The Formal Rules of Algebra Summary of the formal rules of algebra on the set of real numbers 1. The axioms of "equality" a = a Reflexive or 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. 2. The commutative rules of addition and multiplication + = + 풂 · 풃 = 풃· 풂 3. The associative풂 rules풃 of풃 addition풂 and multiplication ( + ) + = + ( + ) ( ) ( ) 풂 × 풃 × 풄 = 풂 × 풃× 풄 4. The identity풂 elements풃 풄 of addition풂 풃 and풄 multiplication: + = + = 풂 · ퟎ= ퟎ· =풂 풂 0 and 1are풂 theퟏ identityퟏ elements풂 풂 for addition and multiplication respectively 5. The additive inverse of a is −a + ( ) = + = 6. The multiplicative inverse or 풂reciprocal−풂 of a− is풂 symbolized풂 ퟎ as (a 0) ퟏ 풂 × = = , The product of a number and its reciprocal is 1 ퟏ 풂 Two풂 numbers풂 풂 areퟏ called reciprocals of one another if their product is 1. 1/a and a are reciprocal to each other. The reciprocal of p/q is q/p. 7. The algebraic definition of subtraction = + ( ) Subtraction, in algebra, is defined풂 as− addition풃 풂 of the− inverse풃 . 1 8. The algebraic definition of division ÷ = = × , 풂 ퟏ Division,풂 풃 in algebra,풃 풂 is defined풃 풃 ≠ asퟎ multiplication by the reciprocal. Hence, algebra has two fundamental operations: addition and multiplication. 9. The inverse of the inverse ( ) = 10. The− relationship−풂 풂 of = ( 풃 − 풂) 풕풐 풂 − 풃 11. The풃 Rule− 풂of Signs− for풂 multiplication,− 풃 division, and fractions ( ) = . ( ) = . ( )( ) = . 풂 −=풃 =−풂풃 and− 풂 풃= − 풂풃 −풂 −풃 풂풃 풂 −풂 풂 −풂 풂 Note: "Like−풃 signs풃 produce− 풃 a positive−풃 number;풃 unlike signs, a negative number." 12. Rules for 0 · = · = If풂 ퟎ , thenퟎ 풂= ,ퟎ but is not defined ퟎ 풂 풂 ≠ ퟎ 풂 ퟎ ퟎ 13. Multiplying/Factoring ( + ) = + The distributive rule/ Common factor 14. The풎 same풂 operation풃 on풎풂 both sides풎풃 of an equation If = , then + = + If = , then = 풂 풃 풂 풄 풃 풄 We may풂 add풃 the same풂풄 number to풃풄 both sides of an equation; we may multiply both sides by the same number. 15. Change of sign on both sides of an equation If = , then = . We may change−풂 every풃 sign on풂 both sides−풃 of an equation. 2 16. Change of sign on both sides of an inequality: Change of direction (sense) If < , then > . When풂 we change풃 the− 풂signs on− both풃 sides of an inequality, we must change the direction (sense) of the inequality. 17. The Four Forms of equations corresponding to the 16. Four Operations and their inverses If + = , then = . If – = , then = + . 풙 풂 풃 풙 풃 − 풂 If = , then = 풙 풂 풃 풙 풂 풃 풃 If 풂풙 = 풃 , then 풙 = �풂 18. Change풙 of sense when solving an inequality ⁄풂 풃 풙 풂풃 If < , then > 풃 19. Multiplication−풂풙 of풃 fractions 풙 − 풂 × = and × = 풂 풄 풂풄 풄 풂풄 20. Division풃 풅 of fractions풃풅 풂 (Complex풅 풅 fractions) ÷ = × = or equivalently 풂 풄 풂 풅 풂풅 Division풃 풅 is multiplication풃 풄 풃풄 by the reciprocal. 21. Addition/Subtraction of fractions ± ± = Same denominator 풂 풃 풂 풃 풄 풄 풄 ± ± = Different denominators 풂 풃 풂풅 풃풄 풄 풅 풄풅 22. Power and exponents Let n be a natural number, then = × × × . × 풏 풂 풂����풂���풂��������풂 풏−풇풂풄풕풐풓풔 Here, is called power, n is called exponent and a the 풏 풂 3 Laws of Exponents Laws Examples 1) = = ퟏ ퟏ 2) 풙 = 풙 ퟔ = ퟔ ퟎ ퟎ 3) 풙 =ퟏ / ퟕ =ퟏ / −ퟏ −ퟏ 4) 풙 =ퟏ 풙 ퟒ =ퟏ ퟒ = 풎 풏 풎+풏 ퟐ ퟑ ퟐ+ퟑ ퟓ 5) 풙 풙/ =풙 풙 풙/ =풙 =풙 풎 풏 풎−풏 ퟔ ퟐ ퟔ−ퟐ ퟒ × 6) (풙 )풙 = 풙 풙( )풙 = 풙 = 풙 풎 풏 풎풏 ퟐ ퟑ ퟐ ퟑ ퟔ 풙 풙 풙 풙 풙 7) ( ) = ( ) = 풏 풏 풏 ퟑ ퟑ ퟑ 풙풚 풙 풚 풙풚 풙 풚 8) ( / ) = / ( / ) = / 풏 풏 풏 ퟐ ퟐ ퟐ 풙 풚 풙 풚 풙 풚 풙 풚 9) = / = / −풏 풏 −ퟑ ퟑ 풙 ퟏ 풙 풙 ퟏ 풙 And the Laws about Fractional Exponents: / / 10) = = 풏 ퟑ ퟏ 풏 ퟏ ퟑ = = = = 11) 풙 풎 풙 풙ퟐ 풙 풏 √ 풏 풎 ퟑ √ ퟑ ퟐ 풏 풎 ퟑ ퟐ 풙 √풙 �√풙� 풙 √풙 �√풙� = = Proof of 11): 풎 follows from the fact that 풏 풏 풎 풏 풎 풙 = √×풙 ( / )�=√풙( �/ ) × 풎 풏 풎 ퟏ 풏 ퟏ 풏 풎 4 .