But look at the next to last equation at that page, where the series is equal to s.
That should dispel your objection in the OP, as it shows that the nomenclature is in common and mathematically consistent use.
I discovered this way back in college, (which is already decades ago). I was so pleased with myself, I would challenge college mathematics professors to write two real numbers that are exactly equal to each other without sharing any digits in common.
Many people thought it was a problem with decimal representation of real numbers. I went so far as to define this even further, to other bases. That a sequence of repeating digits of the last digit in the base is always equal to one in that base. That is, in hexidecimal, (base 16), 1.0 = 0.FFF~, in octal, (base 8), 1.0 = 0.777~, in base 3, 1.0 = 0.222~. Ah, but what about binary, (base 2), does 1.0 = 0.000~? no, it does not. the rule simply disappears. The explanation would require an essay and a mathematical proof not appropriate for this theorem.
Yes, the two numbers are identical in every way. It is just that our minds don’t really deal with the “~” part very well and mentally truncate that idea, which is good enough for estimates. But it is why most calculators are wrong to a certain degree. Back in High School, I discovered that I could actually get a repeating series of every number 1,2,3,4,5,6,7,8,9… but then you hit “10”, the one carries over on top of the 9, which carries a one over the eight, and you end up with a repeating sequence of “123456790”… The equation for that is 1/81, or 1/9 squared. That’s the first number I try out on calculators because you will see how much a calculator messes up repeating numbers like that.
More fun(?) with bases, I took the 1/81 rule and realized that it was really 1 divided by the last digit in the base, squared. So, in hexidecimal, 1/E1 or 1/F squared produces a similar sequence of ever ascending digits, with the carry messing up the two last ending digits, etc. I could go on…
Sure the equal sign “is in common and mathematically consistent use” in this case. But common and consistent only within the realm of infinite series. Every mathematical sub-field has their own definitions of equal and not equal, so we must always pay strict attention to how equal is defined in the particular realm we are exploring.
Of course, to such as I, such mathematical ramblings just show how silly the human mind can get. 1/3 is 3 three divided into one. 2/3 is 3 divided into 2. (Funny that there is no “division sign” on the keyboard!) Whatever the results are in these two equations, 3 divided by 3 is one, since any number “divided” by itself is 1. “3 thirds” is the the whole pie chart. Anything other than that is mind games, “intellectuals” trying to prove they are “smarter” than the rest of us, but mainly folks with nothing practical to do with their minds. Believe me, if you eat 3/3 of a pie and lick the plate, there is nothing left! I don’t care how much calculus you’ve had!
Sure the equal sign “is in common and mathematically consistent use” in this case. But common and consistent only within the realm of infinite series. Every mathematical sub-field has their own definitions of equal and not equal, so we must always pay strict attention to how equal is defined in the particular realm we are exploring.
Sure the equal sign “is in common and mathematically consistent use” in this case. But common and consistent only within the realm of infinite series. Every mathematical sub-field has their own definitions of equal and not equal, so we must always pay strict attention to how equal is defined in the particular realm we are exploring.
Sure the equal sign “is in common and mathematically consistent use” in this case. But common and consistent only within the realm of infinite series. Every mathematical sub-field has their own definitions of equal and not equal, so we must always pay strict attention to how equal is defined in the particular realm we are exploring.
Damn sorry about this multiple post, my DSL line went bonkers. Have a good weekend everyone!!!
Sure, and as near as I can tell, Cecil did a fine job of that. That’s why I disagree with your claim in the OP that it only makes sense if we use “approximately” instead of “equals.”
If I say 17 = 11 everyone would say … nope, that’s not true. But, if I say 17 = 11 in Hex, then everyone would certainly agree. Is that the same equal sign? Sure it is, we just have to consider the context.
But anyways … it’s been fun … take care folks.
I think I would agree if you said 17 in Decimal = 11 in Hex.
Do you then agree that Cecil’s column was correct?
No, the rule doesn’t disappear, you just didn’t state it right. In binary, 0.111~ = 1.0, in exactly the same manner that in decimal, 0.999~ = 1.0, or in hex, 0.FFF~ = 1.0 .
As for the cheerleaders, the version I heard added a physicist, who ignored the cheerleaders and immediately turned and kissed the secretary administering the test. When you can’t solve the problem at hand, you change it to a different problem which you can solve ;).
The rule doesn’t disappear in binary, you’re just misapplying the rule.
1 = 0.11111… in binary.
I agree with amore that this topic has been discussed too many times - enough is enough !!
byzcath to make the division sign hold the “ALT” Key down and at the same time type 246 and you get ÷
Want to know how to make other special characters? Go to:
http://www.1728.com/altchar.htm
If .999~ <> 1.0 then there exists at least one real number such that .999~ < that number < 1.0.
That number does not exist.
Hence, a contradiction.
Therefore, .999~ = 1.0
Is there any need to keep going on about this? I thought it was covered clearly and succinctly in Cecil’s column.
In case anyone missed it, here is the link again:
Thank you, Cecil, for publishing this question and your answer, correct and entertaining as always (though an explanation of how the hell John Stamos ever got Rebecca wouldn’t have killed you).
I do hereby officially announce that future persons who wish to whine, protest, aver, argue or otherwise opine that “.999~ = 1” is not true are a problem for you and Arnold Winkelried and not for me, Bib or Matrix.
This is a happy, happy day.
What I found most interesting in this whole discussion is how you represent periodic numbers… We represent 1/3 as 0.(3) over here (Romania), which seems to make more sense to me – how do you represent 3/7 for instance? Do you go like 0.428571428571428571~? I think 0.(428571) looks a lot clearer… Cecil hints at the horizontal bar used above the number, but we use that for the representation of arbitrary numbers in fixed format (for instance abc with a bar above would equal 100a+10b+c). Do you use it for this as well? Just curious, obviously…
I don’t know where this 0.999~ representation for 0.999… began, but it has proliferated throughout this discussion. So, I suppose to keep with this newly coined standard of representing repeated decimals I will represent 0.999… as 0.999~ in this thread.
As you can probably tell I was a math major. Big surprise there huh. I’m not sure about this “bar” representation you talk about, it may be used in some area of higher math here in the states. There are certain areas of mathematics that I haven’t had much training in, in particular number theory, perhaps they use the “bar” representation here, I don’t know.
I know I had a statistics professor that crossed his sevens. I think he said they do that in some places in Europe.
One more thing, I like the way Europe represents dates as DD/MM/YYYY. We have the standard MM/DD/YYYY. Your system seems to make more sense. And lets not even mention the metric system (a far superior way of representing weights and measures).
Heh, yes, we are indeed taught to cross our sevens when we learn how to write, but most people don’t nowadays – it was habitual in my granfather’s generation though, at least in Romania.