> 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!
> 10 is the base of our decimal numeric system — it means 10 ones is ten, 10 tens is hundred, 10 hundreds is thousand, etc. We use decimal system on a daily basis.
> Digit “0” was invented in Oriental world but it took a long time to be accepted in the Western hemisphere. For number 10, people initially use a (blank) bar or a circle symbol as there’s nothing in units-place; then it has become a standard to use ‘0’ once ‘0’ emerged. Roman numerals, which did not embrace the idea of “place value”, were used until recent (where we use X, C etc for 10, 100, .. — there never has been there a zero in Roman numeral).
> 10 is a triangle number as well as a centric triangular number.
10 = 1 + 2 + 3 + 4 = 1 + 3 + 6
As shown, 10 is the sum of the first four counting numbers, as well as the sum of the first three triangle numbers [1,3,6 are triangle numbers].
> If a number has exactly two prime divisors, then it is called a semi-prime. Number 10 is a semi-prime as
10 = 2 × 5
The first semi-prime is 6. Any semi-prime will have 4 divisors (including number 1 and itself).
> 10 can be represented as the sum-of-squares as below:
10 = 12 + 32 = (12 + 12) (12 + 22)
> There is something interesting in the entry just above. If two numbers, n and m can be both written as the sum of two squares, then their product can also be written as the sum of two squares. 10 is the first of such examples, apart from the trivial 4 = (12 + 12) (12 + 12) = 22 + 0.
The general formula is illustrated below:
if n = (a2 + b2), m = c2 + d2, then
nm = (ac-bd)2 + (ad+bc)2
Note that the representation of (nm )as the sum of two squares is usually NOT unique.
> Ancient Greeks has suggested to use the square root of 10 as an approximate value of π (where of a circle, π is the ratio of circumference to radius).
> In the metric system, the conversion among different units of measurement are based on the power of 10; for example: 1 metre = 100 centimetre; 1 kilometre (km) = 1000 metre. In monetary system, 1$ = 100 c and 1 dime = 10c (though “cent” has been out of use).
In early history, as a fast calculation mechanism, people resort to logarithms; of which the most commonly used was 10 (therefore base-10 logarithm obtained its name as the common logarithm).
> Considered that a number is raised to the 10th power.
The last digit of a 10th power of an integer is the same as the last digit of the square.
For example: 210 = 1024 and 22 = 4;
For another one, 710 = 282475249, which has the last digit 9, whereas it’s known that 72 = 49 too.
In this post we study “5” again – but with reference specifically to Graphs and Shapes, to which number 5 illustrate many interesting properties.
> Do you know complete graph? Mark any n points, then connect each pair of points, you will have a complete graph. Here is a graph with 4 points.
It’s not necessary to connect all points with straight lines: you can connect them with an arc or a curve – like in the figure shown in this paragraph. What do you note: no connecting line intersect with any other lines. Such a graph is called a planar graph.
What about a complete graph with 5 points? We draw it as a pentagon plus a 5-star (also called pentagram). No matter how you try it, some two lines of all connecting lines must intersect somewhere. There is no way to avoid or escape from that.
Put it simple:
A complete graph with five points is not planar.
(if you want avoid crossing each other in connecting lines), you have to raise one point so that it does not share the same plane with the other four points.
> An equilateral triangle is called a golden triangle if
(I) the ratio of its base to (any of) its sides is 1.618 .. — called golden triangle of type I; [that is: BC : AB = 1.618.. ]
(II) the ratio of (any of) its sides to base is 1.618 .. — called golden triangle of type II.
The number 1.618 .. is called a golden ratio.
> What’s so special about the shapes with golden ratio? It’s visually pleasing to the eye – shown by some experiments. Not only in triangles. In a rectangle, if the ratio of longer side to shorter side is 1.618 .. It also looks pleasing to a human eye.
> In the title we mentioned pentagon-pentagram. A regular pentagon is a polygon with 5 sides, with all the sides being equal and all the interior angles of this pentagon being equal. A pentagram is constructed by connecting straight lines between the vertices — usually being referred to as the (regular) five-star.
In the diagram of regular pentagon-pentagram, we can observe many of these golden ratios.
There are many many golden triangles.
Can you find some of them?
This is the first puzzle we have posted. can you find?
Let us end this post with a verse in Christmas’s popular song,
“In the fifth day of Christmas,
my true love gives me
FIVE GOLDEN RINGS .. .. ..
How about the five golden rings all of which contains the golden ratio?
> 5 is a factor of 10, which links to our decimal system (carry one to the next higher place whenever 10 is reached).
All multiples of 5 ends in digit 0 or 5.
> The square of any integer ending in 5 will also have 5 as the last digit (units place).
–> We can even go one step further: for the square of any integer ending in “5”, the last two digits of the results must be 25. For example: 25 × 25 = 625, and 45 × 45 = 2025.
> When a fraction m/n (both m,n are integers) is converted to a decimal, then:
If n = 5 => the decimal has only one digit after the decimal point
If m/n, after converting to decimals, has only one digit after the decimal point, then n = 2 or n = 5
> 5 is a prime number: 5 = 1 × 5
5 = 221 +1 (this type of numbers are called Fermat numbers)
The next Format number is 17 = 222 +1.
> [this item is a bit hard] The numbers in Fermat’s number sequence increases very rapidly: as
221 +1, 222 +1, 223 +1, 224 +1, … …
We calculate these numbers as:
5, 17, 257, 65537, ..
So the number increase very rapidly. Interesting enough, the first four prime numbers are all prime numbers, yet the fifth one (225 +1) is a composite number, with the smaller divisor being 641.
And once started, it keeps going this way: it has been known that from the fifth Fermat number to the thirty-second Fermat number (which is 2232 +1), all of them are composite numbers. It remains open whether there is any prime number in this sequence down the way.
There are lots of facts regarding prime 5: let’s be simple not to mention everything.
> Look at the identity: 4 = 2 × 2.
Conclusion: 4 is the smallest composite number (a composite number can be written as the product of smaller positive integers). Number 4 is a perfect square; it is also a power of 2.
> Quadrilaterals (4-sided polygons) include many common shapes, like rectangles, parallelograms, etc. Can you give more examples?
> In a plane there are 4 directions: up, down, left and right.
> A tetrahedron (4-faced polyhedron) has 4 vertices, 6 edges and 4 faces.
A tetrahedron – also called a triangular pyramid, is the simplest polyhedron (in terms of number of faces is smallest).
There is a favourite topic in science fiction: 4-dimensional space.
> Number 4 inspired the imagination of mathematicians and their enthusiastic exploration: and yield fruitful results: the most famous are “four-colours theorem” and the “sum-of-four-squares” [this item is a bit hard].
About Four-colours Theorem: On a plane or a sphere, it suffice to employ only four colours to seperate all neighbouring regions apart.
About Sum-of-four-squares theorem: Any integer can be written as the sum of at most four squares.
For example: 5 = 22 + 12, 25 = 42 + 32 [As Sum of two perfect squares]
7 = 22 + 12 + 12 + 12 [As Sum of four perfect squares]
> If we list all powers of 4: 4, 16, 64, 256, 1024, 4096, etc.
Have you noticed: the last digit (units-place) is either 4 or 6.
By the way, the complimentary number of 4 is 6, and 4 × 6 = 24.
Conclusion: ________________ (The reader may summarize by himself).
> The sum of four consecutive integers is an even number, however, it is NEVER a multiple of 4. For example: 1 + 2 + 3 + 4 = 10.
Since ten resembles “perfect”, so as number 4 is loved in some cultures (for example, the Pythagorean scholars in ancient Greece).
Last Spring, Jonah’s Math Corner hosted a column called “Prime Fun Numbers”. Lots of feedback were received that the column were welcomed by math kids and their parents as well.
That’s why we kick off 2017 with this same column, and updated contents.
The first post will be on number 17 (seventeen). What have you thought of when hear 17? Stay tuned ..
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.
One is the beginning of everything.
Such a graph is called a planar graph.
