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III. Understand --PROPERTIES OF REAL NUMBERS
Commutative, Associative, Identity, Multiplication Property of Zero, Division by Zero, Inverse, Distributive, Reflexive, Symmetric, Transitive, Addition Property of Equations, Multiplication Property of Equations
COMMUTATIVE PROPERTY FOR ADDITION AND MULTIPLICATION
The order in which numbers are added or multiplied does not matter. Adding 2 + 3 or 3 + 2 results in the same answer. Likewise, multiplying 2(3) or 3(2) gives the same result.
The commutative property of addition may be stated: For every number a and b; a + b = b + a.
The commutative property of multiplication may be stated: For every number a and b; ab = ba.
The commutative property does not apply to division problems nor to subtraction problems.
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ASSOCIATIVE PROPERTY FOR ADDITION AND MULTIPLICATION
The associative property allows us to regroup adjacent numbers in addition and multiplication problems. For example, (1 + 2) + 3 = 1 + (2 + 3) and 2 × (3 × 4) = (2 × 3) × 4.
The associative property of addition may be stated: For every number a, b and c; a + (b + c) = (a + b) + c.
The associative property of multiplication may be stated: For every number a, b and c; a(bc) = (ab)c
The associative property does not apply to division problems nor to subtraction problems.
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IDENTITY PROPERTIES FOR ADDITION AND MULTIPLICATION
0 + 4 = 4 illustrates the identity property of addition. Notice that zero was added to four and the four did not change.
The additive property of 0 or the additive identity may be stated:
For every number a; 0 + a = a.
1 × 4 = 4 illustrates the identity property of multiplication. Notice that four was multiplied by one and the four did not change.
The multiplicative property of 1 or the multiplicative identity may be stated:
For every number a; 1• a = a.
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THE MULTIPLICATION PROPERTY OF ZERO
Zero times any number is zero. 0 × 5 = 0 illustrates the Multiplication Property of Zero. Notice that when five was multiplied by zero the five was eliminated. This is NOT an identity property.
The multiplicative property of 0 may be stated: For every number a; 0 • a = 0.
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DIVISION BY ZERO
The multiplication property of zero affects division by zero.
6 ÷ 3 = 2 because 2 • 3 = 6; 6 ÷ 0 does not exist because 0 • any number = 0, not 6.
This also means that a fraction having a denominator of 0 contradicts the multiplication property of 0, since the fraction bar is also a symbol for division.
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INVERSE PROPERTIES FOR ADDITION AND MULTIPLICATION
Inverse operations undo each other. For example, addition and subtraction are inverse operations since one undoes the other. Multiplication and division are inverse operations since the results of one operation are reversed with the other operation.
The additive inverse may be stated: For every number a; a + (–a) = 0 and –a + a = 0.
The multiplicative inverse may be stated: For every number a; except a = 0, a • 1/a = 0 and 1/a • a = 1.
The inverse properties state that a number added to its opposite results in the answer of 0 and a number multiplied by its reciprocal results in the answer of 1.
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DISTRIBUTIVE PROPERTY
The distributive property allows an addition problem to be part of a multiplication problem. Multiplication is then distributed over the addition.
The distributive property may be stated: For every number a, b, and c; a(b + c) = ab + bc.
Use the distributive property to multiply addition problems that cannot be simplified due to the fact that the addends are unlike terms. For example: 2( 3x + 6) = 2(3x) + 2(6) = 6x + 12
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REFLEXIVE PROPERTY
The reflexive property states that any number is equal to itself. This is the simplest of all the properties.
The reflexive property may be stated: For every number a; a = a.
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SYMMETRIC PROPERTY OF EQUALITY
The symmetric property of equality allows equality to be reversed.
The symmetric property of equality may be stated; For every number a and b; if a = b, then b = a.
3x + 5 = 8 and 8 = 3x + 5 are the same because of the symmetric property of equality.
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TRANSITIVE PROPERTY OF EQUALITY
The transitive property of equality allows for the transfer of equality. Two amounts are equal if they both are equal to the same third amount. If y = 3x and 10 = 3x, then we know that y = 10 because of the transition property of equality.
The transitive property of equality may be stated:
For every number a, b, and c; if a = c and b = c, then a = b.
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ADDITION PROPERTY OF EQUATIONS
The Addition Property of Equations allows the same number to be added to both sides of the equals sign in an equation. The appearance of the equation is transformed. The solution to the equation remains unchanged.
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MULTIPLICATION PROPERTY OF EQUATIONS
The Multiplication Property of Equations allows both sides of the equals sign in an equation to be multiplied by the same number. The appearance of the equation is transformed. The solution to the equation remains unchanged.
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