Question

Докажите , что минус на минус даёт плюс.

Answer 1

-1 - это число, противоположное числу 1, то оно таково, что -1 + 1 = 0. Его существование следует из соответствующей аксиомы. Из другой несложной теоремы следует, что -1 и 1 взаимно противоположны, то есть противоположное к -1 это 1.
Теперь посчитаем:
(-1) +(-1)*(-1) = (-1)*1+(-1)*(-1)=(-1)(1+(-1))=-1*0=0 (каждая операция выполняется по аксиомам)
В результате получаем, что (-1) +(-1)*(-1)=0, то есть (-1) и (-1)*(-1)-противоположные. Но мы знаем, что к -1 противоположна 1, и знаем, что противоположное число всегда единственно, значит (-1)*(-1)=1

Answer 2

Introduction

In mathematics, the concept of negative numbers is well-defined. When we multiply two negative numbers, the result is positive. However, to provide a more detailed explanation and proof, let's explore the concept further.

Understanding Negative Numbers

Negative numbers are numbers less than zero. They are denoted by a minus sign (-) placed before the number. For example, -3 is a negative number.

Multiplication of Negative Numbers

To understand why multiplying two negative numbers gives a positive result, let's consider the following examples:

1. (-1) x (-1): When we multiply -1 by -1, we get 1 as the result. This can be seen as a simple case of multiplication, where a negative number multiplied by a negative number gives a positive result.

2. (-2) x (-3): When we multiply -2 by -3, we get 6 as the result. This can be understood by breaking down the multiplication as follows: - (-2) x (-3) = (-1) x 2 x (-1) x 3 - Rearranging the terms: (-1) x (-1) x 2 x 3 - Multiplying the positive numbers: 1 x 2 x 3 = 6

We can see that the negative signs cancel out, resulting in a positive product.

Proof

To provide a more formal proof, we can use the concept of additive inverses. The additive inverse of a number is the number that, when added to the original number, gives a sum of zero.

Let's consider two negative numbers, a and b. The additive inverse of a is -a, and the additive inverse of b is -b.

When we multiply a by b, we can express it as the sum of a multiplied by -b: a x b = a x (-b)

Now, let's consider the additive inverse of a, which is -a. We can express it as the sum of -a multiplied by -b: -a x (-b)

Using the distributive property of multiplication over addition, we can expand the expression: -a x (-b) = (-a) x (-b) + a x (-b)

Since a x (-b) is the same as -a x b, we can substitute it in the equation: -a x (-b) = (-a) x (-b) + (-a) x b

Now, let's rearrange the terms: -a x (-b) = (-a) x (-b) - a x b

Since the left side of the equation is equal to the right side, we can simplify it: 0 = (-a) x (-b) - a x b

Rearranging the terms, we get: a x b = (-a) x (-b)

This equation shows that the product of two negative numbers, a and b, is equal to the product of their additive inverses, -a and -b. Since the product of two positive numbers is positive, we can conclude that the product of two negative numbers is also positive.

Conclusion

In conclusion, multiplying two negative numbers gives a positive result. This can be understood intuitively or proven using the concept of additive inverses. The negative signs cancel out during multiplication, resulting in a positive product.