# Math

Although the study of gemology requires no formal prior training, a highschool diploma would make it easier to understand basic math. Especially knowledge of trigonometry might serve you well.

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

## Contents

## Cross-multiplication

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

- <math>5 = \frac{10}{2}</math>

which is the same as <math>\frac{5}{1} = \frac{10}{2}</math> because 5 divided by 1 = 5.

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

- <math>\frac{5}{1}\swarrow \frac{10}{2}</math> we multiply 10 with 1 to get it to the left side and:

- <math>\frac{5}{1}\searrow \frac{10}{2}</math> we multiply 5 with 2, so we get: 10 x 1 = 5 x 2 or
__10 = 5 x 2__

We can put all of this in a simple diagram. In Fig.1 you see a triangle with the equation 10 = 5 x 2 (the "x" 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.

Say that you need 2 to be on the left side of the equation. Then you simply isolate the 2 and see what the positions of the other two numbers are.

As 2 is below the double lines, the double lines must serve as a division line, so <math>\frac{10}{5}=2 \Rightarrow 2=\frac{10}{5}</math>

The same applies for 5, try to see what 5 is in this diagram.

This is ofcourse not much fun because all the answers are given. But this simple knowledge is basic when you want to solve equations as:

- <math>2.417 = \frac{300}{x}</math>

Give it a try, 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 vaccuum).

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

## Sine, cosine and tangent

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

In fig.2 the 3 sides of a right triangle (seen from corner A) are labeled Adjacent side, Opposite side and Hypothenuse. The hypothenuse is always the slanted (and largest side) in a right triangle.

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

### Sine

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 hypothenuse. For this you need to know two values:

- 1. the value of the opposite side
- 2. the value of the hypothenuse.

In Fig.3 those values are 3 and 5, the sine of A or better sin(A) is 3/5 = 0.6

- <math>\sin = \frac{opposite\ side}{hypothenuse} = \frac{3}{5} = 0.6</math>

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. Luckely 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°

- <math>\arcsin \left(\sin A\right) = \arcsin \left(0.6\right) = 36.87</math>

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:

- <math>index\ of\ refraction = \frac{sin\ i}{sin\ r}</math>

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

- <math>\sin r = \frac{\sin i}{n} = \frac{\sin 30}{2.417} = \frac{0.5}{2.417} = 0.207</math>

so using the inverse sine:

- <math>\arcsin \left(\sin r \right) = \arcsin \left(0.207 \right) = 11.947 \Rightarrow angle\ of\ refraction = 11.947^\circ</math>

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.

### 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 hypothenuse. The cosine is abbreviated as "cos"

In Fig.3 that would be 4 divided by 5 = 0.8

- <math>\cos = \frac{adjacent\ side}{hypothenuse} = \frac{4}{5} = 0.8</math>

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 aswell, so the angle remains 36.87° (as expected).

- <math>\arccos \left(\cos A\right) = \arccos \left(0.8\right) = 36.87</math>

### Tangent

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

- <math>\tan = \frac{opposite\ side}{adjacent\ side}</math>

For Fig.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°.

- <math>\arctan \left(\tan A\right) = \arctan \left(0.75\right) = 36.87</math>

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

- SOH = Sine-Opposite-Hypothenuse
- CAH = Cosine-Adjacent-Hypothenuse
- 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 (we name them 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 23/360 = 0.063°

This totals in 24 + 0.433 + 0.063 = 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 atleast know of its excistance as you may get confused when reading articles. Sometimes values are given in decimal degrees, at other times in radian values.