Summary of algebra - Topics in precalculus

 

Topics in P R E C A L C U L U S Table of Contents | Home 1 The Formal Rules of Algebra ALGEBRA  is a method of written calculations.  Now what is a calculation? It is replacing one set of symbols with another? In arithmetic we may replace the symbols  '2 + 2'  with the symbol  '4.' In algebra we may replace  'a + (−a)'  with '0.' a + (−a) = 0. A formal rule, then, shows how an expression written in one form may be rewritten in a different form.  The = sign means  "may be rewritten as"  or  "may be replaced by." If p and q are statements (equations), then a rule If p, then q, or equivalently p implies q, means:  We may replace statement p with statement q.  For example, x + a = b  implies  x = b − a. That means that we may replace the statement  'x + a = b'  with the statement  'x = b − a.' Algebra depends on how things look.  We can say, then, that algebra is a system of formal rules.  The following are what we are permitted to write. (See the complete course, Skill in Algebra.) 11.  The axioms of "equals" a = a Identity   If a = b, then b = a. Symmetry   If a = b  and  b = c, then a = c.   Transitivity These are the "rules" that govern the use of the = sign. 12.  The commutative rules of addition and multiplication a + b  =  b + a   a· b  =  b· a 13.  The identity elements of addition and multiplication: 3.  0 and 1 a + 0 = 0 + a = a a· 1 = 1· a = a Thus, if we "operate" on a number with an identity element, it returns that number unchanged. 14.  The additive inverse of a:  −a a + (−a) = −a + a = 0 The "inverse" of a number undoes what the number does. For example, if you start with 5 and add 2, then to get back to 5 you must add −2.  Adding 2 + (−2) is then the same as adding 0 -- which is the identity.   15.  The multiplicative inverse or reciprocal of  a,   5.    symbolized as  1 a  (a 0) a·  1 a   =   1 a · a   =  1 Two numbers are called reciprocals of one another if their product is 1. Thus, 1/a symbolizes that number which, when multiplied by a, produces 1.   The reciprocal of   p q  is  q p . 16.  The algebraic definition of subtraction a − b = a + (−b) Subtraction, in algebra, is defined as addition of the inverse. 17.  The algebraic definition of division a b   =   a·  1 b Division, in algebra, is defined as multiplication by the reciprocal. Hence, algebra has two fundamental operations: addition and multiplication. 18.  The inverse of the inverse −(−a) = a 19.  The relationship of  b − a  to  a − b b − a = −(a − b) Now,  b + a  is equal to a + b.  But  b − a  is the negative of a − b. 10.  The Rule of Signs for multiplication, division, and 10.  fractions a(−b) = −ab.    (−a)b = −ab.    (−a)(−b) = ab.    a −b  = −  a b . −a   b  = −  a b . −a −b  =  a b . "Like signs produce a positive number; unlike signs, a negative number." 11.  Rules for 0 a· 0 = 0· a = 0 If a 0, then 0 a   =  0.   a 0   =  No value.   0 0   =  Any number. Division by 0 is an excluded operation.  (Skill in Algebra, Lesson 5.) 12.  Multiplying/Factoring m(a + b) = ma + mb The distributive rule/   Common factor   (x − a)(x − b) = x² − (a + b)x + ab     Quadratic trinomial   (a ± b)² = a² ± 2ab + b² Perfect square trinomial   (a + b)(a − b) = a² − b² The difference of   two squares   (a ± b)(a² ab + b²) = a³ ± b³     The sum or difference of   two cubes 13.  The same operation on both sides of an equation If      If        a  =  b,   a  =  b,   then      then              a + c  =  b + c.   ac  =  bc. We may add the same number to both sides of an equation; we may multiply both sides by the same number. 14.  Change of sign on both sides of an equation If         −a  =  b,   then         a  =  −b. We may change every sign on both sides of an equation. 15.  Change of sign on both sides of an inequality:  15.  Change of sense If         a  <  b,   then         −a  >  −b. When we change the signs on both sides of an inequality, we must change the sense of the inequality. 16.  The Four Forms of equations corresponding to the 16.  Four Operations and their inverses If     If         x + a  =  b,         x − a  =  b,   then     then     x  =  b − a.     x  =  a + b. * * * If     If         ax  =  b,        x    a  =  b,   then     then     x  =  b a .    x  =  ab. See  Skill in Algebra, Lesson 9. 17.  Change of sense when solving an inequality If         −ax  < b,       then         x  > − b a . 18.  Absolute value If  |x| = b,  then  x = b  or  x = −b. If  |x| < b  then  −b < x < b. If  |x| > b  (and b > 0), then  x > b  or  x < −b. 19.  The principle of equivalent fractions x y  =  ax ay   and symmetrically, ax ay  =  x y We may multiply both the numerator and denominator by the same factor; we may divide both by the same factor. 20.  Multiplication of fractions a b ·    c d  =   ac bd   a ·    c d  =   ac d 21.  Division of fractions (Complex fractions) Division is multiplication by the reciprocal. 22.  Addition of fractions a c  +  b c  =  a + b    c Same denominator   a b  +   c d  =  ad + bc    bd Different denominators with no common factors    a  bc  +   e  cd  =  ad + be    bcd Different denominators with common factors The common denominator is the LCM of denominators. 23.  The rules of exponents aman  =  am+n   Multiplying or dividing   am an   =  am−n   powers of the same base     (ab)n  =  anbn   Power of a product of factors     (am)n  =  amn   Power of a power 24.  The definition of a negative exponent a−n  =   1  an 25.  The definition of exponent 0 a0 = 1 26.  The definition of the square root radical The square root radical squared produces the radicand. 27.  Equations of the form  a² = b If a²  =  b,   then a  =  ±. 28.  Multiplying/Factoring radicals  =    and symmetrically,    =  29.  The definition of the nth root 30.  The definition of a rational exponent It is more skillfull to take the root first. 31.  The laws of logarithms log xy  =  log x  +  log y. log  x y   =  log x  −  log y. log xn  =  n log x. log 1 = 0.   logbb = 1. 32.  The definition of the complex unit i i ² = −1 Next Topic:  Rational and irrational numbers Table of Contents | Home Please make a donation to keep TheMathPage online. 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