\[ \]

\[ \]

\[ \]

*Issue No 28, 22 May 2023 *

*By: Anthony O. Ives*

Intergers are whole numbers such as 1,2,3,4,5,.....120,....,etc. Floating numbers are not whole numbers such as 0.7, 2.5, 3.4, etc otherwise described as decimal numbers. Negative numbers are numbers less than zero such as -1,-2,-3,-4.....-120,.....,etc. Negative numbers in particular have a specific arithmetic in how they are added, subtracted, multiplied and divided. Floating or decimal numbers are really added, subtracted, multiplied and divided in the same way as intergers if you don't let the decimal point confuse you.

Floating or decimal numbers describe numbers that are not whole numbers, the alternative way to describe them is through using fractions. However, decimal numbers are usually simplier to carry out arithmetic operations with and for computers to display and store. Typical decimal and fraction number equivalents are given below:

\[\frac{1}{2} = 0.5 \]

\[\frac{1}{4} = 0.25 \]

\[\frac{3}{4} = 0.75 \]

\[\frac{1}{3} = 0.3333...... \]

\[\frac{2}{3} = 0.6666...... \]

\[\frac{1}{100} = 0.01 \]

\[\frac{1}{1000} = 0.001 \]

The example of one third \(\frac{1}{3}\) is a good example of when a fraction is more accurate to use than decimal number as the decimal number of a one third does not give a precise number like way it does for something like a half \(\frac{1}{2}\). One third as decimal gives a never ending set of 3s so depending on the accuracy you need you have decide how many signicant figures you need.

For example one third for 1 significant figure is 0.3, for 2 significant figures it is 0.33, for 3 significant figures it is 0.333, etc. A fraction is self explanatory it is just simply the number on the top line (known as numerator) divided by the number on the bottom line (known as denominator), see [1]. This may help you to understand how you get the decimal number equivalent, lets look at the below example for one half, dividing ten by 2 and dividing one hundred by 2.

\[\frac{1}{2} = 0.5 = 0.5 \times 10^0 \]

\[\frac{10}{2} = 5 = 0.5 \times 10^1 \]

\[\frac{100}{2} = 50 = 0.5 \times 10^2 \]

The above example should show you how the floating or decimal number is just a way of describing numbers that are not whole numbers. The example also gives the scientific number format. Dividing 10 by 2 is fairly straightforward as it gives 5, however dividing 1 by 2 is very similar so you include decimal point to show it not whole number therefore 0.5. The below example should also explain why decimal numbers are sometimes called floating numbers as it shows a half represented in many scientific notations:

\[\frac{1}{2} = 0.5 = 0.5 \times 10^{0} \]

\[\frac{1}{2} = 0.5 = 5.00 \times 10^{-1} \]

\[\frac{1}{2} = 0.5 = 50.0 \times 10^{-2} \]

Scientific notation makes use of indices to give a more compact way of displaying numbers, indices are explained in more detail in reference [1]. The floating term comes from the fact that the decimal appears to float back and forth depending on the value of the index in the scientific notation.

Negative numbers are used when you have situation where a number has a larger number subtracted from it such as in the below example:

2 - 7 = -5

There are specific arithmetic operations for negative numbers however, most of them are intuitive with the exception of maybe the multiplying and dividing operations. The multiplying, dividing and other arithmetic operations are as follows:

\[(-) \times (+) = (-) \]

\[(-) \times (-) = (+) \]

\[\frac{(-)}{(+)} = (-)\]

\[\frac{(+)}{(-)} = (-)\]

\[\frac{(-)}{(-)} = (+)\]

\[(-) + (-) = (-) \]

\[(-) - (+) = (-) \]

Examples with numbers are as follows:

\[-7 \times +2 = -14 \]

\[-7 \times -2 = +14 \]

\[\frac{-8}{+2} = -4\]

\[\frac{+8}{-2} = -4\]

\[\frac{-8}{-2} = +4\]

\[-7 + (-2) = -9 \]

\[-7 - (+2) = -9 \]

\[-7 - (-2) = -5 \]

\[-7 + (+2) = -5 \]

Hopefully you find arithmetic operations with negative numbers fairly straightforward to understand and remember. The picture below shows how you can use a scientific calculator to convert a number to scientific notation as well as using it to do arithmetic operations with negative numbers.

Negative numbers create a another problem in that you cannot take the square root of a negative number, the below example should explain why this is the case:

\[(+4)^2 = +4 \times +4 = +16 \]

\[(-4)^2 = -4 \times -4 = +16 \]

\[\sqrt {+16} = +/-4\]

\[-4 \times +4 = -16 \]

\[\sqrt {-16} = \sqrt{16} \sqrt {-1} = 4 \sqrt {-1} = ?\]

As the above example demonstrates there is no number that can be multipied by itself to give a negative number as a negative number must be multiplied by a positive number to give a negative number hence its impossible to take the square root of a negative number. However, there is a solution for this problem which generates another set of numbers known as imaginary and complex numbers [2] which will be the topic of a future article.

Please leave a comment on my facebook page or via email and let me know if you found this blog article useful and if you would like to see more on this topic. Most of my blog articles are on:

Mathematics

Helicopters

VTOL UAVs (RC Helicopters)

Sailing and Sailboat Design

If there is one or more of these topics that you are specifically interested in please also let me know in your comments this will help me to write blog articles that are more helpful.

References:

[1] http://www.eiteog.com/EiteogBLOG/No8EiteogBlogIndices.html

[2] Engineering Mathematics, K. A. Stroud, Fourth Edition, 1995, Macmillan

*Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will
not be responsible for any liability, loss or risk incurred as a result of the use and application of information on
its website or in its products. None of the information on Eiteog's website or in its products supersedes any information
contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the
operators of aircraft, UAVs. *

*For any inquires contact: [email protected] copyright © Eiteog 2023*