10.1: Math

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gemology
Gemology Project

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Although the study of gemology requires no formal prior training, a high school diploma would make it easier to understand basic math. Knowledge of trigonometry especially might serve you well.

Below are some basic calculations that you may want to understand.

Cross-multiplication

Some people have trouble with cross multiplications, while it is fairly easy if you keep a simple equation in mind.

\[5 = \frac{10}{2}\]

which is the same as \[\frac{5}{1} = \frac{10}{2}\] because 5 divided by 1 = 5.

Let's say you want to bring the 10 to the left of the equation. Obviously 10 = 5 times 2, so you cross-multiply.

\[\frac{5}{1}\swarrow \frac{10}{2}\] we multiply 10 with 1 to get it to the left side and:

\[\frac{5}{1}\searrow \frac{10}{2}\] we multiply 5 with 2, so we get: $10 * 1 = 5 * 2$ or $10 = 5 * 2$

This would probably make more sense in the following equation:

\[\frac{6}{3}=\frac{4}{2}\]

If you would cross-multiply, you would get $4*3 = 12$ and $6*2 = 12$, so $12 = 12$.

$File:Math.png$

Figure $\PageIndex{1}$

We can do it easier with the aid of a simple diagram. In Figure $\PageIndex{1}$, you see a triangle with the equation $10 = 5 * 2$ (the "$*$" is left out). The double horizontal bars serve as the "$=$" sign OR as the "$/$" (division) sign.
With this simple diagram in mind, you can solve most simple cross multiplications.

How to read the triangle:

You start at one number and then go the next and then to the 3rd
You work your way up first, then down

Examples:

Say you start at 2
Then you go up and see the "$=$" sign. Now you have "$2=$"
Then you move up further, you meet the 10, so you have "$2=10$"
You can't go any further up, so you must go down. You meet the double lines again, but they can't be a second "$=$", so they serve as a division. Now you have "$2=10/$"
You go down further and see the 5, that makes "$2=10/5$"

It works the same when you start with 5.

Now let's start with 10:

You start with 10, so "$10$"
You can't go further up, so you must go down. You encounter the double lines. As they are the first time you see them, they are the "$=$", now you have "$10=$"
Then you meet the 5 (or the 2 depending if you go clockwise or anti-clockwise), making "$10=5$"
You can't go further down, so must go sideways. You see the 2, making the odd-looking "$10=52$". This is actually good math style but is confusing, so we need to place an "$*$" between them. The result is "$10=5x2$", which any prep school kid would agree on.

This is of course not much fun because all the answers are given. But this simple knowledge is basic when you want to solve an equation such as:

\[2.417 = \frac{300}{x}\]

Simply replace "$10=5*2$" with the numbers and the unknown "$x$" of the new equation in the triangle. (Hint: the "$x$" takes the place of the "$2$")

Give it a try and see if you can calculate the speed of light inside a diamond with the above equation (the 300 is short for 300,000 km/s, which is the speed of light in a vacuum).

If all else fails, keep $5 = \frac{10}{2}$ in mind and substitute the numbers for the unknowns in the equation you need to solve.

Sine, cosine, and tangent

File:Triangle.png

Figure $\PageIndex{2}$

The sine, cosine, and tangent are used to calculate angles.

In Figure $\PageIndex{2}$, the 3 sides of a right triangle (seen from corner A) are labeled Adjacent side, Opposite side and Hypotenuse. The hypotenuse is always the slanted (and longest) side in a right triangle.

The opposite and adjacent sides are relative to corner A. If A would be at the other acute corner, they would be reversed.

Sine

File:Pythagoras.png

Figure $\PageIndex{3}$

Sine is usually abbreviated as sin.

You can calculate the sine of a corner in a right triangle by dividing the opposite side by the hypotenuse. For this you need to know two values:

1. the value of the opposite side

2. the value of the hypotenuse.

In Figure $\PageIndex{3}$ those values are 3 and 5, the sine of A or better sin(A) is 3/5 = 0.6

\[\sin = \frac{opposite\ side}{hypotenuse} = \frac{3}{5} = 0.6\]

Now that you have the sine of corner A, you would like to know the angle of that corner.
The angle of corner A is the "inverse sine" (denoted as sin^-1 or arcsin) of the sine and is done by complex calculation. Luckily we have electronic calculators to do the dirty work for us:

type in 0.6
press the "INV" button
press the "sin" button

This should give you approximately 36.87, so the angle of corner A is 36.87°

\[\arcsin \left(\sin A\right) = \arcsin \left(0.6\right) = 36.87\]

When you know the angle a corner makes, lets say 30°, you can calculate the sine as follows:

type in 30
press sin

That should give you 0.5

Practical use

If you know the angles of incidence and refraction in a gemstone, you can calculate the refraction index of that gemstone. Or do other fun things like:

\[index\ of\ refraction = \frac{sin\ i}{sin\ r}\]

Diamond has a refraction index of 2.417, so if the angle of incidence is 30°, the angle of refraction can be calculated as:

\[\sin r = \frac{\sin i}{n} = \frac{\sin 30}{2.417} = \frac{0.5}{2.417} = 0.207\]

so using the inverse sine:

\[\arcsin \left(\sin r \right) = \arcsin \left(0.207 \right) = 11.947 \Rightarrow angle\ of\ refraction = 11.947^\circ\]

It's not all rocket science. Read the page on refraction if you don't know what is meant by angle of incidence and angle of refraction.

Calculating the critical angle

Calculating the critical angle of a gemstone is pretty easy although the formula might scare you.

\[critical\ angle = \arcsin\left(\frac{1}{n}\right)\]

Where n is the refractive index of the gemstone.

The actual formula is $\arcsin(n_2 / n_1)$, but as we gemologists usually are only concerned with the critical angle between air and the gem, n2 = 1.

The calculation of this formula is easy, we'll use quartz with n = 1.54 as an example.
When you use a windows calculator, make sure you are in scientific mode. Then press the following buttons:

1
/
1.54
=

Then check the "inv" checkbox and press the "sin" button. That should give you an approximate value of 40.493, so the critical angle for quartz is 40.5° (rounded down to one decimal).

Cosine

The cosine of a corner in a right triangle is similar to the sine, yet now calculation is done with the division of the adjacent side by the hypotenuse. The cosine is abbreviated as "cos"

In Figure $\PageIndex{3}$, that would be 4 divided by 5 = 0.8

\[\cos = \frac{adjacent\ side}{hypotenuse} = \frac{4}{5} = 0.8\]

Again as with the sine, the inverse of the cosine is the arccos or cos^-1:

type in 0.8
press INV
press cos

This should give you 36.87 as well, so the angle remains 36.87° (as expected).

\[\arccos \left(\cos A\right) = \arccos \left(0.8\right) = 36.87\]

Tangent

The 3rd way to calculate an angle is through the tangent (or shortened to "tan"). The tangent of an angle is opposite side divided by adjacent side.

\[\tan = \frac{opposite\ side}{adjacent\ side}\]

For Figure $\PageIndex{3}$, that will be 3/4 = 0.75
Calculation of the angle is as above, but using the arctan or tan^-1:

type in 0.75
press INV
press tan

This should give you 36.87, so through this method of calculation the angle of corner A is again 36.87°.

\[\arctan \left(\tan A\right) = \arctan \left(0.75\right) = 36.87\]

A simple bridge to remember which sides you need in the calculations is the bridge SOH-CAH-TOA.

SOH = Sine-Opposite-Hypotenuse
CAH = Cosine-Adjacent-Hypotenuse
TOA = Tangent-Opposite-Adjacent

Degrees, minutes and seconds

When we think of degrees we usually associate it with temperature and we consider minutes and seconds as attributes of time. However, in trigonometry, they are used to describe angles of a circle and we refer to them as the radian values.

A full circle has 360 degrees, or 360°.
Every degree can be divided into 60 minutes (like in a clock) instead of the 10 decimal subdivisions.
Minutes are notated with a ', as in 26'.
The individual minutes are further divided into 60 seconds and they are described with '', as in 23''.

This may look odd at first, but it's not very hard to understand.

If you have an angle of 24°26'23'' (24 degrees, 26 minutes and 23 seconds), this means that the decimal value is:

24°
26 divided by 60 or 26/60 = 0.433°
23/(60 * 60) or 23/3600 = 0.0063°

This totals as 24 + 0.433 + 0.0063 = 24.439° in the decimal value (which is the decimal value of the critical angle of diamond).

When you want to calculate the radian value of 24.439°, you do the following:

the 24 stays 24 (because that doesn't change)
you try to find how many times 0.439 times 60 fits in the degree by: 60 times 0.439 = 26.34, so that is 26 full minutes (0.34 left over)
you calculate the seconds through 60 times 0.34 = 20.4 (or 20 full seconds because we don't count lower than seconds).

This gives 24°26'20'' (24° + 26' + 20'') instead of the 24°26'23''. The 3-second difference is caused by the rounding down to 3 decimals in the prior calculation. In gemology, we usually don't even mention the seconds, so it will be rounded down to ≈ 24°26'.

Even though you may not need this knowledge often, it is important that you at least know of its existence as you may get confused when reading articles. Sometimes values are given in decimal degrees, at other times in radian values.

External links

Introduction to mathematics
Sine definition with Java demo
Cosine definition with Java demo
Tangent definition with Java demo

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Practical use

Calculating the critical angle