BoostGSoC18 · adarshguptacse18 · Oct 5, 2019
diff --git a/README.md b/README.md
@@ -15,14 +15,14 @@ Boost.Real numerical data type for real numbers representation using range arith
 ## Introduction
 
 ### The problem addressed by boost::real
-Several times, when dealing with complex mathematical calculus, numerical errors can be carried from one operation to the next and after several steps, the error may significantly increase obtaining a non-trustworthy result. Normally, in these situations, the error is generated by the representation precision limit that truncates those numbers that do not fit in the representation precision generating errors without at least keeping track of the magnitude of the error. When this is the case, the real result is equal to the obtained approximated result plus the error (which is unknown).
+Several times, when dealing with complex mathematical calculus, numerical errors can be carried from one operation to the next.  After several steps, the error may significantly increase obtaining a non-trustworthy result. Usually, in these situations, the error is generated by the representation precision limit that truncates those numbers that do not fit in the representation precision generating errors without at least keeping track of the magnitude of the error. When this is the case, the real result is equal to the obtained approximated result plus the error (which is unknown).
 
-Another major problem when dealing with real numbers is the representation of the irrational number as the number π or e<sup>π</sup>, they are not handled by the native number data types causing limitations when calculations are based on those numbers. Normally we define a truncation of those numbers that are good enough for our purposes, but many times, the needed precision depends on the operation to do and to the composition of multiple operations, therefore, we are unable to determine which is the correct precision until we run the program.
+Another major problem when dealing with real numbers is the representation of the irrational number as the number π or e<sup>π</sup>, and they are not handled by the native number data types causing limitations when calculations are based on those numbers. We usually define a truncation of those numbers that are good enough for our purposes. Still, many times, the needed precision depends on the operation to do, and to the composition of multiple operations, therefore, we are unable to determine which is the correct precision until we run the program.
 
 ### The boost::real solution
-Boost::real is a real number representation data type that address the mentioned issues using range arithmetic [1] and defining the precision as dynamical to be determined in run-time. The main goal of this data type is to represent a real number x as a program that returns a finite or infinite set of intervals containing the number x x(k) = [m<sub>k</sub> - e<sub>k</sub>, m<sub>k</sub> + e<sub>k</sub>], K ∈ N ≥ 0, e<sub>k</sub> ≥ 0. Where K1 < K2 ⇒ |x(k2)| < |x(k1)|. For this purposes, any Boost::real number has a precision const iterator that iterates the serie of intervals representing the number. The intervals within the serie are sorted from larger to smaller, thus, each time the iterator is increased, the number precision increases due the new calculated interval is smaller than the previous one until the interval is the number itself (if possible).
+Boost::real is a real number representation data type that addresses the mentioned issues using range arithmetic [1] and defining the precision as dynamical to be determined in run-time. The main goal of this data type is to represent a real number x as a program that returns a finite or infinite set of intervals containing the number x x(k) = [m<sub>k</sub> - e<sub>k</sub>, m<sub>k</sub> + e<sub>k</sub>], K ∈ N ≥ 0, e<sub>k</sub> ≥ 0. Where K1 < K2 ⇒ |x(k2)| < |x(k1)|. For this purposes, any Boost::real number has a precision const iterator that iterates the series of intervals representing the number. The intervals within the serie are sorted from larger to smaller, thus, each time the iterator is increased, the number precision increases due the new calculated interval is smaller than the previous one until the interval is the number itself (if possible).
 
-Also, to allow representing irrational numbers as π or e<sup>π</sup>, boost::real has a constructor that takes as parameter a function pointer, functor (function object with the operator ()) or lambda expression that for any integer n > 0, the function returns the n-th digit of the represented number. For example, the number 1/3 can easily be represented by a program that for any input n > 0, the function returns 3.
+Also, to allow representing irrational numbers as π or e<sup>π</sup>, boost::real has a constructor that takes as a parameter a function pointer, functor (function object with the operator ()) or lambda expression that for any integer n > 0, the function returns the n-th digit of the represented number. For example, the number 1/3 can easily be represented by a program that for any input n > 0, the function returns 3.
 
 ## The boost::real numbers representation
 In boost::real, a number has one of the next three representations:
@@ -31,10 +31,10 @@ In boost::real, a number has one of the next three representations:
 2. Algorithmic number: This representation is equal to the Explicit number but instead of using a vector of digits, a lambda function must be provided. The lambda function takes an unsigned integer "n" as parameter and returns the n-th digit of the number.
 3. A number is a composition of two numbers related by an operator (+, -, *), the number creates pointers to the operands and each time the number is used, the operation is evaluated to return the result.
 
-Because of the third representation type, a number resulting from a complex calculus is a binary tree where each internal vertex is an operation and the vertex children are its operands. The tree leaves are those numbers represented by either (1) or (2) while the internal vertex are those numbers represented by (3). More information about the used number representation can be found in [3]
+Because of the third representation type, a number resulting from a complex calculus is a binary tree where each internal vertex is an operation, and the vertex children are its operands. The tree leaves are those numbers represented by either (1) or (2) while the internal vertex are those numbers represented by (3). More information about the used number representation can be found in [3]
 
 ## The boost::real precision iterator.
-The boost::real::const_precision_iterator is a forward iterator [4] that iterates through the number interval precisions. The iterator returns two numbers, a lower and an upper boundary that represent the [m<sub>k</sub> - e<sub>k</sub>, m<sub>k</sub> + e<sub>k</sub>] limits of the number approximation interval for a given precision. Each time the iterator is incremented, the interval approximation_interval is decreased and a new interval with a better precision is obtained. Normally, there is no need to interact with the precision iterator and it is used by the boost::real operators <<, < and >.
+The boost::real::const_precision_iterator is a forward iterator [4] that iterates through the number interval precisions. The iterator returns two numbers, a lower and an upper boundary that represent the [m<sub>k</sub> - e<sub>k</sub>, m<sub>k</sub> + e<sub>k</sub>] limits of the number approximation interval for a given precision. Each time the iterator is incremented, the interval approximation_interval is decreased, and a new interval with a better precision is obtained. Usually, there is no need to interact with the precision iterator, and it is used by the boost::real operators <<, < and >.
 
 ## Interface