Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like a b 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'. For example, 2 3 is 8. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: 1,1,1 1,1,2 1,2,1 2,1,1 2,1,2 2,2,2 1,2,2 2,2,1.
So what does 3 0 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table? Just one I know this sounds a little fishy since we started with a rule I could have just made up which is why I gave the other reason first , but these formulas are all consistent and there is never any magic step, I promise! One rule for exponents is that exponents add when you have the same base.
Now, remember that if you have a negative exponent, it means you have one divided by the number to the exponent:. If you had trouble understanding it all with variables, let's look at it again,but this time as an example with numbers:. Let's look at what it means to raise a number to a certain power: it means to multiply that number by itself a certain number of times.
Let's look at a few examples:. If you look at the pattern, you can see that each time we reduce the power by 1 we divide the value by 3. Using this pattern we can not only find the value of 3 0 , we can find the value of 3 raised to a negative power! Here are some examples:. No matter what number we use when it is raised to the zero power it will always be 1. Suppose instead of 3 we used some number N, where N could even be a decimal. Heres a quick demonstration of why any number except zero raised to the zero power must equal 1.
As an example we will let that any number be the number 3. You might want to look at this post. Community Bot 1. It can only be approached from the positive side. I saw somebody defined it, can I now say it is defined?
See my answer also posted here. They are two different concepts. It is strange that people keep forgetting that. Vincenzo Oliva Vincenzo Oliva 6, 1 1 gold badge 16 16 silver badges 40 40 bronze badges. Look at the implicit limit you took and you'll find that I covered that case already without dividing by zero. Don't divide by zero.
Why don't you provide a citation validating your claim. From Concrete Mathematics p. Graham, D. Knuth, O. Published by Addison-Wesley, 2nd printing Dec, References Knuth. Two notes on notation. AMM 99 no. Mufasa Mufasa 5, 1 1 gold badge 16 16 silver badges 24 24 bronze badges. Meni Rosenfeld Meni Rosenfeld 3, 2 2 gold badges 12 12 silver badges 25 25 bronze badges. This is one of those things that unfortunately too few people understand. That's what happens when people limit themselves to thinking of real numbers and fail to realize that math is much richer than that - with all kinds of structures which look nothing like numbers, and yet where multiplication and integer exponentiation make sense.
But FWIW, problematic in my view. Saying that the integer exponent zero is somehow different than the real number exponent zero is tricky at best. How would you even write that symbolically? However, this is more of a conceptual matter, and I believe that in any case where the distinction matters, the correct interpretation will be clear from context.
So zero miles equals zero gallons equals zero light-year kilotons per microsecond equals a zero wavelength equals zero time and zero space Zero is the most important number to be "typed. In my intuition zero siriometers equals zero cow's grasses equals zero shakes, etc. They all signify no-thing. At least in realms concerned with dimensional analysis. In usage outside of such realms, it would be strange to say, "We've got zero kinematic viscosity in the fuel tank - we're not going anywhere.
You can rewrite dimensional equations to be dimensionless. Additionally Wikipedia says: Scalar args to transcendental functions must be dimensionless.
Show 6 more comments. Yatharth Agarwal Yatharth Agarwal 6 6 silver badges 19 19 bronze badges. JJacquelin JJacquelin Without that rule it doesn't make sense to look at graphs and limits when there are other ways to obtain a value.
Emilio Novati Emilio Novati Bryan Yocks Bryan Yocks 2, 15 15 silver badges 16 16 bronze badges. Where is the related question?
See if this gives you any clarification. Tyler Clark Tyler Clark 1, 1 1 gold badge 14 14 silver badges 19 19 bronze badges. Only positive numbers to positive powers. Nothing dodgy. There is no experimental error involved. Show 14 more comments.
Dan Christensen Dan Christensen Any value will work, as I show at my blog. I am not aware of any logically compelling reason to choose any particular value. So what? Show 8 more comments. Sid Sid 1, 5 5 silver badges 18 18 bronze badges.
Tanner 1. Hulkster Hulkster 1, 10 10 silver badges 27 27 bronze badges. Upcoming Events. However, in the case of -1 0 , the negative sign does not signify the number negative one, but instead signifies the opposite number of what follows. So we first calculate 1 0 , and then take the opposite of that, which would result in Another example: in the expression - -3 2 , the first negative sign means you take the opposite of the rest of the expression.
Why does zero with a zero exponent come up with an error?? Please explain why it doesn't exist. In other words, what is 0 0? Answer: Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.
But we could also think of 0 0 having the value 0, because zero to any power other than the zero power is zero. So laws of logarithms wouldn't work with it. So because of these problems, zero to zeroth power is usually said to be indeterminate.
However, if zero to zeroth power needs to be defined to have some value, 1 is the most logical definition for its value. This can be "handy" if you need some result to work in all cases such as the binomial theorem. See also What is 0 to the 0 power?
How is that proved?
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