It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value;  a profound and important idea which appears so simple to us now that we ignore its true merit.  We shall appreciate the grandeur of the achievement to be more when we ember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. 

        Piere Simon Laplace A     The ability to reason and think clearly is extremely useful in our daily life.  For example, suppose a politician tells you, ‘If you are interested in a clean government, then you should vote for me.’ What he actually wants you to believe is that if you do not vote for him, then you may not get a  clean government.  

Similarly, if an advertisement tells you, ‘The intelligent wear XYZ shoes’, what the company wants you to conclude is that if you do not wear XYZ shoes, then you are not intelligent enough. 

        An algorithm is a series of well-defined steps which gives a procedure for solving a type of problem.  The word algorithm comes from the name of the 9th-century Persian mathematician al-Khwarizmi.  In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala.  A lemma is a proven statement used for proving Muhammad ibn Musa al-KE another statement.  (C.E 780 – C.E.850

         You can observe yourself that both the above statements may mislead the general public.  So, if we understand the process of reasoning correctly, we do not fall into such traps unknowingly The correct use of reasoning is at the core of mathematics, especially in constructing proofs.  

In Class IX, you were introduced to the idea of ​​proofs, and you actually proved many statements, especially in geometry.  Recall that a proof is made up of several mathematical statements, each of which is logically deduced from a previous statement in the proof, or from a theorem proved earlier, or an axiom, or the hypotheses.  

        An equivalent version of Theorem 1.2 was probably first recorded as Proposition 14 of Book IX in Euclid’s Elements, before it came to be known as the Fundamental Theorem of Arithmetic.  However, the first correct proof was given by Carl Friedrich Gauss in his Disquisitiones Arithmeticae.  

Carl Friedrich Gauss is often referred to as the ‘Prince of Mathematicians and is considered one of the three greatest mathematicians of all time, along with Archimedes Newton.  He has made Carl Friedrich Gauss fundamental contributions to both mathematics and science.  (C.E.1777 – C.E.1855)

        The main tool, we use in constructing a proof, is the process of deductive reaspning We start the study of this chapter with a review of what a mathematical statement is.  Then, we proceed to sharpen our skills in deductive reasoning using several examples.  We shall also deal with the concept of negation and finding the negation of a given statement.  

Then, we discuss what it means to find the converse of a given statement.  Finally, we review the ingredients of a proof learned in Class IX by analyzing the proofs several theorems.  Here, we also discuss the idea of ​​proof by contradiction, which you have come across in Class IX and many other chapters of this book.