Number 8 in Math.

A page of the Numeropedia - the Special Encyclopedia of Numbers

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 1×8+1 = 9

12×8+2 = 98

123×8+3 = 987

1234×8+4 = 9876

12,345×8+5 = 98,765

123,456×8+6 = 987,654

1,234,567×8+7 = 9,876,543

12,345,678×8+8 = 98,765,432

123,456,789×8+9 = 987,654,321

1,234,567,890×8+90 = 9,876,543,210

      (start of a new cycle)

12,345,678,901×8+901 = 98,765,432,109

123,456,789,012×8+9,002 = 987,654,321,098

1,234,567,890,123×8+90,003 = 9,876,543,210,987

12,345,678,901,234×8+900,004 = 98,765,432,109,876

123,456,789,012,345×8+9,000,005 = 987,654,321,098,765

1,234,567,890,123,456×8+90,000,006 = 9,876,543,210,987,654

12,345,678,901,234,567×8+900,000,007 = 98,765,432,109,876,543

123,456,789,012,345,678×8+9,000,000,008 =

= 987,654,321,098,765,432

1,234,567,890,123,456,789×8+90000000009 =

= 9,876,543,210,987,654,321

12,345,678,901,234,567,890×8+ 900000000090 =

= 98,765,432,109,876,543,210

... (end of cycle) ...

 

8×8+13 = 77      

88×8+13 = 717

888×8+13 = 7117

8888×8+13 = 71117

88,888×8+13 = 711,117

888,888×8+13 = 7,111,117

8,888,888×8+13 = 71,111,117

88,888,888×8+13 = 711,111,117

888,888,888×8+13 = 7,111,111,117

8,888,888,888×8+13 = 71,111,111,117

...

 

The smallest cube number: 23. The only 1-digit cube number.

 

Figure Eight curve; its Cartesian equation is: x4 = a2 (x2y2). Its polar equation is: r2 = a2 cos(2q)sec4(q).

 

The last digit of an even perfect number is either 8 or 6.

Perfect number is a number equal to the sum of all of its proper divisors, including 1 (or also called aliquot divisors). Every even number of the form 2n–1(2n–1) is a perfect number if and only if (2n–1) is a prime number. It must be a triangular number: sum of all integers from 1 to (2n–1) and its last digit is either 6 or 8. The first perfect numbers are: 6, 28, 496, 8128, 212(213–1) = 33550336, 216(217–1) = 8,589,869,056 and 218(219–1) = 137,438,691,328…

 

The number of vertices of a cube (having 8 vertices/12 edges/6 faces).

 

A number is divisible by 8 if and only if its last 3 digits also form a number divisible by 8.

 

The 6th Fibonacci number. The largest cube number in the Fibonacci sequence.

 

The 8-point circle goes through 4 midpoints of the sides of a quadrilateral with perpendicular diagonals and 4 feet-points of the perpendiculars dropped from these 4 midpoints to the opposite sides.

 

The number of vertices of a regular 8-pointed star polygon or also called the star Lakshmi, used by the Hindus to symbolize the 8 forms of wealth (Ashtalakshmi). It can be formed by two squares placing diagonally.

  

8 = 13+22+31.

 

83 = 512 and 5+1+2 = 8.

 

Two cube numbers using the same digits: 53 = 125 and 83 = 512.

Also: 353 = 42875 and 383 = 54872.

 

1033 = 81+80+83+83.

 

48 = 82–42 = (4+8)×4, where 8 = 4×2.

 

3784 = 8×473, using the same digits in both sides.

 

All the cubic powers 23 = 8, 223 = 10,648, 2223 = 10,941,048, 22223 = 10,970,645,048, 22,2223 = 10,973,607,685,048 and 222,2223 = 10,973,903,978,085,048 do not use digits 2.

 

98,765,432 = 12,345,679×8.

 

8,101,265,822,784 = 8×1,012,658,227,848.

 

Each ratio 76328/9541 = 76248/9531 = 76184/9523 = 75368/9421 = 75328/9416 = 74816/9352 = 74568/9321 = 74528/9316 = 73456/9182 = 73248/9158 = 73248/9156 = 71632/8954 = 71624/8953 = 71536/8942 = 71456/8932 = 67512/8439 = 67352/8419 = 67152/8394 = 65432/8179 = 65392/8174 = 63528/7941 = 63152/7894 = 59368/7421 = 59328/7416 = 58912/7364 = 58496/7312 = 56984/7123 = 54712/6839 = 54328/6791 = 54312/6789 = 53928/6741 = 51832/6479 = 47368/5921 = 47328/5916 = 47136/5892 = 46712/5839 = 46328/5791 = 46312/5789 = 42968/5371 = 41896/5237 = 38152/4769 = 37528/4691 = 37512/4689 = 36728/4591 = 36712/4589 = 25496/3187 =  8  use each of 9 digits 1-9 once and show how to arrange a 9-book set on 2 shelves to mark the book #8.

 

The set of 4 numbers (1, 3, 8, 120) has a property that the product of any 2 numbers is always equal a square number minus 1.

 

A pan-digital factorization: 291,548,736 = 8×92×531×746.

 

Belgian mathematician Eugénie Charles Catalan in 1844 that 23 and 32 are the only consecutive powers (except 0 and 1), i.e., 32–23 = 1 is the only nontrivial solution to Catalan’s Diophantine problem xayb= ±1. The Catalan’s conjecture was finally proved by Preda Mihailescu in a manuscript privately circulated on April 18, 2002.

 

The first (even) number in the list of numbers sorted by the English alphabetical order is eight. The first odd number in this list is: eight billion eight hundred eight million eight hundred eight thousand eight hundred eighty five.(Mistakenly claimed earlier as 8,018,018,885, which is spelled as: eight billion eighteen million eighteen thousand eight hundred eighty five).

 

Every integer can be a sum of at most 8 cube numbers, except number 23 and 239, which requires 9 cube numbers

 

[Language & Math.]  The first numbers in this list are: eight (8), eight billion (8,000,000,000), eight billion eight (8,000,000,008), eight billion eight hundred (8,000,000,800), eight billion eight hundred eight (8,000,000,808)…  (See also Numbers 8,800,000,000s and 8,000,000,000s and and References [3])

 

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References [3]: Conway & Guy, the Book of Numbers, p.15, 1996.