Math Problem Statement

prove that -a = -1 times a

Solution

To prove that a=(1)×a-a = (-1) \times a, 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 aa is defined as the number that, when added to aa, gives zero:

a+(a)=0a + (-a) = 0

This is the definition of the additive inverse of aa.

Step 2: Associative Property of Multiplication

We now use the associative property of multiplication, which states that:

x×(y+z)=(x×y)+(x×z)x \times (y + z) = (x \times y) + (x \times z)

Let’s substitute x=1x = -1, y=ay = a, and z=az = -a into this property:

(1)×(a+(a))=(1)×a+(1)×(a)(-1) \times (a + (-a)) = (-1) \times a + (-1) \times (-a)

Step 3: Simplifying Using the Definition of Negative

Since we know that:

a+(a)=0a + (-a) = 0

we substitute this into the equation:

(1)×0=(1)×a+(1)×(a)(-1) \times 0 = (-1) \times a + (-1) \times (-a)

And since (1)×0=0(-1) \times 0 = 0, the equation becomes:

0=(1)×a+(1)×(a)0 = (-1) \times a + (-1) \times (-a)

Step 4: Understanding the Product (1)×(a)(-1) \times (-a)

Now, by the definition of multiplication with negative numbers, we know that multiplying two negative numbers results in a positive number. Hence:

(1)×(a)=a(-1) \times (-a) = a

Substitute this into the equation:

0=(1)×a+a0 = (-1) \times a + a

Step 5: Factor Out aa

0=a×(1+1)0 = a \times (-1 + 1)

Since 1+1=0-1 + 1 = 0, we have:

0=a×00 = a \times 0

Which holds true, proving that:

(1)×a=a(-1) \times a = -a

Thus, we have shown that:

a=(1)×a-a = (-1) \times a

Conclusion

We have used the properties of numbers and multiplication to prove that a=(1)×a-a = (-1) \times a.


Would you like further details or any clarification?

Here are 5 related questions for you to explore:

  1. How do we prove that (a)=a-(-a) = a?
  2. What is the distributive property of multiplication over addition?
  3. Can you show how multiplication of negative numbers works with examples?
  4. How do we apply the concept of additive inverses in equations?
  5. What is the difference between the additive inverse and the multiplicative inverse?

Tip: Always remember the fundamental property that multiplying by 1-1 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