Basic Properties of Addition and Multiplication
p.1. Basic Laws of Addition and Multiplication
1. Commutative Law
It doesn't matter what order you add up numbers: a + b = b + a.
2. Associative Law
Changing the grouping of numbers that are added together does not change their result sum: (a + b) + c = a + (b + c) = a + b + c
3. Identity Property of Zero
For any number being added to zero, the sum is the number itself: 0 + a = a
4. Property of Opposites
The sum of a number and its opposite is zero: a + (-a)=0
1. Commutative Law
It doesn't matter what order you multiply numbers: ab = ba.
2. Associative Law
Changing the grouping of numbers that are multiplied together does not change their result product: (ab)c = a(bc) = abc
3. Distributive Law
Any number which is multiplied by the sum of two or more numbers is equal to the sum of that number multiplied by each of the numbers separately: $$a(b +c )= ab + ac$$
4. Identity Property of One
Any number multiplied by 1 keeps its identity: $1 \cdot a = a$
5. Property of Zero
The product of any number and zero is zero: $0 \cdot a = 0$
6. Property of Reciprocals
The product of a number and its reciprocal is 1: $a \cdot a \frac 1a = 1 (a \neq 0)$
Being applied to numerical expressions, the commutative and combinational laws of addition and multiplication simplify calculations greatly.
Example 1. Evaluate the expression by choosing a convenient order of calculations
$(3\frac {17}{25} + 4\frac {7}{9}) + (2\frac {8}{25} - 1\frac {4}{9}) + \frac 23 \cdot 0,2 \cdot 0,8 \cdot 5 \cdot 1,25 =$
$= (3\frac {17}{25} + 2\frac {8}{25}) + (4\frac {7}{9} - 1\frac {4}{9}) + \frac 23 \cdot (0,2 \cdot 0,5) \cdot (0,8 \cdot 1,25) =$
$= (3 + 2 \frac {17 + 8}{25} + 4 - 1) + \frac {7-4}{9} + \frac 23 \cdot 1 \cdot 1 = 9 + (\frac 13 + \frac 23) = 10$
Answer: 10
Example 2. Calculate by choosing a convenient order:
$(74,7 \cdot \frac {2}{21} + (-105,3) \cdot 2 \frac 37 - (-105,3) \cdot \frac {2}{21} - 2 \frac {3}{7} \cdot 74,7) : 10 =$
$( \frac {2}{21} (74,7 + 105,3) - 2 \frac 37 (105,3 + 74,7)) : 10 = ( \frac {2}{21} - 2 \frac 37 ) \cdot (74 + 105 + 1) : 10 = $
$( \frac {2}{21} - \frac {9}{21} - 2) \cdot (180 : 10) = \frac {-7 - 42}{21} \cdot 18 = \frac {-49}{7} \cdot 6 = -7 \cdot 6 = 42$
Answer: 42
p.2. Combining Like Terms
Being applied to expressions with variables, the laws of addition and multiplication allow simplification, primarily by combining like terms.
Like terms are terms in a variable expression that have the same literal part (any literal expression); numbers without a literal part are considered like terms.
Note that the expression 3ab+2ba consists of like terms, since 2ba=2ab in accordance with the commutative law of multiplication.
Therefore, 3ab+2ba=3ab+2ab=(3+2)ab=5ab.
We can introduce the following algorithm.
Algorithm of Combining Like Terms
1. Reorder the terms so that like terms are next to each other, group them using brackets.
2. Bracket out the literal part of like terms.
3. Evaluate the numeric expression in the brackets. This is the final numerical factor.
4. Replace like terms in the expression with the result obtained.
Example 3. Simplify the expression:
$3(a + 4b - 1) - 4(2a - b + 4) + a = 3a + 3 \cdot 4b - 3 - 4 \cdot 2a + 4b - 16 + a=$
$= (3a - 8a + a) + (12b + 4b) - (3 + 16) = (3 - 8 + 1)a + (12 + 4)b - 19=$
$= -4a + 16b - 19$
Example 4. Simplify the expression and evaluate it, given x=8:
$2x(x - 4) + 4(18 - x^2 ) + x(2x - 1)=2x^2 - 8x + 72 - 4x^2 + 2x^2 - x=$
$=(2 - 4 + 2)x^2 + (-8 - 1)x + 72 = -9x + 72$
For x=8, we obtain: -9∙8+72=0
Answer: 0
