Math Problem Statement
prove that -a = -1 times a
Solution
To prove that , we can break it down step by step using basic properties of numbers and multiplication:
Step 1: Definition of Negative Numbers
The negative of a number is defined as the number that, when added to , gives zero:
This is the definition of the additive inverse of .
Step 2: Associative Property of Multiplication
We now use the associative property of multiplication, which states that:
Let’s substitute , , and into this property:
Step 3: Simplifying Using the Definition of Negative
Since we know that:
we substitute this into the equation:
And since , the equation becomes:
Step 4: Understanding the Product
Now, by the definition of multiplication with negative numbers, we know that multiplying two negative numbers results in a positive number. Hence:
Substitute this into the equation:
Step 5: Factor Out
Since , we have:
Which holds true, proving that:
Thus, we have shown that:
Conclusion
We have used the properties of numbers and multiplication to prove that .
Would you like further details or any clarification?
Here are 5 related questions for you to explore:
- How do we prove that ?
- What is the distributive property of multiplication over addition?
- Can you show how multiplication of negative numbers works with examples?
- How do we apply the concept of additive inverses in equations?
- What is the difference between the additive inverse and the multiplicative inverse?
Tip: Always remember the fundamental property that multiplying by changes the sign of a number.
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Math Problem Analysis
Mathematical Concepts
Algebra
Properties of Negative Numbers
Multiplication
Formulas
a + (-a) = 0
(-1) × (a + (-a)) = (-1) × a + (-1) × (-a)
(-1) × (-a) = a
Theorems
Additive Inverse
Associative Property of Multiplication
Suitable Grade Level
Grades 7-9