Problem 10 - Intermediate Algebra

Question

Complete each property of addition. Then give its name. a. \(a+(-a)=\square\) b. \(a+0=\square\) c. \(a+b=b+\square\) d. \((a+b)+c=a+\square\)

Step-by-Step Solution

Answer

a: 0 (Additive Inverse), b: a (Identity), c: a (Commutative), d: (b+c) (Associative).

Step 1: Solving Part a

The expression given is \(a + (-a)\). The property being described here is the additive inverse property, which states that the sum of a number and its opposite is zero. Therefore, \(a + (-a) = 0\).

Step 2: Identify the Property for Part a

The answer from the previous step utilized the additive inverse property, which states that \(a + (-a) = 0\). This property shows that adding a number to its inverse results in zero.

Step 3: Solving Part b

The expression given is \(a + 0\). The property here is the identity property of addition, which states that adding zero to any number does not change the value of that number. Therefore, \(a + 0 = a\).

Step 4: Identify the Property for Part b

The identity property of addition was used in the previous step, showing that \(a + 0 = a\). This property indicates that zero is the identity element in addition.

Step 5: Solving Part c

The expression given is \(a + b = b + \square\). The property here is the commutative property of addition, which states that the order of addition does not matter. Therefore, \(a + b = b + a\).

Step 6: Identify the Property for Part c

The commutative property of addition was used to solve part c, indicating that \(a + b = b + a\) because addition is independent of order.

Step 7: Solving Part d

The expression given is \((a+b) + c = a + \square\). The property here is the associative property of addition, which allows us to regroup numbers in addition without affecting the sum. Therefore, \((a+b) + c = a + (b+c)\).

Step 8: Identify the Property for Part d

The associative property of addition was used in the previous step, showing that \((a+b) + c = a + (b+c)\). This property states that how numbers are grouped in addition does not change their sum.

Key Concepts

Additive Inverse

In the realm of numbers, every number has an "opposite" that, when added together, results in zero. This is known as the additive inverse. The concept is simple: for any number "a," its additive inverse is "-a." When you add these two together, the result is zero: \[ a + (-a) = 0 \].
The additive inverse is a helpful concept in everything from basic arithmetic to advanced mathematical ideas. It ensures that every number can be "cancelled out" by its additive counterpart. For example:

• 3 + (-3) = 0
• -5 + 5 = 0
• 0 + 0 = 0

This property is invaluable in solving equations and simplifying expressions.

Identity Property

The identity property of addition is about discovering that special number which, when added, doesn’t alter other numbers. That number is zero. The identity property tells us that adding zero to any number leaves the number unchanged. For instance:

• 8 + 0 = 8
• 0 + 15 = 15
• -7 + 0 = -7

In equation form, it is expressed as:\[ a + 0 = a \]
Zero acts as a neutral element or the identity element in addition because it doesn’t disturb the original number. This property is commonly used in simplifying and solving equations.

Commutative Property

The commutative property of addition comes in handy by stating that the order in which you add numbers doesn’t affect their sum. Changing the sequence of numbers won't change the result:

• a + b = b + a
• 4 + 2 = 2 + 4 = 6
• -3 + 5 = 5 + (-3) = 2

This property helps simplify calculations and lets us rearrange terms for easier computation. The commutative property is a foundational aspect of arithmetic, providing flexibility with the arrangement of numbers.

Associative Property

The associative property of addition tells us that how numbers are grouped doesn’t change the sum. Whether you add numbers in pairs or change grouping, the sum remains the same. For example:

• (a + b) + c = a + (b + c)
• (2 + 3) + 4 = 2 + (3 + 4) = 9
• (1 + 7) + 2 = 1 + (7 + 2) = 10

This property enables us to regroup numbers in a way that's most convenient for computation. It's important in simplifying equations, especially when dealing with more than two numbers. This property showcases the flexibility and simplicity of addition.