see that in order to obtain a multiplier that is greater than or equal to 1 and less than 10, we need to displace the decimal point four (4) places to the right. Note that the zero in front of the decimal point is placed there to indicate that a whole number has not been omitted. Therefore, the number in index form now becomes             2.45 X 10^–4. Notice that for numbers less than 1.0, 

we use a negative index. In other words, all decimal fractions represented in index form have a negative index and all numbers greater than 1.0, represented in this way, have a positive index.

Every step in our argument up till now has been perfectly logical, but how would we deal with a mixed whole number and decimal number such as 8762.87412355? Well again to represent this number exactly, in index form, we proceed in the same manner, as when dealing with just the whole number. So, displacing the decimal point three places to the left to obtain our multiplier gives 8.76287412355 X 10³. This is all very well but one of the important reasons for dealing with numbers in index form is that the manipulation should be easier! In the above example we still have 12 numbers to contend with plus the powers of ten.

In most areas of engineering, there is little need to work to so many places of decimals. In the above example for the original number, we have eight decimal place accuracy this is unlikely to be needed, unless we are dealing with a subject like rocket science or astrophysics! So, this leads us in to the very important skill of being able to provide approximations or estimates to a stated degree of accuracy.