Groups of Numbers
As we already discussed in the last lesson, Pythagoras view of numbers was that all numbers have their own characteristic which makes the number unique, not just because it represents a certain sum, but because of the personality of the number. Because of this philosophical aspect to the understanding of numbers it mattered a great deal which numbers came out, more precisely, it mattered that all numbers could be expressed as a ratio between two natural numbers.
However, Pythagoras own theorem would sometimes calculate results that weren't.
This very simple triangle illustrates the problem. The hypotenuse in this example is √2 which comes out as a 1,41421... and continues indefinitely. This makes the number special because there is no end to the decimals no matter how many are calculated. Because of this difference, numbers are split into groups. |
Natural Numbers
The first group is the Natural Numbers, which are signified with the letter N. This group contains all whole positive numbers and are the first numbers most people encounter and start calculating with. They easily allow for calculations such as 7-3=4 or 8/2=4 since all numbers here are whole and positive.
If these are turned around however, things become tricky: 3-7=-4 or 2/8=0,25
The solutions to these two calculations move beyond Natural Numbers and into Integers (or Whole Numbers), and Rational Numbers respectively.
If these are turned around however, things become tricky: 3-7=-4 or 2/8=0,25
The solutions to these two calculations move beyond Natural Numbers and into Integers (or Whole Numbers), and Rational Numbers respectively.
Integers
The group of Integers contains all Natural Numbers, as well as zero and negative numbers. This allows us to operate with the type of calculation where a larger number is subtracted from a smaller number. Integers are given the symbol Z for the german Zahl, meaning Number.
Today we don't think anything of the possibility of a result of a calculation being a negative number, but it took quite a long time to come to this point and for negative numbers to become accepted as real. The first time these numbers are mentioned as a solution to an equation in Europe is int he writings of Nicolas Chuquet (ca. 1480). Even he speaks of these numbers as absurd and meaningless however and many people at this time would simply ignore negative numbers as results.
Today we don't think anything of the possibility of a result of a calculation being a negative number, but it took quite a long time to come to this point and for negative numbers to become accepted as real. The first time these numbers are mentioned as a solution to an equation in Europe is int he writings of Nicolas Chuquet (ca. 1480). Even he speaks of these numbers as absurd and meaningless however and many people at this time would simply ignore negative numbers as results.
Rational Numbers
The third group is the Rational Numbers which includes all Natural Numbers and Integers, as well as all fractions of Natural Numbers and Integers. This group is given the symbol Q and enables us to calculate the types of problems that don't result in an integer, but do still have a finite amount of decimals.
Real Numbers
Real Numbers include the Irrational Numbers into the group, while still including the previous groups. Irrational numbers are numbers that cannot be expressed as a ratio between two integers and therefore have an infinite amount of decimals. A lot of square roots fall into this category, but they are not the only ones: the golden ratio and pi being two notable examples.
Notice how the circles lie inside each other. Each circles contain the numbers inside of itsself as well as the numbers inside all circles closer to the center.
The Golden Ratio
The Golden Ratio is an incredibly irrational number, because unlike other irrational numbers that are quite close to being a ratio of integers, the golden ratio is about as far away from that as it is possible to be. The square root of two is by the way also quite far from being a ratio of integers while pi is relatively close to 22/7.
One approximation of the golden ratio is: 34/21
In nature the golden ratio is quite prevelant. Flowers will often place their seeds according to the golden ratio, because it saves space so the plant is able to fit as many seeds into as small an area as possible. If a plant were to place its seeds in ratios of integers it would waste a lot of space as in this example.
One approximation of the golden ratio is: 34/21
In nature the golden ratio is quite prevelant. Flowers will often place their seeds according to the golden ratio, because it saves space so the plant is able to fit as many seeds into as small an area as possible. If a plant were to place its seeds in ratios of integers it would waste a lot of space as in this example.
These seeds are placed at a ratio of 1/8 which creates these columns of seeds. This will always be the pattern with ratios of integers. Irrational numbers however create spiral patterns, and the further an irrational number is from being a ratio of an integer (and thereby creating this column or spokes pattern), the better it is at conserving space and packing as many seeds in there as possible. Notable examples of plants that do this are sunflowers, pinecones and cauliflowers. Petals are also often seen to be arranged in this pattern since it allows the plant to absorb as much sunlight as possible from each petal because the petals will not get in the way of each other.
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The Golden Rectangle
Constructing the Golden Rectangle:
First of all, start out with a square where both sides are 2 and draw a line down the center of it, so there are now two rectangles with the sides 2 and 1. Draw in a line in one of these new rectangles to create two triangles. The hypotenuse will now be defined as the radius of a circle. Draw the circle with its center at the smallest angle. Extend the original square along the diameter of the circle to create a new rectangle. Erase the helping lines and you are left with a Golden Rectangle. |
The Euclidian Algorithm
In this lesson we've been speaking quite a lot about ratios, but sometimes the fractions we are presented with are not expressed in the simplest way possible.
If for instance there were 75 people in a class and 45 of them got a perfect score on their homework. The fraction we would be presented with would be: 45/75
That's a bit cumbersome to work with though. The Euklidian algortihm can help reduce that fraction to its simplest form.
Here is how:
If for instance there were 75 people in a class and 45 of them got a perfect score on their homework. The fraction we would be presented with would be: 45/75
That's a bit cumbersome to work with though. The Euklidian algortihm can help reduce that fraction to its simplest form.
Here is how:
1. Divide the largest of the numbers with the smallest
2. If the resulting sum is not an integer, calculate how much is left over
3. Take the smaller of the numbers used in the last division, and divide by the amount left over
4. Repeat step two and three until an integer is reached. If you don't, then the fraction is already as small as it gets. When this is reached the greatest common divisor has been found.
5. Divide both original numbers with the greatest common divisor.
6. The fraction has now been reduced.
2. If the resulting sum is not an integer, calculate how much is left over
3. Take the smaller of the numbers used in the last division, and divide by the amount left over
4. Repeat step two and three until an integer is reached. If you don't, then the fraction is already as small as it gets. When this is reached the greatest common divisor has been found.
5. Divide both original numbers with the greatest common divisor.
6. The fraction has now been reduced.
In this example we start out with the fraction 45/75
Step 1: 75/45=1,667
Step 2: Since the result is one point something, we first subtract the 45 from 75 since the result of one point something tells us that we are able to do that once but not twice. 75-45=30. That shows us that the left over amount is 30.
Step 3: 45/30=1,5
Step 4: We repeat step two: 45-30=15, and repeat step three: 30/15=2. We've now reached an integer which tells us that 15 is a divisor for both numbers which will yield an integer.
Step 5: 45/15=3 & 75/15=5
Step 6: We can therefore conclude that 3/5 of the students scored top marks on their homework.
Step 1: 75/45=1,667
Step 2: Since the result is one point something, we first subtract the 45 from 75 since the result of one point something tells us that we are able to do that once but not twice. 75-45=30. That shows us that the left over amount is 30.
Step 3: 45/30=1,5
Step 4: We repeat step two: 45-30=15, and repeat step three: 30/15=2. We've now reached an integer which tells us that 15 is a divisor for both numbers which will yield an integer.
Step 5: 45/15=3 & 75/15=5
Step 6: We can therefore conclude that 3/5 of the students scored top marks on their homework.