Nought for a thought

Number Theory Level 1

$\Large\dfrac { { \color{#D61F06}{0} }^{ \color{#3D99F6}{0!} } }{ { \color{#3D99F6}{0!} }^{ \color{#D61F06}{0} } } =\,?$ .

1 0 Undefined ∞

5 solutions

A Former Brilliant Member
Dec 12, 2015

$\Rightarrow\large \dfrac{\color{#3D99F6}{0}\color{#D61F06}{^{0!}}}{\color{#D61F06}{0!}\color{#3D99F6}{^0}}$

$=\large \dfrac{\color{#3D99F6}{0}\color{#D61F06}{^1}}{\color{#D61F06}{1}\color{#3D99F6}{^0}}$

$=\large \dfrac{\color{#3D99F6}{0}}{\color{#D61F06}{1}}$

$=\large \color{#3D99F6}{0}$

NOTE: $\color{#D61F06}{0!}=\color{#D61F06}{1}$ and $\color{#D61F06}{1}\color{#3D99F6}{^0}=\color{#D61F06}{1}.$

I was careless when assuming 0! = 0

Evan Huynh - 5 years, 6 months ago

But if 0! = 0, it breaks the pattern, because n! times n+1 is always (n+1)!, and zero times anything is zero.

Whitney Clark - 5 years, 6 months ago

With this surely (n+1) = 1 because you have said that n = 0 and 0+1=1 and 1! = 1. Therefore 0! = 0

Dennis Acreman - 5 years, 5 months ago

@Dennis Acreman – No, no, you don't add, you multiply. 3! = 2! x 3, 2! = 1! x 2, and 1! = 0! x 1. Since 1! = 1, 0! = 1 also.

Whitney Clark - 5 years, 5 months ago

Though it was to the 0th power. To the best of my knowledge, anything to the 0th power is always 1.

Andy Whitmire - 5 years, 5 months ago

Same is the case with me... Hasty Nut, me

NarasimhaRao Kurapati V L - 5 years, 5 months ago

same here :(

Bassel Safwat - 5 years, 5 months ago

This is my new knowledge 1!=0!

Jet Sirilim - 5 years, 6 months ago

IF 1 ^ ௦ = 1 THEN 1^௦ = 1^1 GIVES ௦ =1 which is ABSURD. THE General Rule is x^௦ = 1 if and only if x not equal to 1.

Ramamurthy TG - 5 years, 6 months ago

So, if 1^1=1 and 1^2=1, then 1^1=1^2 gives 1=2? Your reasoning is absurd, 1^0=1.

Sergio Cárdenas - 5 years, 6 months ago

One to the power of ANYTHING is one, just like zero times anything, such as two or three, is zero - but two does not equal three.

Whitney Clark - 5 years, 6 months ago

@Whitney Clark – FROM your POINT of VIEW 1^௦= 1=2^௦ HENCE 1 = 2. is TRUE

Ramamurthy TG - 5 years, 6 months ago

@Ramamurthy Tg – That is not my point of view. Zero times anything is zero, so in the equation 0x2 = 0x3, you cannot cancel zeroes and get 2=3. Similarly, one to ANY power is one, even though the exponents are different:

$1^2 = 1 \times 1 = 1$ $1^3 = 1 \times 1 \times 1 = 1$ but $2 \neq 3$ .

Similarly, anything (other than zero) to the power of zero is one.

Whitney Clark - 5 years, 6 months ago

@Ramamurthy Tg – This is exactly why x/0 is undefined! Going with Whitney's example real quick, 0x2=0x3, yes, but to get the equation down to 2=3, we'd need to divide both sides by zero! Which is why zero is undefined! Similarly, 2^0 = 1 = 3^0, but taking the log of every side, and pulling down the exponents, we get 0log2 = log1 = 0log3. Well, 0 times anything is 0, so we get 0 = log1 = 0. And log1 = 0 ! So there are no mathematical inconsistencies here; we're only proving the abstract nature of 0, and the fact that x/0 is undefined.

Clinton Kunhardt - 5 years, 5 months ago

You missed the ! as did I. But I do digress this exponent is 0 by the power of NON divided by NON by the power of 0. 0x0div0x0 = 0 since you can't divide 0 zero times the answer remains the same.

Cori Ravenheart - 5 years, 5 months ago

according to your logic 2^0 = 1; 1^1 = 1; 0 = 1 So, does not this seem absurd as well ????? The rule X^0 = 1 should fit in every shoe

Fred Burger - 5 years, 6 months ago

@Fred Burger – Yes, $2^0=1$ and $1^1=1$ . But how does this show that 0 = 1?

Whitney Clark - 5 years, 6 months ago

@Whitney Clark – Read Mr. Ramamurthy Tg's comment; I just quoted his/her hypothesis and tried to prove it wrong.

Fred Burger - 5 years, 6 months ago

I missed it too, I unconsciously miscalcalculated

Azeez Yusuf - 5 years, 5 months ago

if 0/1=0 0=1*0 0/0=1?

kiran kumar - 5 years, 5 months ago

But 0/2=0; 0=2*0; 0/0=2? So 1=2? But in reality, division by zero is undefined.

Whitney Clark - 5 years, 5 months ago

0^0 also equals to 1 according to X^0=1 axiom hence 1/1 = 1

Fred Burger - 5 years, 6 months ago

Luca Righetti
Dec 12, 2015

0! = 1
Therefore the equation can be rewritten as: (0^1)/(1^0)
Anything to the power of 0 (except 0) equals 1
Hence: 0/1 = 0

Moderator note:

Great!

FYI To start a new line, just leave 3 (or more) empty spaces at the end of the line and hit enter. I've edited your solution for your reference.

Calvin Lin Staff - 5 years, 5 months ago

Thank you very much

Luca Righetti - 5 years, 5 months ago

Pranshu Aggarwal
Jan 11, 2016

Just take care of the factorial sign(!) ie. 0!=1 and not ZERO and anything to the power zero is 1, so 1 to the power 0 is 1 itself!!

Arun Venkatesan
Dec 27, 2015

0! = 1; anything to the power 0 is also 1

Satyam Pandey
Dec 15, 2015

0!=1 And 0^1=0

I don't know much about factorials. Could you explain why 0! =1?

Robert White - 5 years, 6 months ago

Factorials are products of all positive integers from 1 to n. 0! is by convention 1 to account for empty product. An empty set is a set by itself.

Linden Marquez - 5 years, 6 months ago

I had the same question. A quick google sea ch revealed this. http://mathforum.org/library/drmath/view/57128.html

Todd Sauve - 5 years, 5 months ago

Nought for a thought

5 solutions

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