Issue No 8, 3 October 2022
By: Anthony O. Ives
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Mathematical notation is just a shorthand way of describing mathematical operations to perform on a quantity or set of quantities to determine another quanitity or set of quantities. The lift coefficent equation as used in ref [1] is good example of this. The lift coefficent equation describes how lift is related to aircraft velocity, wing area, lift coefficient and air density in shorter more concise, convient, clear way. Try describing the relationship in words and it would be a lot longer and probably very confusing plus not that easy to manipulate if for you want to determine stall velocity is related to lift, etc.
The multiplication sign × generally is not used in mathematical formulae to keep them concise hence generally multiplication is represented like so:
\[2x=2 \times x\]
\[ab=a \times b\]
So generally a symbol or letter representing a quantity beside one another like above implies multiplication of the quantities. Division is usually represented represented by a fraction as below:
\[\frac 2x=2 \div x\]
\[\frac ab=a \div b\]
So you divide what is on the top line by what is on the bottom line. Just by the way the top line is known as the numerator and the bottom line is the denominator:
\[\frac {numerator}{denominator} =numerator \div denominator\]
When you come across a symbol or letter with a superscript number or letter (referred to as an indice), etc this is describing that you are muliplying that number of the quantities together which is described simpler by notation below:
20=1
21=2
22=2×2=4
23=2×2×2=8
24=2×2×2×2=16
etc....
Just to note a superscript or indice of 0 always gives 1 no matter what the number is, with exception of 0 which is up for debate! See below for better explanation:
20=1
30=1
40=1
00=Not agreed upon but can be assumed to be 1 in some cases, some computer programs may return NaN (Not a Number). Just a general note the power to 2 is referred to as square because it can calculate the area of a square. Likewise the power to 3 is referred to as cube because it calculate the volume of a cube.
When a number is to power of a fraction or demical number, it the reverse procedure and represents a root of the number depending on what the bottom number of the fraction is. See below for an example and a better explanation, for easier understanding we are reversing the same examples as previously used hence getting 2 which is the root in each case.
\[4^{\frac12}=4^{0.5}=\sqrt{4}=2\]
\[8^{\frac13}=8^{0.333...} =\sqrt[3]{8}=2\]
\[16^{\frac14}=16^{0.25}=\sqrt[4]{16}=2\]
A number to power of a negative means 1 over the power of the indice, again this better explained with examples which includes whole number indices and fractions see below:
\[2^{-2}=\frac{1}{2^2} =\frac14\]
\[2^{-3}=\frac{1}{2^3} =\frac18\]
\[4^{-\frac12}=4^{-0.5}=\frac{1}{\sqrt{4}} =\frac12\]
\[8^{-\frac13}=8^{-0.333...}=\frac{1}{\sqrt[3]{8}} =\frac12\]
You can also simplfy your expression if carrying out arithmetic on number with indices. Dividing two expressions with the same base number with the bottom subtracted from the top indice. Likewise multiplication means you can add the indices. However none these apply if the base numbers are different, see below for examples.
22×23=4×8=22+3=25=32
\[2^3 \div 2^2=\frac84=2^{3-2}=2^1=2\]
(22)3=43=22×3=26=64
22×23=4×8=22+3=25=32
\[2^3 \div 4^2 \neq 2^{3-2} \neq 4^{3-2} \]
\[2^3 \times 4^2 \neq 2^{3+2} \neq 4^{3+2} \]
The relationships for general indice arithmetic is summarised below:
\[x^{\frac{n}{m}}=^m\sqrt{x^n} \]
\[x^{-n}=\frac{1}{x^n}\]
\[x^n \times x^m=x^{n+m}\]
\[x^n \div x^m=x^{n-m}\]
\[\left(x^n\right)^m=x^{nm}\]
\[x^n \times y^m \neq x^{n+m} \neq y^{n+m} \]
\[x^n \div y^m \neq x^{n-m} \neq y^{n-m} \]
The picture gives an example of scientific calculator buttons which can be used to carry out the indices functions. The square (power to 2), square root, cube (power to 3), cube root functions can usually be buildin. Most scientific calculators also which can evaluate any indice or any corresponding root.
I hope this article has been helpful in explaining indices more information can be found in Ref [2].
Please leave a comment on my facebook page or via email and let me know how if you understand how indices simply mathematical notation and the obivous places they can be used.
References:
[1] http://www.eiteog.com/No1EiteogBlogLiftCL.html
[2] Engineering Mathematics, K. A. Stroud, Fourth Edition, 1995, Macmillan
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