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Sagot :
Certainly! Let's explore the properties of real numbers step-by-step with examples for both addition and multiplication.
### 1. Commutative Property
The commutative property states that the order in which you add or multiply two numbers does not affect the result.
#### Addition Example:
Let [tex]\( a = 3 \)[/tex] and [tex]\( b = 5 \)[/tex].
[tex]\[ a + b = 3 + 5 = 8 \][/tex]
[tex]\[ b + a = 5 + 3 = 8 \][/tex]
Both expressions result in 8, demonstrating the commutative property of addition.
#### Multiplication Example:
Let [tex]\( a = 4 \)[/tex] and [tex]\( b = 6 \)[/tex].
[tex]\[ a \times b = 4 \times 6 = 24 \][/tex]
[tex]\[ b \times a = 6 \times 4 = 24 \][/tex]
Both expressions result in 24, demonstrating the commutative property of multiplication.
### 2. Associative Property
The associative property states that the way in which numbers are grouped when adding or multiplying does not affect the result.
#### Addition Example:
Let [tex]\( a = 2 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 6 \)[/tex].
[tex]\[ (a + b) + c = (2 + 4) + 6 = 6 + 6 = 12 \][/tex]
[tex]\[ a + (b + c) = 2 + (4 + 6) = 2 + 10 = 12 \][/tex]
Both expressions result in 12, demonstrating the associative property of addition.
#### Multiplication Example:
Let [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 5 \)[/tex].
[tex]\[ (a \times b) \times c = (1 \times 3) \times 5 = 3 \times 5 = 15 \][/tex]
[tex]\[ a \times (b \times c) = 1 \times (3 \times 5) = 1 \times 15 = 15 \][/tex]
Both expressions result in 15, demonstrating the associative property of multiplication.
### 3. Identity Property
The identity property states that adding zero to a number does not change the number, and multiplying a number by one does not change the number.
#### Addition Example:
Let [tex]\( a = 9 \)[/tex].
[tex]\[ a + 0 = 9 + 0 = 9 \][/tex]
This shows the identity property of addition, where 9 remains unchanged.
#### Multiplication Example:
Let [tex]\( a = 7 \)[/tex].
[tex]\[ a \times 1 = 7 \times 1 = 7 \][/tex]
This shows the identity property of multiplication, where 7 remains unchanged.
### 4. Inverse Property
The inverse property states that every number has an additive inverse (a number that, when added to it, results in zero) and a multiplicative inverse (a number that, when multiplied to it, results in one).
#### Addition Example:
Let [tex]\( a = 12 \)[/tex].
[tex]\[ a + (-a) = 12 + (-12) = 0 \][/tex]
This shows the inverse property of addition, where the sum is 0.
#### Multiplication Example:
Let [tex]\( a = 8 \)[/tex].
[tex]\[ a \times \frac{1}{a} = 8 \times \frac{1}{8} = 1 \][/tex]
This shows the inverse property of multiplication, where the product is 1.
### 5. Distributive Property
The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.
Let [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 4 \)[/tex].
[tex]\[ a \times (b + c) = 2 \times (3 + 4) = 2 \times 7 = 14 \][/tex]
[tex]\[ (a \times b) + (a \times c) = (2 \times 3) + (2 \times 4) = 6 + 8 = 14 \][/tex]
Both expressions result in 14, demonstrating the distributive property.
Here is a summary in table form:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Property} & \text{Addition Example} & \text{Multiplication Example} \\ \hline \text{Commutative} & 3 + 5 = 8 & 4 \times 6 = 24 \\ \hline \text{Associative} & (2 + 4) + 6 = 12 & (1 \times 3) \times 5 = 15 \\ \hline \text{Identity} & 9 + 0 = 9 & 7 \times 1 = 7 \\ \hline \text{Inverse} & 12 + (-12) = 0 & 8 \times \frac{1}{8} = 1 \\ \hline \text{Distributive} & 2 \times (3 + 4) = 14 & 2 \times (3 + 4) = 14 \\ \hline \end{array} \][/tex]
### 1. Commutative Property
The commutative property states that the order in which you add or multiply two numbers does not affect the result.
#### Addition Example:
Let [tex]\( a = 3 \)[/tex] and [tex]\( b = 5 \)[/tex].
[tex]\[ a + b = 3 + 5 = 8 \][/tex]
[tex]\[ b + a = 5 + 3 = 8 \][/tex]
Both expressions result in 8, demonstrating the commutative property of addition.
#### Multiplication Example:
Let [tex]\( a = 4 \)[/tex] and [tex]\( b = 6 \)[/tex].
[tex]\[ a \times b = 4 \times 6 = 24 \][/tex]
[tex]\[ b \times a = 6 \times 4 = 24 \][/tex]
Both expressions result in 24, demonstrating the commutative property of multiplication.
### 2. Associative Property
The associative property states that the way in which numbers are grouped when adding or multiplying does not affect the result.
#### Addition Example:
Let [tex]\( a = 2 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = 6 \)[/tex].
[tex]\[ (a + b) + c = (2 + 4) + 6 = 6 + 6 = 12 \][/tex]
[tex]\[ a + (b + c) = 2 + (4 + 6) = 2 + 10 = 12 \][/tex]
Both expressions result in 12, demonstrating the associative property of addition.
#### Multiplication Example:
Let [tex]\( a = 1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 5 \)[/tex].
[tex]\[ (a \times b) \times c = (1 \times 3) \times 5 = 3 \times 5 = 15 \][/tex]
[tex]\[ a \times (b \times c) = 1 \times (3 \times 5) = 1 \times 15 = 15 \][/tex]
Both expressions result in 15, demonstrating the associative property of multiplication.
### 3. Identity Property
The identity property states that adding zero to a number does not change the number, and multiplying a number by one does not change the number.
#### Addition Example:
Let [tex]\( a = 9 \)[/tex].
[tex]\[ a + 0 = 9 + 0 = 9 \][/tex]
This shows the identity property of addition, where 9 remains unchanged.
#### Multiplication Example:
Let [tex]\( a = 7 \)[/tex].
[tex]\[ a \times 1 = 7 \times 1 = 7 \][/tex]
This shows the identity property of multiplication, where 7 remains unchanged.
### 4. Inverse Property
The inverse property states that every number has an additive inverse (a number that, when added to it, results in zero) and a multiplicative inverse (a number that, when multiplied to it, results in one).
#### Addition Example:
Let [tex]\( a = 12 \)[/tex].
[tex]\[ a + (-a) = 12 + (-12) = 0 \][/tex]
This shows the inverse property of addition, where the sum is 0.
#### Multiplication Example:
Let [tex]\( a = 8 \)[/tex].
[tex]\[ a \times \frac{1}{a} = 8 \times \frac{1}{8} = 1 \][/tex]
This shows the inverse property of multiplication, where the product is 1.
### 5. Distributive Property
The distributive property states that multiplying a sum by a number gives the same result as multiplying each addend by the number and then adding the products.
Let [tex]\( a = 2 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 4 \)[/tex].
[tex]\[ a \times (b + c) = 2 \times (3 + 4) = 2 \times 7 = 14 \][/tex]
[tex]\[ (a \times b) + (a \times c) = (2 \times 3) + (2 \times 4) = 6 + 8 = 14 \][/tex]
Both expressions result in 14, demonstrating the distributive property.
Here is a summary in table form:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Property} & \text{Addition Example} & \text{Multiplication Example} \\ \hline \text{Commutative} & 3 + 5 = 8 & 4 \times 6 = 24 \\ \hline \text{Associative} & (2 + 4) + 6 = 12 & (1 \times 3) \times 5 = 15 \\ \hline \text{Identity} & 9 + 0 = 9 & 7 \times 1 = 7 \\ \hline \text{Inverse} & 12 + (-12) = 0 & 8 \times \frac{1}{8} = 1 \\ \hline \text{Distributive} & 2 \times (3 + 4) = 14 & 2 \times (3 + 4) = 14 \\ \hline \end{array} \][/tex]
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