Aman Learning

  Algebraic Expressions An algebraic expression is an expression made up of variables and constants along with mathematical operators. An algebraic expression is a sum of terms, which are considered to be building blocks for expressions. A term is a product of variables and constants. A term can be an algebraic expression in itself. Examples of a term – 3 which is just a constant. – 2x, which is the product of constant ‘2’ and the variable ‘x’ – 4xy, which is the product of the constant ‘4’ and the variables ‘x’ and ‘y’. – 5x 2 y, which is the product of 5, x, x and y. The constant in each term is referred to as the coefficient. Example of an algebraic expression: 3x 2 y+4xy+5x+6 which is the sum of four terms: 3x2y, 4xy, 5x and 6. An algebraic expression can have  any number of terms . The  coefficient  in each term can be  any real number . There can be  any number of variables  in an algebraic expression. The  exponent  on the variables, however, must be  rational numbers. Polynomia

  Real Numbers Real numbers constitute the union of all rational and irrational numbers. Any real number can be plotted on the number line. Euclid’s Division Lemma Euclid’s Division Lemma states that given two integers  a  and  b , there exists a unique pair of integers  q  and  r  such that  a = b × q + r   a n d   0 ≤ r < b . This lemma is essentially equivalent to :  dividend = divisor  ×  quotient + remainder In other words, for a given pair of dividend and divisor, the quotient and remainder obtained are going to be unique. Euclid’s Division Algorithm Euclid’s Division Algorithm is a method used to find the  H.C.F  of two numbers, say  a  and  b  where a> b. We apply Euclid’s Division Lemma to find two integers  q  and  r  such that  a = b × q + r   a n d   0 ≤ r < b . If r = 0, the H.C.F is b, else, we apply Euclid’s division Lemma to b (the divisor) and r (the remainder) to get another pair of quotient and remainder. The above method is repeated until a remainder of zer