## abdul_shabeer 3 years ago 100000000000................ continues infinitely. Is it a number?

1. KaylaBrewington

yes because the number line never stops

2. sauravshakya

|dw:1349359330358:dw| Is infinity a number?

3. experimentX

not a natural number and not real number either.

4. chandhuru

infinity is something which cant be defined...

5. experimentX

no .. irrational number is a real number.

6. abdul_shabeer

Is it a number?

7. sauravshakya

IS infinity a number?

8. experimentX

it's a number is http://en.wikipedia.org/wiki/Hyperreal_number and http://en.wikipedia.org/wiki/Surreal_number ... I don't know much about this ... but this is not a real number and natural number.

9. experimentX

*in

10. sauravshakya

If 100000000000000... is not a natural number then the set of natural number is finite. right?

11. chandhuru

@experimentX how do u say that it is not a real number??

12. experimentX

no ... the set of natural number is infinite but does not contain infinity.

13. TuringTest

Infinity is a concept, not a number (I know I'm basically just restating, but it is worth emphasizing)

14. experimentX

you always write |dw:1349359774760:dw|

15. experimentX

you should have noted.

16. abdul_shabeer

then why 1/3 is considered a number?

17. TuringTest

0.3333.... as the 3's continue the number is not getting any bigger, as opposed to making an infinitely number before the decimal which would create an infinitely large number.

18. TuringTest

hm... I phrased that poorly :/

19. experimentX

natural number is defined as inductive set that begins with 1 and |dw:1349359861034:dw|

20. TuringTest

indeed what would 10000.....+1 be? could you write it? where would you put the 1 ?

21. abdul_shabeer

Why 0.333333333333333333....... is a number?

22. TuringTest

that is different from 0.3333....+1=1.3333... not problem there

23. experimentX

1/3 is repetitive ... if you take 1m string ... fold it thrice, you can always pinpoint this is 1/3 . Therefore it lies inside real number line.

24. TuringTest

you can add any number to 0.333.... and create a new number on the real line where would 1000...+1 be on the real line if you can point it out then I will let you call it a number ;)

25. TuringTest

I like this video to help with understanding repeating decimals, though it does not directly answer your question http://www.youtube.com/watch?v=TINfzxSnnIE

26. chandhuru

@abdul_shabeer u can find the number 0.333333333 or atleast u can locate the point 0.33333 number between 3 and 4 , but u cant locate the infinity .......

27. abdul_shabeer

We can't locate 0.333333.......

28. TuringTest

yes we can

29. chandhuru

atleast u know that 0.33333 is present between 0.3 and 0.4 .... in case of infinity u cant ...... can u ????

30. TuringTest

|dw:1349360426152:dw|

31. TuringTest

now you show me what two numbers 1000..... lies between :)

32. abdul_shabeer

What is (0.333333......)* 10?

33. TuringTest

3.333333...

34. abdul_shabeer

0.333..... * 10 = 0.333...+0.3333....+0.3333....( 10 times) Can we add this?

35. TuringTest

I think it is a lot easier to see it as 1/3+1/3+1/3+...(ten times) =10/3

36. abdul_shabeer

I want to add it without converting it into 1/3

37. TuringTest

why?

38. abdul_shabeer

As 0.3333.... = 1/3, in whichever way i add I must get the same result

39. Razzputin

100000000000......... lies between 99999999999...... and 11000000000000...... where all three of the numbers stretch out infinitly.

40. TuringTest

0.333...+0.333...=0.666... 0.666...+0.333...=0.999... 0.999...+0.333...=1.222... do that ten times and you will get 3.333...

41. TuringTest

and @Razzputin that is wrong, how can you show me that 9999... is not bigger than 1000... ?

42. abdul_shabeer

How 0.999...+0.333...=1.222...?

43. TuringTest

are there more digits in one number or the other? no

44. Razzputin

what if you where to state that 999.... was 1 less than infinite?

45. TuringTest

ad a 3 to each 9

46. TuringTest

47. TuringTest

48. TuringTest

the 1's carry over

49. abdul_shabeer

|dw:1349364346874:dw|

50. abdul_shabeer

and the last digit would be 2

51. TuringTest

yeah but there are an infinite number of 3's and 9's so every digit carries a one from the previous decimal place

52. TuringTest

there is no last digit, that's why it's an infinitely repeating decimal

53. abdul_shabeer

There is no last digit, I just took a small value

54. TuringTest

which makes it not the same as 1/3+1 which is what you are trying to do

55. TuringTest

$0.333\neq0.333...=1/3$

56. TuringTest

this is why we don't add numbers with infinite decimals very often; It's ugly, confusing, and entirely unnecessary

57. abdul_shabeer

what do you get when you add 0.9999...+0.3333... without taking 0.99999...=1 and 0.333... = 1/3

58. TuringTest

math is about making things as simple as possible, not over-complicating things and I told you 0.999....+0.333...=1.333... ^ why is this not a 2? because there is another 3 after it, though we did not write it and another after that, and another after that, ad infinitum

59. TuringTest

at no point does the decimal terminate, hence EVERY digit carries a 1 from the digit after it

60. abdul_shabeer

But still you just can't neglect a 2

61. TuringTest

yes you can because there is absolutely no 2 how many decimal points down would you expect to find it?

62. abdul_shabeer

Which means it is not equal to 1+ 0.3333....

63. TuringTest

it would have to be in the same decimal place as the remaining 1 you might expect by subtracting 1-0.999... you might say, "well where is the remaining 1?" well, it is literally *infinitely* far down the decimal, which means it ain't there

64. TuringTest

how do you figure that 0.999...+0.333... is not the same as 1+0.333... ?

65. abdul_shabeer

66. TuringTest

so?

67. TuringTest

you want to start at the far right side I presume? but what is the farthest right digit, and what decimal place is it in?

68. TuringTest

you are trying to use math skill you learned in grade school to deal with simple, finitely long numbers on an infinitely long decimal representation

69. abdul_shabeer

How do you add two numbers?

70. TuringTest

either by grasping the concept that there is no last digit and recognizing that the 1 will ALWAYS carry over from the previous digit, or by representing it as a fraction which is perfectly valid an makes life way easier

71. TuringTest

I suggest you stop trying to hurt your brain dealing with infinitely long decimals and and work on deepening your understanding as to why these infinite decimals are exactly equivalent to fractions

72. abdul_shabeer

0.9999.... = 1, I want to disprove this

73. TuringTest

you can't disprove something that is true

74. abdul_shabeer

Can you prove it?

75. TuringTest

again, what is 1-0.999... ?

76. TuringTest

prove it? yes would you like me to?

77. abdul_shabeer

yes

78. TuringTest

$x=0.\overline9$$10x=9.\overline9$$10x-x=9x=9$$x=1$

79. abdul_shabeer

0.999...*10 = 0.999...+0.999...+0.999...(10 times) How do you add this?

80. TuringTest

again, the way I showed you adding 0.333... every time we get the following succession 0.333... 0.666... 0.999... 1.333... 1.666... 1.999... 2.333... 2.666... 2.999... 3.333...

81. abdul_shabeer

But what about the last digit. Though we don't get till last digit we observe that the last digit gets a different value

82. TuringTest

oh I did it with 0.333... but the idea is the same

83. TuringTest

where is this mythical last digit? what decimal place is it in? the tens, the thousands, the ten-millions?

84. TuringTest

what is the last digit of pi?

85. abdul_shabeer

It is an irrational number

86. TuringTest

but the point is the same, where is the last digit of 0.999... ? what decimal place is it in?

87. abdul_shabeer

88. TuringTest

normally, from right to left but this is not a comparable problem because each number has a last digit, which is where you start adding from can you do that if there is *no last digit*???

89. TuringTest

how can you start at the far right when there is no far right? you can't

90. abdul_shabeer

But you say that 0.3333..... is a rational number

91. TuringTest

yes and I can prove it what is your point?

92. abdul_shabeer

When you can't start at the far right, how do you take 0.999...*10 = 9.999...

93. TuringTest

by having a deeper understanding of what it means to add two infinitely long decimal representations, or by understanding that this is the same as 1*10 your grade school adding methods are powerless here, you must accept that

94. abdul_shabeer

Why it is same as 1*10?

95. TuringTest

because 0.999...=1

96. abdul_shabeer

What is the standard way of adding two numbers?

97. abdul_shabeer

98. TuringTest

brb @experimentX feel free to take over if you like I will be back in a sec

99. experimentX

i lost track of it ... looks like conversation reached almost infinity ... where do you have problem?

100. abdul_shabeer

What is the standard way of adding two numbers?

101. experimentX

102. abdul_shabeer

103. experimentX

draw a real line .. |dw:1349367528137:dw|

104. experimentX

|dw:1349367566686:dw|

105. abdul_shabeer

How would you add 0.999.... and 0.333...?

106. TuringTest

I would agree that is a good standard way to think of addition^ but you want to do 0.333...+0.999... in which case you have to observe a pattern, which you seem to be reluctant to accept

107. experimentX

you can always locate these points on the real line. and if you continue 0.999.... .... to up infinity this is 1 and for 0.33333.... you might think that you can never locate this point on real line. you can actually locate it. and there is ONE and only ONE point on the line i drew. add these distances.

108. abdul_shabeer

This problem started in the proof of 0.999... = 1

109. experimentX

yeah ... it seems that this is not equal to 1 ... but if you continue this up to infinity ... still this is non intuitive. here a short reason to believe. 0.9 ~ 1.1 <--- let's find a pair of points 0.99 ~ 1.01 0.999 ~ 1.001 <--- 0.9999 ~ 1.0001 <--- in similar fashion you put infinite zeros between 1 and the last 1 what would you get 1.00000000000.. infinite zeros.....................1

110. TuringTest

to do 0.999...*10 you can either 1) accept that multiplication by 10 moves the decimal place over by one space (it is perfectly okay to utilize that, if we did not utilize powerful concepts as givens then math would be a monstrosity to do!) or 2) add each corresponding digit ten times and recognize that there is not last digit, hence every digit will carry a one from the digit to its right, of which there are infinitely many I strongly suggest option 1

111. abdul_shabeer

Multiplication by 10 is nothing but adding it 10 times

112. experimentX

if you say this is 1 1.00000000000.. infinite zeros.....................1 then this must be equal to 1 too 0.99999999999999.... infinite nines

113. TuringTest

or as I put it, 1-0.999...=? if we say 1-0.999...=0.000...1 in what decimal place would the 1 be? infinitely far out, i.e. there isn't one

114. experimentX

I think you are misunderstanding infinity with undefined, infinity is greater than you imagine.

115. experimentX

1.00000000000.. infinite zeros.....................1 this means 1.00000000000.. here are more zeros than you imagine ....................1

116. TuringTest

@abdul_shabeer if you want to utilize option 2 that is fine, but be careful about how the digits carry over from infinitely far to the right

117. abdul_shabeer

You mean something which is at infinity is not there.

118. TuringTest

don't play fast and loose with ideas like "infinity means it's not there" this is a subtle and tricky issue, and cannot be captured in a phrase like that

119. experimentX

yep. kinda something like that http://math.stackexchange.com/questions/11/does-99999-1 also that video by ViHart is very nice ... but few the argument she used had been downvoted quite badly on MSE. Let the experts know.

120. abdul_shabeer

Do you remember the question on whether a point is dimensionless? If we compare the size of earth with universe, how big do you think the earth would be?

121. TuringTest

the same size it would be if the universe was only as big as the solar system comparison does not make a thing bigger or smaller

122. abdul_shabeer

if you are out of Milky way and you look at earth, would you be able to find it?

123. estudier

If it isn't 1 * 10^n, what is it?

124. TuringTest

with the right equipment, theoretically yes

125. abdul_shabeer

With no equipments

126. TuringTest

with the human eye (which is a piece of equipment itself in many respects)? no of course not, why does that matter?

127. abdul_shabeer

If we see the space out of earth, would it appear like a 2D image?

128. TuringTest

the ancients thought it was, so I guess you could argue yes, but that is just a matter of the limitations of human perception

129. TuringTest

there is no obvious perspective point in the night sky, so our brains don't do so well gauging distance why are we talking about this all of a sudden?

130. abdul_shabeer

Okay Thank You Max and ExperimentX

131. TuringTest

very welcome!

132. experimentX

no probs at all ...