is 0.77777778 the same as 0.8 ? ( Square root problem )

Question

I have holes in my math because I didn't pay attention when I was a kid.( so please explain in detail if possible <3 )
While relearning everything I found my self stuck not understanding how this works : $$\sqrt{\frac{49}{81}}$$ my steps to solve this were :
1- divide 49 by 81
2- square root the result $$\sqrt{0.6049382716049383}$$ 3-the result is 0.7777777777777778

so what i need to understand is :
1- isnt 0.777777777777778 = 0.8 ?
2- is 0.7777777777777778 ( my result ) = $\frac{7}{9}$ ( which is the correct result according to the book )

So there's an error in my steps ? I saw someone square root both numbers first .. but cant I square root after I divide ? I think that's where the error lies. — Moath, Sep 11 '23 at 06:32
You can square root after you divide, that is fine. And the answer you obtain on your calculator is equal to $7/9$, at least to the precision that your calculator is able to compute. — littleO, Sep 11 '23 at 06:36
For (2), you need to remember that most calculators cannot always give exact answers and instead give a numerical approximation. Therefore, while 0.7777777777777778 is not the same as 7/9, it is the same as 7/9 to the calculator. — Brian Moehring, Sep 11 '23 at 06:36
The numbers are not even close. Why do you think they are equal ? — Peter, Sep 11 '23 at 06:37
The comments that are expressing incredulity are to your first question. I similarly am confused why you'd ever think 0.7777777777777778 is .8. Without an explanation of why you think that might be, that incredulity is all we can give you. (for one, note .79 lies strictly between 0.7777777777777778 and .8, so they can't be the same) — Brian Moehring, Sep 11 '23 at 06:38
when i do 7\9 in my calc it gives me 0.7777777777777778, I thought if a number is over 5 in decimals you increase the number to its left by one therefore 0.78 should be 0.8 but seems like I am a few zeros off and thats the error ? — Moath, Sep 11 '23 at 06:42
There are some numbers like 0.781 or 0.795 between your number and 0.8 — Lourrran, Sep 11 '23 at 06:45
yeah this is an issue, English isn't my first language either, so searching for stuff online didn't give me the result I was looking for. so if it was 0.79 I can make it 0.8 but not 0.78 ? or does it have to be 7.9 repeating 9's ? — Moath, Sep 11 '23 at 06:48
While it’s true that .78 rounded to the nearest tenth is .8, this does not mean that .78 is equal to .8. They are different numbers. — littleO, Sep 11 '23 at 06:52
Typically, you cannot just round .79 or .78 to .8 without changing the intended value. Sometimes we deal with "fuzzy" numbers but for most people that only shows up in scientific measurements in the form of "significant figures/digits". In most cases in math, rounding changes the intended value, so you cannot round and still call it equal. (The fact that .8=.79999... with the 9s infinitely repeated is a true but unrelated anomaly with how we define real numbers that we otherwise don't need to concern ourselves with) — Brian Moehring, Sep 11 '23 at 06:56

MathTrain · Accepted Answer · 2023-09-11T06:55:54.173

6

The short answer is that for question 1, the answer is no, absolutely not. For question 2, technically also no, but it's so close that for most practical purposes, yes.

In more detail, you have to think about what these decimals actually mean. Apologies for the lack of repeating notation, I'm writing on my phone.

0.77777777777778 is the result you get from a calculator, but that's only because your calculator can only do so much decimal precision. The actual answer is 0.7777... repeating infinite sevens forever. This is equal to 7/9.

But what you've done with the decimals is far too much work and over-relies on the calculator (imagine how hard this would be to do by hand). The problem can be done mentally. Just take the square root of the numerator and denominator. $$\sqrt{49/81}=\sqrt{49}/\sqrt{81}=7/9. $$

I'd also like to address why 0.7... repeating doesn't equal 0.8. That would be 0.8=0.799999999... with infinite repeating 9s. If you subtract these numbers, you'd get infinite repeating zeros, that is, 0. For two numbers to subtract to get 0, they must be equal. However, if you subtract 0.8 - 0.77777777... you'd get 0.0222222222... repeating. Since their difference isn't 0, they're not the same.

Hope this helps!

edited Sep 11 '23 at 06:55

answered Sep 11 '23 at 06:43

MathTrain

2,147

9

I'd really like to know why someone downvoted my answer here. I am a high school teacher and get questions like this all the time. My description is accurate (at the very least to the level of understanding of the OP). What is wrong here? – MathTrain Sep 11 '23 at 06:59
1

Lol, it's probably because someone is jealous about answers to easy questions getting lots of upvotes – Johnny Lenin Sep 11 '23 at 13:08
@MathTrain Based on this chat message in CURED, I think Peter downvoted your post. At least, Peter was calling for people to delete your answer. – Mike Earnest Sep 13 '23 at 21:54
@MikeEarnest Thanks. As I suspected initially, I think it has to do with the question being easy relative to most on here. However, this isn't stack overflow, and the site is for questions "at any level" – MathTrain Sep 14 '23 at 16:30

score 1 · Answer 2 · answered Sep 11 '23 at 07:07

1

$0.77777777778 \approx 0.8$

$0.777777778 \approx \frac{7}{9}$ rounded to 9 decimal places

$0.\bar7 = \frac{7}{9}$

answered Sep 11 '23 at 07:07

Agent Smith

951

is 0.77777778 the same as 0.8 ? ( Square root problem )

2 Answers2