Irrational Numbers
Some Irrational Numbers and their Ratios - and their transformation capabilites
Definition: Irrational Number
- A number that cannot be exactly expressed as a ratio of two integers. It is not rational and cannot be written as a simple fraction. When writing out an irrational number as a decimal, the numbers in the decimal would go on forever without repeating.
Of course, the number "1" is a rational number. But we often forget the beauty of a square... the perfect four-sided polygon of symmetry; the ratio of 1:1. It's so elegant, I don't even have to explain it... you already know what a square is.
Let us introduce the host of famous irrational numbers:-
φ
The "Golden Section", also called "Phi". As a ratio, it has wonderful properties if you add or minus squares of its sides. Loved for it's beauty, especially by architects... and other non-mathematical people, who don't know how to draw spirals. It's value is about 1.6180339 [a] [b] [c] Also called Golden Mean and Golden Ratio.
There are more irrational numbers but we're concentrating on these few... for now.
Before we start, just to explain...
We will be using the format...
a > b > 0
which means that, in all the rectangles here,
"a" is the longer length
and "b" is the shorter length.
Also "h" is the diagonal.
But with "h" in any right-angled triangle,
"h" is called the Hypotenuse.
Ratio just refers to "a" divided by "b".
Ratio = a / b
So if a = 4 and b = 3
then the ratio a/b = 4/3
or 1.333∙
This is also a rational number.
Definition: Pythagoras' Theorem
- In a right-angled triangle, the square (of the length) of the Hypotenuse is equal in area to the sum of the squares (of the lengths) of the other two sides.
On the left is a "3 4 5" triangle. It's a right-angled triangle with lengths 3 and 4 while the hypotenuse happens to be 5.
So a = 4, b = 3 and we will show that h = 5.
Using Pythagoras' Theroem,
a2 + b2 = h2
(4 × 4) + (3 × 3) = h2
16 + 9 = 25 = h2
Therefore h = 5
The 4/3 ratio is not irrational. But the "3 4 5" triangle is somewhat elegant because the lengths are just simple integer numbers. There are many others (called Pythagorean Triples) like "5 12 13" and "8 15 17" but "3 4 5" is the most famous.
The Square
In a square, a = 1 and b = 1.
The Ratio a / b is
1 / 1 = 1
...which is not surprising
since a = b in a square.
Let's apply Pythagoras' Theorem to a square.
So a = 1, b = 1
(1 × 1) + (1 × 1) = h2
1 + 1 = 2 = h2
Therefore h = √2
Say hello to "Square Root of 2"
√2 { Square Root of 2 } 1.4142...
If you take a square piece of paper and fold it along the diagonal, that diagonal length is the "square root of 2".
The ratio √2 is elegantly convenient for dividing in half and doubling. The ratio √2 is used in the "A" Series paper standard (ie A3, A4 and A5 paper).
If you halve it (fold the long length in half) -or- if you double it (join at the long lengths side-by-side), the ratio remains the same... still at √2. This is a big deal!
The ratio √2 = a / b = 2b / a
Also, one-half of √2 is the same as the reciprocal of √2.
√2 / 2 = (√2 × √2) / (2 × √2) =
2 / (2 × √2) = 1 / √2
So when you halve it, you get the same ratio but the sideways version... like when you halve an A4 paper, you get A5 paper
Similarly, doubling A4 paper gets you A3 (made of 4 x A5 paper).
2 × √2 = 2 × √2 × √2 / √2 =
2 × 2 / √2 = 4 / √2
φ { the Golden Section } 1.6180...
The Golden Section, φ, has the properties as follows:-
- if you add a Square (to the long side), the resultant ratio is still φ (but sideways).
- if you minus a Square (from the short side), the new ratio is still φ (but sideways).
Yes, this is also a big deal!
The ratio φ = a / b = (a + b) / a
The diagram on the left shows how to draw the Golden Section. You take a square, folding it in half, draw a diagonal (in the half section), unfold the square and now extending the square (from the middle fold) by the length of the diagonal.
You can use Pythagoras'Theorem to solve
ratio φ = (1 + √5) / 2
It's quite easy if you make the shorter length 2 and then work out the longer length to be 1 + √5.
The Golden Section φ can also be arrived at in the Fibonacci series as a ratio of a number and the preceding number.
Fibonacci sequence : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610... basically where the next number is the sum of the last 2 numbers... so next is 377 + 610 = 987.
As the sequence gets higher, the ratio between the numbers becomes more accurate to arrive at φ : eg. 987 / 610 = 1.61803...
The φ ratio is always maintained when we keep adding or subtracting Squares (See φ end diagram).
In comparison, the √2 ratio is always maintained by doubling or halving itself (Compare with √2 end diagram).
Thinking about φ, what if we add 2 squares? Well, we get the lesser-known pauper-cousin of φ called the "Silver Ratio". It's so less-known that I didn't even bother to mention it (Oh wait, did I just...).
δS { Silver Ratio }
Very briefly, the Silver Ratio δS is much like the golden one except it adds or subtracts 2 Squares (not just 1).
δS works out to be 2.4142... or much simply as (1 + √2).
δS = a / b = (2a +b) / a
π { Pi } 3.1416...
π = c / 2r
where c = Circumference
and 2r = 2 × Radius = Diameter
As such, c = 2πr
Also, the Area of the Circle = πr2
Because Pi is defined from a circle, anything based on circles will involve π. This includes ellipses, cylinders, cones, spheres and tori (donuts).
π is a transcendental number. Other than measuring the circumference of a circle, Pi cannot be arrived at through root equations nor algebra nor by drawing out rectangles with arcs. π is what it is.
Middle diagram shows the π ratio as a rectangle. There are no tricks to manipulate it.
e { Euler's number } 2.7183...
Year 1
$100 × (1 + 100%) 1
Year 2
$100 × (1 + 100%) 2
Year 3
$100 × (1 + 100%) 3
Year n
$money (1 + 1)n
1 time event:
(1 + 1/1)1
2 time events:
(1 + 1/2)2
12 time events:
(1 + 1/12)12
n time events:
(1 + 1/n)n
To solve exponential equations, you will need to use loge(x) or "logarithm to the base e" which is called natural logarithm or ln(x).
The irrational number e is defined from Growth. In nature, growth is exponential; in finance, growth is a compound interest calculation. On its own, e does not seem meaningful visually or in a diagram because growth is an expansion over a period of time. Also e is transcendental.
Imagine a pond with algae but the algae is just constrained surrounded by a floating rope. Let's say that the algae occupies a space of 1 sq.metre. After a year, you loosen the rope to measure the growth of algae to be 2 sq.metres. You then conclude that the algae growth rate is 100% per year.
In financial terms, this would be where you invest $100 in a bank deposit and, after a year, get back $200 (interest rate of 100% per year). Then you invest that $200 and, after another year, get back $400. Then $400, invested after another year becomes $800 and so on. This accelerated earning is called "compounding" because whatever is earned in re-invested back in.
We can express that as $100 × (1 + 100%) years or more simply as $money (1 + 1)n where n is the number of years.
What if at the pond, we loosen the constraining rope every half year? Then after the first year, having done the release twice, the calculation would be (1 + 1/2)2 = 2.25 sq.metres. It's more than 2 sq.metres although the growth rate is the same because the loosening allowed a wider area of algae to grow as well.
What if at the bank, they pay the interest monthly? Then after the first year, having compounded 12 times, the calculation would be (1 + 1/12)12 = $261.30 which is more because of the numerous re-investments compounded.
Let n be the number of discrete events of rope loosening or investment compounding.
Then the calculation is (1 + 1/n)n
As n increases, the result increases but slows down as it converges towards a limit, e. When n reaches infinity, the the result is e, 2.71828...
e = limn→∞ (1 + 1/n)n
At infinity (n→∞), the pond has no rope to constrain the algae. Here, the algae's expansion is continuous natural growth. The number e is the ratio for all continuous natural growth.
exponential formula: growth = ex
where x is (rate × time). Typically, x is graphed over time with a fixed rate; but it can be a combination of both. The algae & interest examples used r = 1 to simplify the solution (ie growth at 100%/year).
e also makes an appearance in something called the Normal Distribution.
There are events with only 2 possible outcomes. This could be the toss of a coin; where the outcomes are only heads or tails. This could be a plant creeping vertically up a brick wall; having to creep left or right following the grooves. Pascal's Triangle shows the results when this is repeatedly done {compound events}.
Let's use "H" for Heads and "T" for Tails. Using the Pascal's Triangle chart on the left, let's use the left-side (white) for Heads and the right-side (green) for Tails. The vertical box on the far-left are just the row numbers.
● Let's start at row 1. This indicates 1 toss of a coin, It can be either H (heads) or T (tails). The diagram shows 1 for H and 1 for T.
● Row 2 shows 2 tosses. The possible combinations are HH, HT, TH, TT. So we count the permutations as 1 of 2H, 2 of 1H1T, 1 of 2T. The chart shows "1 2 1".
● Row 3 shows 3 tosses. The combinations are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. So the permutations are 1 of 3H, 3 of 2H1T, 3 of 1H2T, 1 of 3T. The chart shows "1 3 3 1".
If we go further down the diagram, we can see that any particular number is simply the sum of the two numbers above it. That's how the chart is made.
● Row 4 shows "1 4 6 4 1" where 6 = 3 + 3 (from the row above) and so there are 6 permutations of 2H2T (ie 2Heads,2Tails).
Let's use "n" to indicate the number of events in a sequence (same as row number).
The second chart displays the values for n = 12 as vertical discrete blocks. This is called a Binomial Distribution. It uses the word "Bi" because there were only 2 starting outcomes: For more possible outcomes, it would be called "Poly"-nomial.
The third chart is when the sequence of events reaches infinity [ n = ∞ ] and display as a continuous bell-curve. This is called the Normal Distribution (or Laplace-Gauss curve). You may know this from Statistics.
The probability density function for the standard normal distribution is...
P(x) = e(-x2/2) / √2π
Notice that both e and π are in the formula.
End of Document