0^0, 1/0, -1/0, 0/0 and infinity in GeoGebra
- Author:
- Michael Borcherds
- Topic:
- Numbers
For 0^0 see here:
mathforum.org/dr.math/faq/faq.0.to.0.power.html
and https://en.wikipedia.org/wiki/Zero_to_the_power_of_zero
GeoGebra example where this is useful:
Sum(Sequence(x^i/i!,i,0,n))
Try dragging points A and B. Observe what happens to a when
- A is directly above B
- B is directly above A
- A and B are in the same place
Further reading:
https://en.wikipedia.org/wiki/Extended_real_number_line
http://functions.wolfram.com/Constants/ComplexInfinity/introductions/Symbols/ShowAll.html
https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html
The division of 0 by 0 results in a NaN. A nonzero number divided by 0, however, returns infinity: 1/0 = ∞, -1/0 = -∞. The reason for the distinction is this: if f(x) ↦ 0 and g(x) ↦ 0 as x approaches some limit, then f(x)/g(x) could have any value. For example, when f(x) = sin x and g(x) = x, then f(x)/g(x) ↦ 1 as x ↦ 0. But when f(x) = 1 - cos x, f(x)/g(x) ↦ 0. When thinking of 0/0 as the limiting situation of a quotient of two very small numbers, 0/0 could represent anything. Thus in the IEEE standard, 0/0 results in a NaN. But when c > 0, f(x) ↦ c, and g(x) ↦ 0, then f(x)/g(x) ↦ ±∞, for any analytic functions f and g. If g(x) < 0 for small x, then f(x)/g(x) ↦ -∞, otherwise the limit is +∞. So the IEEE standard defines c/0 = ±∞, as long as c ≠ 0. The sign of ∞ depends on the signs of c and 0 in the usual way, so that -10/0 = -∞, and -10/-0 = +∞