Undecidability of equality of real numbers
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, formula_1 and exponential and sine functions. It was proved in 1968 by the mathematician and computer scientist Daniel Richardson of the University of Bath.
Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable "x", the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions.
For some classes of expressions generated by other primitives than in Richardson's theorem, there exist algorithms that can determine whether an expression is zero.
Statement of the theorem.
Richardson's theorem can be stated as follows: 
Let "E" be a set of expressions that represent formula_2 functions. Suppose that "E" includes these expressions:
Suppose "E" is also closed under a few standard operations. Specifically, suppose that if "A" and "B" are in "E", then all of the following are also in "E":
Then the following decision problems are unsolvable:
Extensions.
After Hilbert's tenth problem was solved in 1970, B. F. Caviness observed that the use of "ex" and ln 2 could be removed.
Wang later noted that under the same assumptions under which the question of whether there was "x" with "A"("x")  ; 0 was insolvable, the question of whether there was "x" with "A"("x") = 0 was also insolvable.
Miklós Laczkovich removed also the need for π and reduced the use of composition. In particular, given an expression "A"("x") in the ring generated by the integers, "x", sin "xn", and sin("x" sin "xn") (for "n" ranging over positive integers), both the question of whether "A"("x")  ; 0 for some "x" and whether "A"("x") = 0 for some "x" are unsolvable.
By contrast, the Tarski–Seidenberg theorem says that the first-order theory of the real field is decidable, so it is not possible to remove the sine function entirely.
References.
 ;templatestyles src="Reflist/styles.css" / ;