Listening to teachers of mathematics, most students perceive the material as an axiom.But few people trying to get to the bottom and find out why the "minus" to "plus" gives the "minus" sign, and the multiplication of two negative numbers comes out positive.
laws of mathematics
Most adults can not explain to themselves or to their children why this is so.They firmly grasp this stuff in school, but did not even try to figure out where did these rules.And for good reason.Often, today's children are not so gullible, they need to get to the bottom and understand, for example, why the "plus" to "negative" gives a "minus".And sometimes urchins specifically ask tricky questions, in order to enjoy the time when adults can not give a clear answer.And it really matter if a young teacher gets trapped ...
way, it should be noted that the above-mentioned rule is effective for both multiplication and division to.The work of negative and positive numbers give only a "minus.If there are two numbers with the sign "-", the result is a positive number.The same applies to the division.If one of the numbers is negative, then the quotient will also be with the sign "-".
to explain the correctness of the law of mathematics, it is necessary to formulate the axiom rings.But first need to understand what it is.In mathematics, the ring is called a set, which involved two operations with two elements.But to understand it better with an example.
axiom rings
There are several mathematical laws.
- commutative first of these, according to him, C + V = V + C.
- second called associative (V + C) + D = V + (C + D).
He also obeys and multiplication (V x C) x D = V x (C x D).
Nobody canceled and the rules by which the opening brace (V + C) x D = V x D + C × D, it is also true that C × (V + D) = C x V + C x D.
Furthermore, it was found that the ring can enter a special neutral by addition of an element, the use of which the following is true: C + 0 = C. In addition, for each C has the opposite element, which can be designated as (-C).This C + (-C) = 0.
Withdrawal axioms for negative numbers
Taking the above statements, it is possible to answer the question: "" plus "to" negative "gives a sign?" Knowing the axiom about the multiplication of negative numbers,you must confirm that indeed (-C) x V = - (C x V).And that is true equality: (- (- C)) = C.
It will have to first prove that each element has only one opposite him "brother."Consider the following evidence.Let's try to imagine what the C opposite are two numbers - V and D. From this it follows that C + V = 0 and C + D = 0, ie C + V = 0 = C + D. Recalling the commutative law andon the properties of the numbers 0, we can consider the sum of the three numbers: C, V, and D. Let's try to figure out the value of V. Logically, V = V + 0 = V + (C + D) = V + C + D, because the value of the C +D, as has been made above, equals 0. Thus, V = V + C + D.
Similarly, output and value for D: D = V + C + D = (V + C)+ D = 0 + D = D. On this basis, it is clear that V = D.
In order to understand why all the "plus" to "negative" gives a "minus" sign, it is necessary to understand the following.Thus, for an element (-C) are opposite and C (- (- C)), i.e. they are equal to each other.
then obvious that 0 x V = (C + (-C)) = C x V x V + (-C) x V. From this it follows that C x V opposite to (-) C x V, therefore,(-C) x V = - (C x V).
For complete mathematical rigor must also confirm that V = 0 x 0 for any element.If you follow the logic, 0 x V = (0 + 0) x V = 0 V + x 0 x V. This means that the addition of the product 0 × V does not change the prescribed amount.After all this work is zero.
Knowing all of these axioms can be derived not only as the "plus" to "negative" provides, but that is obtained by multiplying negative numbers.
multiplication and division of two numbers with the sign «-»
If you do not go into the mathematical nuances, you can try a simple way to explain the rules of operations with negative numbers.
Assume that C - (-V) = D, on the basis of this, C = D + (-V), that is, C = D - V. We transfer V and get that C + V = D. That is, C+ V = C - (-V).This example explains why the expression, where there are two "minus" in a row, said the signs should be changed to "plus".Now let's deal with multiplication.
(-C) x (-V) = D, in the expression, you can add and subtract two identical pieces that do not change its value: (-C) x (-V) + (C × V) - (C × V) = D.
to remember the rules of work with parentheses, we get:
1) (-C) x (-V) + (C × V) + (-C) x V = D;
2) (-C) x ((-V) + V) + C x = V D;
3) (-C) + C x 0 x = V D;
4) V = C x D.
From this it follows that C x V = (-C) x (-V).
Similarly, we can prove that as a result of the division of two negative numbers come out positive.
general mathematical rules
Of course, this explanation is not suitable for primary school children who are just beginning to learn abstract negative numbers.They'd better explain to the visible objects, manipulating them familiar term through the mirror.For example, invented, but there are toys there.They can be displayed and the sign "-".Multiplication of two objects transmirror transfers them to another world, which is equal to the present, that is, as a result, we have positive numbers.But the multiplication of abstract negative number to a positive only provides all the familiar result.After all, the "plus" multiplied by "minus" gives the "minus".However, in the primary school age children are not too try to understand all the nuances of mathematics.
Although, face it, for many people, even with higher education and many of the rules remain a mystery.All take it for granted that teachers teach them, will not complicate to delve into the complexities inherent in the mathematics."Negative" to "negative" gives "plus" - know about it all, without exception.This is as true for the whole, and for fractional numbers.
