BASIC PROPERTIES LIST:
There are two fundamental operations, addition and multiplication, that have the following basic properties, listed with their commonly designated names ( a,b,c are real numbers):
MORE DETAILED VIEW OF THE BASIC AND SOME DERIVED PROPERTIES, DEFINITIONS AND EXAMPLES:
Closure law: The sum a + b and the product a.b or ab are unique real numbers; more specifically if a and b are positive (belong to R+), then a + b and a . b are positive Commutative law: a + b = b + a: order does not matter in addition, ab = ba: order does not matter in multiplication. Associative law: a + (b + c) = (a+ b)+ c = a + b + c: grouping does not matter in addition. a(bc) = (ab)c = abc: grouping does not matter in multiplication. Distributive law: a(b+ c) = ab + ac; (a+ b)c = ac + bc: multiplication is distributive over addition. Identity law: There is a number 0 with the property that 0 + a = a + 0 = a; There is a number 1 with the property that 1.a = a .1 = a
that aa; ^{-1} = a^{-1}a = 1-a is called the negative of a ; ais called the ^{-1} reciprocal of a
(2 + x) + 7 = (x + 2) + 7 Commutative Law = x + (2 + 7) Associative Law = x+9
(a+ b)(c + d) = a(c + d)+ b(c + d) Distributive law = ac + ad + bc + bd Distributive Law This rule is referred to as
Definition of substraction: a -b = a + (-b)Definition of division: a / b = a . b^{-l}. therefore b^{-l} = 1 . b^{-l} = 1 / b ; Note that since the number 0 has no reciprocal, a / 0 is not defined.Properties of division: -(a/b) = (-a)/b=a/(-b)= - (-a)/(-b) ; (-a)/(-b) = a/b ; a/b=c/d if and only if a.d = b.c ; a/b = (p.a)/(p.b) for any non-zero p
1. If 2. For every real number If The number written
The following may be deduced from these definitions:
If If If For any real number a, either If
| example: |10| = 10 ; |-37.98| = 37.98
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Fractions properties: The fraction properties follow from the basic properties and the definition of division. The most commonly used fraction properties are listed below:Order of operations: In expressions involving multiple operations of different types, the fol-lowing order is followed: 1. operations within grouping symbols are performed first. If grouping symbols are nested inside other grouping symbols, operations are performed in order from the innermost outward. 2. exponents are performed before multiplications and divisions (unless rule 1 indicate otherwise) 3. perform multiplications and divisions from left to right, before additions and subtractions, also from left to right) ( unless unless rule 1 and 2 indicate otherwise). The above rules for the order of operations are known under the abbreviation
10 - 8[(23 – 6)(3 + 4)] = 10 - 8[(17) . (7) ] = 10 – 8.[ 119]
= 10 - 952 = -942 |