Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Are you maybe from a culture that reads from right to left? – Raskolnikov Apr 9 '14 at 15:40
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 '14 at 15:51
@ChrisH Get out – bwv869 Apr 9 '14 at 16:51
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 '14 at 18:44
This is a bizarre question from someone who has previously asked about finding the domain of an expression:… -- 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 '14 at 19:52

$$1.73205 = 1.7 + 0.03205$$

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

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

share|cite|improve this answer
1.73205; thank you so much. – user137452 Apr 9 '14 at 15:46
You are welcome! – amWhy Apr 9 '14 at 15:47
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 '14 at 6:45
$0 = \dfrac{0}{100,000}.\quad$ But $0.03205 = \dfrac{3205}{100,000}>0$ – amWhy Apr 10 '14 at 11:41
Rawling: Remember that $0 = 0.00000$. Compare with $0.03205$:$$\begin{align} &0\,.\,0\,3\,2\,0\,5 \\ -\,&0\,.\,0\,0\,0\,0\,0 \\ &\hline\\ \\ = &0\,.\,0\,3\,2\,0\,5\end{align}$$ – amWhy Apr 10 '14 at 11:53

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.

share|cite|improve this answer
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 '14 at 17:20
Real-world explanation. :) – Anton Babenko Apr 9 '14 at 17:40
@Spencer According to some people, money isn't math. I don't understand them. – Izkata Apr 9 '14 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 $3$.

share|cite|improve this answer
Wow!Thank you so much. My god. Never thought about it like that. – user137452 Apr 9 '14 at 15:40
You have now increased the number of significant digits. In $1.7$ has 2 sig fig while 1.70000 has 6 sig fig. :) – Sufyan Naeem May 10 '15 at 12:01

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.

share|cite|improve this answer

A simple way to understand this may be

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

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


Hence 1.73205 is larger than 1.7.

share|cite|improve this answer
This answer isn’t really on the same level as the question being asked… – bdesham Apr 9 '14 at 16:59
@bdesham: You mean at a lesser level or...? – legends2k Apr 9 '14 at 18:21
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 '14 at 18:33
+1 on my defence, I thought the OP was confused only about real numbers and is probably comfortable handling integers :) – legends2k Apr 9 '14 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 decimal 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:

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

share|cite|improve this answer

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 $0.8205$.

share|cite|improve this answer
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 '14 at 2:49

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, $a > b$.

share|cite|improve this answer

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.