chapter 3: figurate numbers

CHAPTER 3

Figurate Numbers

Prepared by:Nathaniel T. SullanoBS Math - 3

A figurate number, also known as a figural number or polygonal number, is a number that can be represented by a regular geometrical arrangement of equally spaced points.

source: http://mathworld.wolfram.com/FigurateNumber.html

Example:

Triangular Square Pentagonal Hexagonal

Image source: http://mathworld.wolfram.com/images/eps-gif/PolygonalNumber_1000.gif

SQUARE NUMBERSrepresented by squares

1st 2nd 3rd 4th 5th

Image source: http://www.learner.org/courses/mathilluminated/images/units/1/1094.gif

We can also view the square numbers from a different aspect. Like the figure below

We can observe that a square is partitioned into a smaller square and a carpenter’s square or gnomon.

Image source: http://tonks.disted.camosun.bc.ca/courses/psyc290/figure1002.jpg

In observing the figure on the left, we see that the fourth square number is the sum of the first four odd numbers,

1 + 3 + 5 + 7 = 42

n2= (n – 1)2+(2n – 1)

Image source: http://tonks.disted.camosun.bc.ca/courses/psyc290/figure1002.jpg

Below shows the first few square numbers as sums of odd numbers:

1st 1 = 12

2nd 1 + 3 = 4 = 22

3rd 1 + 3 + 5 = 32

4th 1 + 3 + 5 + 7 = 42

5th 1 + 3 + 5 + 7 + 9 = 52

6th 1 + 3 + 5 + 7 + 9 + 11 = 62

Generalizing, we see that the nth square number is the sum of the first n odd numbers,

1 + 3 + 5 + 7 + … + (2n – 1)

The difference between any two consecutive addends is 2. Since the difference between any two consecutive numbers in the sum which names a square number is 2, we say the common difference of the square numbers is 2.

TRIANGULAR NUMBERSbeing pictured as triangles

Image source: http://mathforum.org/workshops/usi/pascal/images/triang.dots.gif

If we observe the pictures above of the representations of the triangular numbers, we see that the number of dots in the representation of the first triangular number is 1, of the second, 1 + 2, of the third, 1 + 2 + 3, and so on.

Image source: http://mathforum.org/workshops/usi/pascal/images/triang.dots.gif

1st 12nd 1 + 23rd 1 + 2 + 34th 1 + 2 + 3 + 45th 1 + 2 + 3 + 4 + 5

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.

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nth 1 + 2 + 3 + . . . + n where n is any counting number. Notice that the difference between any two addends in this sum is 1, so, 1 is the common difference of the triangular numbers.

nth triangular number is given by

the formula,

PENTAGONAL NUMBERSpictured in the form of a pentagon

represented by a square with a triangle on top

Image source: http://thm-a02.yimg.com/nimage/17d300a4d2a62fa0

In general, the nth pentagonal number is

OO O

O O OO O O OO O O OO O O OO O O O

Refer to the figure below. We see that the fourth pentagonal number is made up of the fourth square number and the third triangular number.

Solution:

1st 1 = 12nd 1 + 4 = 53rd 1 + 4 + 7 = 124th 1 + 4 + 7 + 10 = 225th 1 + 4 + 7 + 10 + 13 = 356th 1 + 4 + 7 + 10 + 13 + 16 = 51

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.

.nth 1 + 4 + 7 + … + (3n – 2) =

Common difference of the addends is 3.

Following the triangular, square, and pentagonal numbers are the figurate numbers with differences: 4, 5, 6, and so on. If the common difference is 4, we have the hexagonal numbers.

1st 1 = 12nd 1 + 5 = 63rd 1 + 5 + 9 = 154th 1 + 5 + 9 + 13 = 28

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.

.nth 1 + 5 + 9 + . . . + (4n – 3) = n(2n – 1)

PATTERNS FROM FIGURATE NUMBERS

Notice that the representation of the

fourth square number is partitioned so that it is

composed of the 3rd and 4th triangular number .

Thus, 6 + 10 = 16.

RELATION OF SQUARE AND TRIANGULAR NUMBERS

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Triangular number 1 3 6

10 ...

Triangular number 1 3 6 …

Square number 1 4 9

16 ...

Table 3.2

DISCOVERIES OF RELATIONS OF FIGURATE AND

ORDINARY NUMBERS

I. Every whole number is the sum of three or less triangular numbers. For Example:

17 = 1 + 6 + 10 26 = 1 + 10 + 1546 = 10 + 36 150 = 6 + 66 + 7864 = 28 + 21 + 15 25 = 10 + 15

II. Every whole number is the sum of four or less square numbers. For Example:

56 = 36 + 16 + 4 = 62 + 42 + 22

150 = 100 + 49 + 1 = 102 + 72 + 12

III. If we multiply each triangular number by 6, add a plus the cube of which triangular number it is, say we get the kth, we get (k + 1)3

Example: 3rd triangular number,

(6 x 6) + 1 + 33 = (3 + 1)3

In general,

IV. Eight (8) times any triangular number add 1 is a square. Example,

(8 x 1) + 1 = 9 = 32

(8 x 3) + 1 = 25 = 52

(8 x 6) + 1 = 49 = 72

In general,

BRIEF HISTORY OF POLYGONAL NUMBERS

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Pythagoras572 – 500 B.C.

The theory of polygonal numbers goes back to Pythagoras, best known for his Pythagorean Theorem.

End of Presentation

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