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I find this a bit confusing. If I have two numbers, lets say 1.7 or 1.73205; which one is bigger and why?

In my mind the 1.7 is larger since the next decimal place to the right is smaller. Should I think of it in a different fashion.

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Are you maybe from a culture that reads from right to left? –  Raskolnikov Apr 9 at 15:40
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I know this is maths.SE, but a valid answer in physical sciences would be that you don't know and/or that they're equal: 1.7 is only specified to 2SF and all you know it doesn't round to 1.8 or 1.6. So 1.73205=1.7 to 2SF. –  Chris H Apr 9 at 15:51
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@ChrisH Get out –  LTS Apr 9 at 16:51
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When you say "In my mind the 1.7 is larger since the next decimal place to the right is smaller.", I really don't understand what your thought process is here. Could you clarify? –  Bruno Apr 9 at 18:44
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This is a bizarre question from someone who has previously asked about finding the domain of an expression: math.stackexchange.com/questions/723720/… -- if you are struggling with the concept of quantifying numbers relative to one another, algebra and calculus are learning to fly before you learn to crawl. –  Chris Apr 9 at 19:52

10 Answers 10

$$1.73205 = 1.7 + 0.03205$$

Put differently, $$1.7 = 1.73205 - 0.03205$$

Now can you answer which of you given numbers is larger?

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1.73205; thank you so much. –  user137452 Apr 9 at 15:46
    
You are welcome! –  amWhy Apr 9 at 15:47
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Simple and beautiful! –  Sami Ben Romdhane Apr 9 at 18:31
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But how do I know which is bigger, 0 or 0.03205? In my mind the 0 is larger since the next decimal place to the right is smaller. Should I think of it in a different fashion. –  Rawling Apr 10 at 6:45
    
$0 = \dfrac{0}{100,000}.\quad$ But $0.03205 = \dfrac{3205}{100,000}>0$ –  amWhy Apr 10 at 11:41

Try thinking in terms of this example. Which amount of money is larger: $ \$1.70$ or $ \$1.73$ ?

The answer is $\$1.73$ becuase this amount has three more cents.

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Not that I'm complaining, but I always find it surprising that the answers which require the least amount of effort on my part get the most upvotes. –  Spencer Apr 9 at 17:20
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Real-world explanation. :) –  Anton Babenko Apr 9 at 17:40
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@Spencer According to some people, money isn't math. I don't understand them. –  Izkata Apr 9 at 20:47

Think of $1.7$ as $1.70000$. Then $1.70000$ is smaller than $1.73205$ because at the first decimal place where they differ, the second one, the digit $0$ is less than the digit $3$.

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Wow!Thank you so much. My god. Never thought about it like that. –  user137452 Apr 9 at 15:40

1.7 is short for 1.7000000000 and keep adding zeroes forever and ever. 1.73205 is short for 1.73205000000 and again keep adding zeroes until you feel like stopping.

If that doesn't make sense think of it like this:

1.7 = 1 + 0.7

1.73205 = 1 + 0.7 + 0.03 + 0.002 + 0.00005

The latter is bigger because it's like taking the former number and adding more numbers to it (however small), thus making the latter larger than the former.

I hope that helps.

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The simplest way to understand this is

a = 1.7

b = 1.73205

multiplying both sides by the same positive number to retain the ordering, say k = 100000, now

ka = 170000

kb = 173205

${\therefore\space kb > ka \implies b > a}$

${\blacksquare}$

Hence 1.73205 is larger than 1.7.

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4  
This answer isn’t really on the same level as the question being asked… –  bdesham Apr 9 at 16:59
    
@bdesham: You mean at a lesser level or...? –  legends2k Apr 9 at 18:21
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The OP was asking a basic arithmetic question—doing any kind of algebraic manipulation to explain it is probably overkill (and likely to go over OP’s head) :-) –  bdesham Apr 9 at 18:33
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+1 on my defence, I thought the OP was confused only about real numbers and is probably comfortable handling integers :) –  legends2k Apr 9 at 19:00

Expanding on @user134824's answer, every positive number can be thought of as consisting of the following parts:

  1. An infinite series of zeros
  2. The integer part
  3. A decimal separator
  4. The fractional part
  5. An infinite series of zeros

Aligning the numbers at the decimal point, you can then visually traverse the numbers left-to-right starting at the first non-zero digit to figure out which one is the larger:

$$\dots01.7\color{red}{3}2050\dots$$ $$\dots01.7\color{red}{0}0000\dots$$

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First you need to agree on a formal and clear definition of greater than relationship.

One definition (for non-negative numbers a, b, c) can be: $$ a > b \leftrightarrow \exists c\: (c \neq 0) \land (a = b + c) $$

Let $a=1.73205, b=1.7$. This leads to $c=0.3205 $, which is non-zero and non-negative. Now using the above definition, $a > b$.

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Maybe these are rounded numbers obtained from measurements of different accuracy. In this model the first hidden number $X$ is uniformly distributed in the interval $[1.65,1.75]$, and the second hidden number $Y$ is uniformly distributed in the interval $[1.732045,1.732055]$, independently of $X$. This means that we have a random point $(X,Y)$ in the rectangle $[1.65,1.75]\times[1.732045,1.732055]$. The probability that this point satisfies $Y>X$ comes out to $0.8205$.

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Rarely would 1.7 represent a uniform distribution over the interval 1.65 to 1.75; it would more typically represent a vaguely-bell-shaped distribution over a slightly larger interval (e.g. a physical attribute which is exactly 1.64 would be expected to read 1.6 most of the time, but sometimes read 1.7). –  supercat Apr 10 at 2:49

A written number like "1.7" can be used to represent one of two things:

  1. The exact fraction 17/10--equivalent to 1.700000000000... (with any number of zeroes)

  2. A quantity whose exact value is unknown, but is probably closer to 17/10 than to 16/10 or 18/10, and very likely closer to 17/10 than to 15/10 or 19/10. Note that when reporting physical measurements, trailing zeroes matter. If reporting a measured time as 1.3 seconds does not say with certainty that it couldn't have really been 1.21 or 1.39; on the other hand, reporting the measured time as 1.20 asserts that it was almost certainly no less than 1.27 and no greater than 1.33.

If the number 1.7 and 1.73 represent exact numerical quantities, the latter is definitely larger. If they represent physical measurements, the latter is probably larger, but the measurements are insufficiently precise to say with certainty. To use an analogy, if someone uses a ruler to determine that a hole is 3 15/16 inches in diameter, and someone else uses a micrometer to determine that a peg is precisely 100.0100mm in diameter, will the peg fit in the hole? If the hole is in fact precisely 3 15/16 inches, that would be exactly 100.0125mm, so the peg should fit, but even if the hole was only 99.8mm, it might still be reported as 3 15/16", so the measurements suggest that a fit is merely likely--not certain.

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Everyone is assuming there are only zeroes to the right of 1.7. To be accurate, you must know whether or not 1.7 is in fact the entire number or is it simply rounded to the nearest tenth. 1.749999999999999999999 rounded to the nearest tenth is 1.7 and is larger than 1.73205 or: 1.74999 - 1.73205 = 0.01794

More information is needed to answer this question accurately.

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2  
The number $1.7$ is $1.700\overline{0}$. What you are referring to isn't a mathematical issue, but rather one of measurement and approximation which is clearly not the OP's issue. –  Spencer Apr 9 at 16:43
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"More information is needed to answer this question accurately." - Seriously??? –  zerosofthezeta Apr 9 at 21:50

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