Natural numbers, Integers, Rationals and Real numbers each define a set. Therefore, they can be represented by the Venn diagram.
Can we draw the diagram in such a way that the area of each circle is proportion to how many numbers are in the set represented by the circle (or whatever shape)? And first things first shall we try that?
Now let’s elaborate.
Natural numbers (or counting numbers) are like 1, 2, 3, .. We use them when counting. Integers include all natural numbers as well as zero (0) and all negative numbers like -1, -2, -3, ..
Rational numbers are those that can be written as the ratio of two integers. Rational numbers includes integers; rational numbers can be either positive or negative.
Examples are like 1 &frac; 2 or 2 &frac; 3.
What’s the real numbers? Well, they includes both Rational and Irrational numbers. Irrational numbers are those that CANNOT be written EXACTLY as the ratio of two integers. Two famous examples are sqrt(2) as well as π. Indeed, as learning more, we find much more IRRATIONALS than RATIONALS — so that if we resize total REAL numbers to 100, then 100% will be IRRATIONAL, and ALMOST NOTHING is RATIONAL.
In Venn diagram, we typically represent one set by a circle. Two circles overlap indicates they have common elements. If one circle is completely contained in another circle, that indicates one set A is completely contained in another set B (that is, any element of set A is an element of set B).
π Ideas for Circles and Trigonometry:
Stories, Facts and Implications
1st Story: A Fun Way of Finding the π Value
Here is some facts found in somebody’s math venture:
He marks a point A on a circle. He then split the radius of the same circle into 4 equal segments – call the length of one segment as b.
He measured along the circle using b (explain the measurement). After 7 steps he stopped and label that point as B. — As he noticed, the arc from A to B is slightly greater than a quarter of the circle.
Now he returns to split the radius of this circle into 5 equal parts – call the length of one segment as d. Then he measured along the circle using d. After 7 steps he stopped and label that point as D. — As he noticed, the arc from A to D is slightly less than a quarter of the circle.
Having a second look, he has noticed as well that the line connecting B and D “almost” passes through the centre of the circle (or line BD is “almost” a diameter).
Pause for a minute. Try to draw a diagram according to this description. If not sure, look into the figure below for a check.
pi-SmartRatio
[As in this figure, N is the center of the circle, NA is the radius of the big circle, and the two small circles are of radius that is (1/4), resp. (1/5) of NA. From A, going through 7 steps to reach B; from A, going the other direction 7 steps to reach D. Pls. note the length of step in the two directions are not the same! ]
After some thought and calculation, he decided the approximate value of π. The value he’s found is 3.15 – which is pretty close to the true value. (For sure you know π is 3.14159, don’t you?)
Now is your chance to take part: how did he finds that value?
For a bonus question: he has decided that (7/4) radians is approximately 100 degrees. Do you agree? How did he come to that conclusion?
About a decade ago, there was a great Tsunami happening in Asia.
What do you know about the Asian tsunami?
Read through the article first. Use the following numbers to fill in the blanks in the story.Think about which numbers make sense.
500 20 8,000; 2004 110,000 30,000 9.0
A tsunami triggered by a very large earthquake off the coast of the Indonesian island of Sumatra on December 26, ____, has left more than 150,000 people dead and millions homeless. Countries hit hardest by the disaster include Sri Lanka, Indonesia, India, Thailand, and the Maldives. Almost 75% of the deaths occurred in Indonesia, estimated at ____. Sir Lanka was second highest with about 20% of the estimated deaths, or ______ people lost that day. The rest of the deaths, approximately ____, occurred in the other nine countries affected by the tsunami.
The ____ foot wall of water, higher than a two-story building, swallowed entire villages. The tsunami waves were not only very high, they moved at a much faster speed than normal. These waves were comparable in size to those you see on some of the surfing movies; but those waves travel at 30 miles an hour, and the tsunami waves were moving more than fifteen times as fast at ____ miles an hour. The velocity of the force is what caused the destruction—a massive force that swept away everything in its path.
The earthquake causing this Tsunami was a destructive earthquake measuring ______ on the Richter scale, the fourth worst earthquake in recorded history. Earthquakes are measured on a Richter scale that has a range from 0 to 12; a 6.0 on the scale is a pretty bad earthquake.
(Story constructed from January 2005 news reports)
> 9 is the largest digit under 10. 9 is a perfect square, 9 = 32.
A whole number multiplied by itself for some times is called a perfect power. In a perfect power, both the base and the exponent are whole numbers.
Since 32 – 23 = 1, we say that 9 = 32 is a perfect power that is one larger than the other perfect power. And it is the only one so satisfying the condition.
> In decimal system, to judge whether an integer is a multiple of 9, one add all the digits and find the sum. If the result of the sum is greater than 10, then from the results you add all the digits again. So finally we can get a digit that has only one place (less than 10). If this final one is 9, then the initial number would be a multiple of 9.
Mathematically speaking, for any multiple of 9, its digit root is 9.
> The “casting-out 9” method is a shortcut method for checking the results of integer operations, based on the principle we just discussed.
> Take 4 points from a circle, and choose to connect pairs of points to form chords. If a restriction is imposed so as not to allow intersection of chords, then there are exactly 9 ways to make the choice. As shown in the illustration below.
To recapitulate, for four points on a circle, there are 9 ways of making the chords through connecting some pairs of points (include no connection at all), when the restriction is not to allow intersection chords.
Since Motzkin (T. Motzkin – an Isralian mathematician) first worked on this problem, so named after him, number “9” is called a Motzkin number.
> Please look at the following division-by-9 operations:
405 ÷ 9 = 45
2025 ÷ 9 = 225
6075 ÷ 9 = 675
45 ÷ 9 = 5
225 ÷ 9 = 25
675 ÷ 9 = 75
Take for example the identity 45 ÷ 9 = 5. Take 45 and strike out from it a digit 4, then what’s left is another digit 5. (Of course, this cannot be true for any number divided by 9.) What’s amazing in the above is, that the trick can be applied twice (for example, in 405 -> 45 -> 5); each time we strike off certain digit and the value of number is reduced to (1/9) of the value before the division operation. So let’s call this a 9-reducible chain of numbers.
Taken another close look. Can you find any other pattern in the above sequence?
> The Arabic digit has several different ways of writing (ancient writing, not in use today). Some looks like a question mark, some looks exactly like digit “3”, and for some others, digit 3 is contained inside; just like, as we know, 9 contains 3 as one factor. The last of these continued to evolve into today’s number “9”.
> In English, we have “cloud line”, and in orient we also have “Nine Layers of Heaven). The meaning does not coincide completely, but the usage seems to be similar.
> Number 4 and 5 naturally, or seems naturally, to go together with number 9. We have 9 = 4 + 5. Let’s look at a few more (some have been mentioned as above).
A. For 4 points on a circle, there are 9 ways of making the chords through connecting some pairs of points;
B. Each 9-reducible chain ends with a multiple of power of 5.
C. 45 = 5 x 9 and 95 = 5 x 19.
> Finally we have certain facts that are interesting and relate to the number “9”
For the precise value of π = 3.14159 .. , starting from the 762th digit, there appears six consecutive “9”s.
1/7 = 142857 / 999999
9 = 3(21)
Random picked facts as they are, it seems number 9 appear universally at many places we’re not expecting!
We can also say that 8 is a perfect power (as 8 is the cubic power of 2; any power, whether square, cubic, or fourth degree, etc, can be called a power).
>
Look at this identity equation:
8 = 23 = 32 – 1
8 is the ONLY power that is one less than another perfect power (other than the trivial case of 0 and 1). This is (proved) Mihailescu’s Theorem.
> An octagon has 8 sides; an octagon can be obtained through “splitting” each side of a square shape into two shorted sides.
An octahedron (3D shape) has 8 faces. You may call it a quadruple tetrahedron as you can split it into four tetrahedrons.
And an octopus has 8 legs.
> The octal numerical system is often used in computer science:
Information is stored in a computer using binary. Each binary is called a bit, and three bits form a 3-bits which is octal. For an example, the decimal number 20 is represented as 010100 in binary, and as 24 in octal.
>
A cube has 8 vertices, 6 faces, and 12 edges.
An octahedron has 6 vertices, 8 faces and 12 edges.
(Notice that the number of vertices and the number of faces swap in a cube and in an octahedron,
and the number of edges is the same for a cube and for an octahedron.)
An octahedron is therefore the dual polyhedron for the cube (by swapping vertices and faces). Dual pair of polyhedrons is an interesting mathematical concept.
> When a number has exactly 3 different prime factors, it is called a Sphenic number. Examples of Sphenic numbers are 30 = 2 × 3 × 5, 105 = 3 × 5 × 7, 231 = 3 × 7 × 11 etc. It is a known fact that
Every sphenic number has 8 exact divisors (include 1 and itself).
And naturally, if we exclude number 1 and the Sphenic number itself, then:
Every sphenic number has 6 exact proper divisors (not counting 1 and itself).
Number 8 is not a Sphenic number as it contains three identical prime numbers (8 = 23 = 2 × 2 × 2).
> How number 8 come to its current shape “8” in arabic number system is an interesting story. The following glyphs shows the evolution of 8.
One of the many possible explanations for the shape of 8 is as follows:
The glyph at the very beginning is showing two overlaying hands (represented by the middle bar), deducted the two thumbs (represented by the two vertical bars), so there are exactly 8 fingers. Later in the glyph, when the two vertical bars become curved (as in ), it looks more like the English letter S (which starts the word “symmetry”). At the end (as written today), the glyph becomes closed completely (and symmetric, with several axis of symmetry).
— When you think of the symmetry, think about the figure “8” shape!
The “figure 8” shape often used in context of sports such as figure skating. We also see figure-eight turns of a rope or cable appearing in some drawings or digital icons.
> Since one week has 7 days, the eighth day just repeat the 1st day of last week. Musically, the Diatonic Scale has 7 notes, which are: 1 (Do) 2 (Re) 3 (Mi) 4 (Fa) 5 (So) 6 (La) 7 (Ti), but the 8th note is Do again. With these facts as the inspiration, “8” often signifies a new beginning, and the hope for the new era.
Super 8, when you read the two words together （liason) without pause, it sounds like “super rate” therefore become business brands, meaning to provide super service with an affordable price.
> A rainbow has 7 colours.
To be precise, rainbows span a continuous spectrum of colours. Any distinct bands perceived are an artefact of human colour vision. The most commonly cited and remembered sequence of the rainbow is Newton’s sevenfold red, orange, yellow, green, blue, indigo and violet,
A week has 7 days. Our world has 7 wonders.
> We start from the reciprocal of 7.
1 / 7 = 142857 / 999999 = 0.142857142857141857 .. .. .. (repeating forever)
Now see the amazing pattern at the side: it always follows the same pattern 1 -> 4 -> 2 -> 8 -> 5 -> 7; (when you’ve come to the “7”, come to “1”), the only difference is starting with different first digit. It’s a full-circle pattern, as shown. It’s also interesting to note that the digits 3, 6, 9 are skipped in each of the strings. One explanation is for the first digits 14, 2 × 14 = 28 and 3 × 14 = 42 so there is no thirty something for the first two digits of any string. Of course 3 × 13 = 39 but no string starting with 13 will have nice pattern like that 142857.
> Suppose you are rolling standard 6-sided dice with numbers 1- 6 on respective faces. If you roll two dices independently, and then add the number of the top face of each dice to get the sum. What would be most likely for this sum to be? You guessed it – “7” (as appeared in combination 1 & 6, 2 & 5, 3 & 4, 4 & 3, 5 & 2, 6 & 1) – a total of (1/6) likelihood.
How many different numbers can we get? From 2 to 12: there are 11 numbers. If you do the experiment 110 times, in about 18 times you’ll get the sum “7”, but on average since there are 11 numbers, each number has about 10 chances. Now you see the significant difference.
In actual experiment, the results may vary as we always in experiment with random events.
>
(the Kissing Solution for a Unit Circle).
The problem is to use several circles of diameter 1 (the smaller circles) to completely cover a perfect circle of diameter 2 (the large circle, which has twice as long radius, and thus is four times in area).
Now at least how many smaller circles do we need to completely cover the large circle? And how?
The answer is that you need at least 7 circles.
Can you figure out how? And Can you show less (say 6) of smaller circles are not enough to cover the large circle?
> 7 is a number in the Mersenne sequence, which goes as:
Of the Mersenne sequence, number 7 is the third number . Both 3 (from the 3rd) and 7 are primes.
It has been proven that if n is NOT a prime number, then 2n – 1 is not a prime either. How about the other side? If n is a prime number, does that guarantee that 2n – 1 is also a prime. This is a famous conjecture about the Mersenne primes; but today we know that it is not so. We refrain from discussing here, but that’s an interesting topic.
>
If we take the quadruple of 7, we get 28. Since
28 = 1 + 2 + 4 + 7 + 14
(Notice that the right hand has included, as the addends, all the proper divisors of 28)
So we can conclude that 28 is a complete number. It is 2nd complete number, coming after 6.
> Number 7 is also be considered as “mystic” or “magic” in both oriental and western cultures. In English, “7” sounds like “heaven”. Number “7” has been used many times in the bible, for example, the great flood lasted 7 weeks (7 x 7 days), and Egypt had 7 abundant years followed by 7 disastrous when all harvest and so bad, etc.
We have mentioned 7 wonders of the world . And have you heard the English phrase “at sixes and sevens”? It is a British phrase used to describe a state of confusion or disarray – – so not good — But whether good or bad, “7” is special.
The 3D space we live has 6 directions: up, down, left, right, front and back.
> 6 is an even number. And it is a complete number.
Note: for any number, we can add all of its proper divisors (except itself), and find the sum. If the sum of all its proper divisors equal to the number, then we call this number as a complete number.
For 6: it has four divisors: 1, 2, 3, 6. We strike out 6 (as it is not proper). Then we find that:
1 + 2 + 3 = 6,
which leads us to the conclusion that 6 is a complete number.
The next complete number is 28.
6 also relates to the integer solution of the following equation:
x + y + z = (x) (y) (z)
Solution to this question, with no particular order, is 1, 2, and 3. Which makes both hands equalling 6.
> For any number ending in “6”, when you square it, the last digit of the square is also 6.
> A hexagon has 6 vertices and 6 sides. If we connect the alternate vertices, we get an equilateral. This equilateral is similar to the equilateral when we divide hexagons up into 6 smaller equilaterals.
A tetrahedron (4 faces) has 6 edges.
A rectangle has four sides; but adding the two diagonals, there are 6 linking lines connecting each pair of vertices of the rectangle.
> 6 is also a triangle number (defined as follows):
1 = 1
3 = 1 + 2
6 = 1 + 2 + 3
.. .. .. ..
15 = 1 + 2 + 3 + 4 + 5
.. .. .. ..
36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8
All numbers appearing at the left-hand side are triangle numbers.
It is interesting to note that 6 and 6 x 6 = 36 both appear (respectively as 3rd and 8th) in this sequence.
> A regular hexagon can be inscribed to a perfect circle, with the side being equal to the radius. With radius being 1, the circumference of the circle is greater than 6: the perimeter of the hexagon. We know that π (for circles) is defined as the ratio of circumference to the diameter. This definition, combined with the fact that the circumference of the circle is greater than 6, has established that π > 3: one of the earliest fact about circles for humankind exploring the shape of perfect circles.
> If you know many 6-letter English words, you may score a high point in scrabble (a game for guessing English words). Pause to think of 6-letter words, your mind may generate some 5-letter or 7-letter words first (or some times 3-letter, 4-letter words) before writing any 6-letter.
There are in fact lots of 6-letter words, such as:
>> The English word “number” has 6 letters. “Figure” also has 6 letters.
There are some 6-letter English words that you use a lot in the exam: e.g. “answer” and “choice”. When you deal with 3D solids, you may be asked to calculate the “volume” – there, another one.
(But English words “shape”, “angle” “graph” “ratio”, “speed”, “digit” each has 5 letters only. You may write “shapes”, “angles”, “graphs”, etc. but this is somewhat cheating!
“Fraction” and “decimal” has respectively 8 and 7 letters.)
A list of compiled 6-letter words (one word per line) may fill over 100 pages, as using double space format. That’s a lot!
The last of these continued to evolve into today’s number “9”.
“Eight”
at the very beginning is showing two overlaying hands (represented by the middle bar), deducted the two thumbs (represented by the two vertical bars), so there are exactly 8 fingers. Later in the glyph, when the two vertical bars become curved (as in 
), it looks more like the English letter S (which starts the word “symmetry”). At the end (as written today), the glyph becomes closed completely (and symmetric, with several axis of symmetry).
It’s also interesting to note that the digits 3, 6, 9 are skipped in each of the strings. One explanation is for the first digits 14, 2 × 14 = 28 and 3 × 14 = 42 so there is no thirty something for the first two digits of any string. Of course 3 × 13 = 39 but no string starting with 13 will have nice pattern like that 142857.