Negative x Negative = Positive? Abstract Proofs – The Math Doctors
Last time we looked at explanations for the product of negative numbers in terms of various concrete models or examples. But it really requires a mathematical proof, as we’ll explain and demonstrate here, first with a couple different proofs, then with the bigger picture, giving the context of such proofs.
It has to be that way
First, from 1998:
Why Does a Negative Times a Negative Equal a Positive? How does a negative number times another negative number equal a positive number?
Doctor Bruce answered:
Hello Jessica, I detect in your question a measure of annoyance at having to learn the rule for multiplying negative numbers. I'll bet it seems like someone just made the rule up out of thin air, with no particular reason why the answer should be positive. I want to reassure you that this rule is not just "made up." There is a chain of reasoning -- a mathematical "argument" -- that shows why the rule *has* to be that negative times negative equals positive.
If someone did just decree this “rule”, then it would be annoying, wouldn’t it? But math is not about arbitrary rules; it’s about reasoning from basic assumptions or known facts, to less obvious facts.
What follows is a proof, presented in a style intended for students who are not familiar with proofs. We have to start with known facts about multiplication (assumptions or axioms):
Mathematical argument takes a little getting used to. This might look rather strange at first. Here's how the reasoning goes: (1) Zero times anything equals zero. (2) Every number has exactly one additive inverse. This means if N is a positive number, then -N is its additive inverse, so that N + (-N) = 0. Likewise, the additive inverse of -N is N. (3) We want negative numbers to obey the distributive law. This says that a*(b+c) = a*b + a*c.
The first and third facts are true of multiplication as we know it; in adding the concept of negative numbers (that is, the additive inverse) to the arithmetic we are already familiar with, we don’t want to change these facts.
Positive times negative
We can show that these facts imply what multiplication of negative numbers has to look like, in two steps. First:
(4) Now, we are forced to accept a new law, that negative times positive equals negative. This is because we can use the distributive law on an expression like 2*(3 + (-3)). This equals 2*(0), which is zero. But by the distributive law, it also equals 2*3 + 2*(-3). So 2*(-3) does the job of the additive inverse of 2*3, and therefore 2*(-3) is the additive inverse of 2*3. But the additive inverse of 6 is just -6. So 2 times -3 equals -6.
There is only one additive inverse of a number; anything that does what \(-6\) does must be \(-6\).
We can write this in the general case, supposing that m and n are any two positive numbers: $$0=m\times 0=m\times(n+(-n))=m\times n+m\times(-n)\\ \Rightarrow\;m\times(-n)=-(m\times n)$$ So a positive times a negative is a negative (and, by the commutative property, a negative times a positive is negative).
Negative times negative
But we can repeat the same process, using the fact just demonstrated:
(5) Next, we are forced to accept another new law, that negative times negative equals positive. It's a lot like the example in (4). We use the distributive law on, say, -3*(5 + (-5)). This is again equal to zero. But by the distributive law, it also equals -3*5 + (-3)*(-5). We know the first thing, (-3*5) equals -15 because of the law in (4). So (-3)*(-5) is doing the job of the additive inverse of -15. We know -15 has exactly one additive inverse, namely 15. Therefore, (-3)*(-5) = 15.
Again, $$0=(-m)\times 0=(-m)\times(n+(-n))\\=(-m)\times n+(-m)\times(-n)\\=-(m\times n)+(-m)\times(-n)\\ \Rightarrow\;(-m)\times(-n)=-(-(m\times n))=m\times n$$
I hope this doesn't frighten you! The main thing is, keep right on questioning the things that don't make sense. In mathematics, you are always entitled to an explanation of WHY things are the way your teacher (or I) say they are.
of negative is positive
Next, a slightly different question from 2001:
Prove That -(-a) = a Why does -(-a) = a? How do you prove this using the properties of real numbers?
Doctor Bruce had passed over this lightly; we’ll go a little deeper and further here.
Doctor Rick answered:
Hi, Eduardo. We can start with how -a is defined. It is "the additive inverse of a" - that is, it is the number that, when added to a, gives 0: a + -a = 0 Therefore -(-a) means the number that, when added to -a, gives 0. But applying the commutative property of addition, the equation above becomes -a + a = 0 Therefore the number that, when added to -a, gives 0 is a; or, -(-a) = a
That’s all it takes; it follows immediately from the definition. But notice that this doesn’t just tell us that the negative of a negative number is positive; our variable a could be negative itself! What we have proved is that changing sign twice returns us to where we started, wherever that was.
Multiplying by negative 1
From that basic fact, we can take a step toward multiplication:
A closely related, but different, question is how we can prove that -1 * -1 = 1 The theorem linking these two is this: -1 * a = -a
We’re working our way up, bit by bit, to the general fact. To some people it can seem obvious that multiplying by -1 changes the sign, but it needs to be proved; and proving this little fact will make everything else easy. We’ll use ideas very much like Doctor Bruce’s:
Let's prove this. Start with the fact that zero times any number is zero: 0 * a = 0 Write 0 as (1 + -1), which follows from the definition of -1. (1 + -1)*a = 0 Apply the distributive property: 1*a + -1*a = 0 Use the fact that 1 times any number is the same number: a + -1*a = 0 Now, the number that, when added to a, gives 0 is -a. Therefore -a = -1*a Using this theorem, we can easily prove that -1 times -1 is 1: -1 * -1 = -(-1) = 1
From here we could go all the way to the general fact about negatives time negatives, by just bringing in the commutative and associative properties: $$(-a)\times(-b)=((-1)\times a)\times((-1)\times b)\\=(-1)\times a\times(-1)\times b\\=((-1)\times(-1))\times(a\times b)\\=1\times(a\times b)\\=a\times b$$
Discarding old models
These proofs can be a little dizzying. The next question, from 2003, asks us to back up and think about what’s happening.
Why Does a Negative Times a Negative Make a Positive? Why is a negative times a negative a positive? When you answered this question to other people, the way you explained it was over my head. When you have a negative integer, in order to multiply by another negative number why would you have to go to the positive side of the number line?
Doctor Ian answered:
Hi Brea, It's really hard to understand this in terms of a number line. One of the ways math works is that we make up simple ways of thinking about some set of numbers; but those break down when we try to think of other sets of numbers. So then we have to make up new ways of visualizing our expanded sets.
We aren’t yet talking about the question itself, but about the need to change our thinking when we apply old ideas to a new kind of number. We’re used to positive numbers; expanding to negative numbers requires a new perspective.
First, another example of such an expansion:
The easiest example is fractions. At first, you learn to think of something like 3/4 as meaning "break an item into 4 pieces, and keep 3 of them". That works okay as long as you're dealing with integers. But what would sqrt(2) ------- pi mean in those terms? How would you break something into 'pi' pieces? And how would you keep sqrt(2) of them? Eventually, you have to let go of the 'dividing into pieces' model, and just start thinking of fractions as divisions that you haven't done yet.
This was discussed in Fractions: What Are They, and Why?, and in How to Convert a Fraction to a Decimal – and Why.
Similarly, the number line is pretty good for thinking about addition and subtraction, but it's not so good for thinking about multiplication. To be honest, I've never been able to think of a way that is both intuitive and rigorously correct, to explain why the product of two negatives is positive. There are nice ways to visualize the result, but they don't explain why it _has_ to be that way.
The examples we looked at last time are attempts to make this idea intuitive, but they can all leave a student unconvinced. What we need is … a proof.
A compact proof
Doctor Ian here gives a proof stripped of the words of explanation we’ve used until now, so you have to think more carefully about why each step is true, and even why he would start the way he does; this is typical of mathematicians’ proofs. From the proofs we’ve seen before, you should be able to see most of the reasons; what you’ll discover is essentially that he’s combined the two steps of Doctor Bruce’s version into one.
There is a pretty simple proof, which goes like this: Let a and b be any two real numbers. Consider the number x defined by x = ab + (-a)(b) + (-a)(-b). We can write x = ab + (-a)[ (b) + (-b) ] (factor out -a) = ab + (-a)(0) = ab + 0 = ab. Also, x = [ a + (-a) ]b + (-a)(-b) (factor out b) = 0 * b + (-a)(-b) = 0 + (-a)(-b) = (-a)(-b). So we have x = ab and x = (-a)(-b) Two things that are equal to the same thing are equal to each other, so ab = (-a)(-b)
This is an interesting style of proof, where we evaluate one expression in two different ways and get different results, which must therefore be equal. But such “elegance” can be very hard to follow; and you could only invent it by first exploring the ideas we’ve seen above.
He followed the proof with an attempted model, which I will omit; I wouldn’t find it very convincing.
But he closes with something useful:
So what about the remaining possibility? Well, so far, we've seen that changing the sign of _either_ factor changes the sign of the product. That is, in the diagram below, (+) | II | I | (-)-------- --------(+) | III | IV | (-) as we move from quadrant I to quadrant II, we change signs; and as we move from quadrant I to quadrant IV, we change signs. So we should also change signs as we move from II to III, or from IV to III, right? But if we do that, we end up with a positive area in quadrant III... which is just another way of saying that the product of two negatives should be a positive.
Changing the sign of one factor changes the sign of the product; so changing the sign of both should change it back to positive. That’s not proof, but plausibility.
So maybe that's another way to think about it. If we take _anything_, and multiply it by -1, we should end up with a different sign, shouldn't we? So let's start with something like a * b = c where a, b, and c are all positive. If we multiply that by -1, we have to end up with a different sign: -1 * (a * b) = -c and if we multiply by -1 again, we have to end up with a different sign again: -1 * (-1 * (a * b)) = c Does that make sense so far? If we _don't_ do this, then sometimes multiplying by -1 would change the sign, and sometimes it wouldn't. That would be pretty strange.
This assumes the facts that Doctor Rick proved in our second proof, both easy to believe.
So if c = -1 * (-1 * (a * b)) then, since we can group and order multiplications any way we want, c = -1 * (-1 * (a * b)) = -1 * -1 * a * b = -1 * a * -1 * b = (-1 * a) * (-1 * b) = (-a) * (-b) Do any of these explanations work for you? If not, I can try to find another one that does.
Model with tiles? Or not
I’ll close with one more question, from 2001, asking for a model like those we used last time:
Algebra Tiles and Negatives How can we use a model (algebra tiles) to demonstrate that a negative times a negative = a positive? I tried to show repeated addition but that doesn't work. For example, -3(-5). How do you represent -3 sets of -5 at a 7th grade level?
Algebra tiles (see here) are little squares representing positive numbers by one color (say, yellow) and negative by another (red). (They also represent variables, but that is not involved at this level.) A \(+1\) cancels a \(-1\) in a way that resembles antimatter annihilation, but with less energy (depending on the kid who is using them). You could represent \(3\times-5\) as three sets of 5 “\(-1\)”s. But how can you make a negative number of groups?
This question would belong in last week’s post on models … but Doctor Ian didn’t keep the focus on the model:
Hi David, The truth is, models only go so far, and it's not clear that there is a way to model the fact that -1 * -1 = +1 that doesn't create more confusion than it clears up. One thing you might try is modeling 'this times that' in the following way: When 'this' is positive, add copies. When 'this' is negative, take away copies. When 'that' is positive, use items. When 'that' is negative, use holes. So, 3 times 4: 1. Start with 4 items * * * * 2. Add copies * * * * * * * * * * * * -3 times 4: 1. Start with 4 items * * * * 2. Take away copies, leaving holes o o o o o o o o o o o o 3 times -4: 1. Start with 4 holes o o o o 2. Add copies o o o o o o o o o o o o -3 times -4 1. Start with 4 holes o o o o 3. Take away holes... by adding copies! * * * * * * * * * * * *
His “items” and “holes” correspond to the yellow and red tiles in the set, and that terminology is a nice way to think of the tiles. But adding and taking away, as in some of our examples last time, can just be confusing. (In the site I linked about algebra tiles, subtraction is not modeled by taking away, but by explicitly adding the opposite; and multiplication is done by changing the sign of the result – “flipping” tiles, if they had red on only one side; so no attempt is made to motivate those rules.)
But even this isn't very satisfying, because it's not any easier to think of removing holes by adding copies than it is to remember that -1 * -1 = +1. (You could use electrical charges as a model -- removing a negative charge has the same effect as adding a positive one -- but many students find electricity even more confusing than math.) And even if it were satisfying, by trying to reduce this to something that can be modeled with physical items, you would be missing the chance to make a very important point about one of the ways in which mathematics grows.
This is a key insight. Math goes beyond models, taking us into the world of the mind.
Don’t break anything!
Rather than trying to justify the rules by the model, he turns to an abstract proof:
If we can agree that a negative number is just a positive number multiplied by -1, then we can always write the product of two negative numbers this way: (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)ab For example, -2 * -3 = (-1)(2)(-1)(3) = (-1)(-1)(2)(3) = (-1)(-1) * 6
This is Doctor Rick’s approach starting with \((-1)\times a = -a\), though without formally proving that. In effect, he is taking it as an axiom, because it is easy to agree to.
So the real question is, (-1)(-1) = ? and the answer is that the following convention has been adopted: (-1)(-1) = +1 This convention has been adopted for the simple reason that any other convention would cause something to break. For example, if we adopted the convention that (-1)(-1) = -1, the distributive property of multiplication wouldn't work for negative numbers: (-1)(1 + -1) = (-1)(1) + (-1)(-1) (-1)(0) = -1 + -1 0 = -2 As Sherlock Holmes observed, "When you have excluded the impossible, whatever remains, however improbable, must be the truth. " Since everything except +1 can be excluded as impossible, it follows that, however improbable it seems, (-1)(-1) = +1.
He emphasizes more strongly than we have done elsewhere, that the “rule” is just a convention, an agreement we make as a way of defining a multiplication that, until we do this, has no definition! But the convention is not arbitrary; it is chosen in order not to break the existing behavior of operations, just as we have said before.
If that seems too abstract, try to think of it in terms of a murder mystery. We've considered all the other possible suspects, and we know that none of them could have committed the murder; and we have no reason for believing that '1' didn't commit the murder; and _someone_ did it; so '1' must be guilty. Also, if you just want to teach your students a good way to remember that a negative times a negative must be positive, they can use this trick: -a(b + -b) = (-a)(b) + (-a)(-b) \________/ \_____/ \______/ 0 neg must be pos
This is a quick version of the distributive property proofs we’ve seen.
Thank you for your response to my question I thought that I was going to go insane. I also thought that I was the only person having problems explaining why a negative times a negative was a positive. It helps me to rest a little easier knowing that there are a lot of people that have wrestled with this question.
We can attest to that! This is one of our more frequently asked questions!
Positive and Negative Numbers | SkillsYouNeed
Standard numbers, anything greater than zero, are described as ‘positive’ numbers. We don’t put a plus sign (+) in front of them because we don’t need to since the general understanding is that numbers without a sign are positive.
Numbers that are less than zero are known as ‘negative’ numbers. These have a minus sign (−) in front of them to indicate that they are less than zero (for example, -10 or ‘minus 10‘).
Visualising Negative and Positive Numbers
Probably the easiest way to visualise negative and positive numbers is using a number line, a tool with which you may well be familiar, especially if you have children at primary school.
It looks something like this:
A number line can help you to visualise both positive and negative numbers and the operations (adding and subtracting) that you can do with them.
When you have an addition or subtraction to calculate, you start at the first number and move the second number of places either to the right (for an addition) or left (for a subtraction).
This number line is a simplified version, but you can draw them with every number included if you wish. The big advantage of a number line is that it is very easy to draw for yourself on the back of an envelope or a piece of scrap paper, and it is also quite hard to go wrong with the calculation. As long as you are careful to count the number of places that you are moving, you will reach the correct answer.
Subtracting Negative Numbers
If you subtract a negative number, the two negatives combine to make a positive.
−10−(−10) is not −20. Instead, you can think of it as turning one of the negative signs upright, to cross over the other, and make a plus. The sum would then be −10+10 = 0.
A Quick Note on Brackets
For clarity, you would never write two negative signs side by side without brackets.
So if you are asked to subtract a negative number, it will always have brackets around it so that you can see that the use of two negative signs was intentional.
-10—10 is incorrect (and confusing)
-10-(-10) is correct (and clearer)
Multiplying and Dividing with Positive and Negative Numbers
When multiplying or dividing with combinations of positive and negative numbers, you can simplify the process by first ignoring the signs (+/-) and just multiplying or dividing the numbers as if they were both positive. Once you have the numerical answer, then you can apply a very simple rule to determine the sign of the answer:
- When the signs of the two numbers are the same, the answer will be positive.
- When the signs of the two numbers are different, the answer will be negative.
(positive number) × (positive number) = positive number
(negative number) × (negative number) = positive number
(positive number) × (negative number) = negative number
As a side issue, this goes some way to explaining why you cannot have the square root of a negative number (there is more about this in our page on Special Numbers and Concepts). The square root is the number that is multiplied by itself to get the number. You cannot multiply a number by itself to get a negative number. To get a negative number, you need one negative and one positive number.
The rule works the same way when you have more than two numbers to multiply or divide. An even number of negative numbers will give a positive answer. An odd number of negative numbers will give a negative answer.
Why does multiplying two negatives give a positive answer?
The fact that a negative number multiplied by another negative number produces a positive result can often confuse and seem counterintuitive.
To explain why this is the case, think back to the number lines used earlier in this article since these help to explain this visually.
- First, imagine standing on the number line at point zero and facing towards the positive direction, i.e. towards 1, 2 and so on. You take two steps forwards, pause, then take two steps more. You have moved 2 × 2 steps = 4 steps.
Hence positive × positive = positive
- Now go back to zero, and face in the negative direction, i.e. towards −1, −2, etc. Take two steps forwards, then another two. You are now standing on −4. You have moved 2 × −2 steps = −4 steps.
Hence negative × positive = negative
In both of these examples, you have moved forwards (i.e. the direction you were facing), a positive move.
- Go back to zero again, but this time you are going to walk backwards (a negative move). Face the positive direction again and take two steps backwards. You are now standing on −2. A positive (the direction you are facing) and a negative (the direction you are moving) result in a negative move.
Hence positive × negative = negative
- Finally, back at zero again, face in the negative direction. Now take two steps backwards, and then another two backwards. You are standing on +4. By facing in the negative direction, and walking backwards (two negatives), you have achieved a positive result.
Hence negative × negative = positive
- Two negatives cancel each other out. You can see this in speech:
- “Just do it!” is a positive encouragement to do something.
- “Don’t do that!” is asking someone not to do something. It’s a negative.
- “Don’t not do it” means “please do it”. Two negatives cancel out and make a positive, in maths as well as in speech.
- The signs add together physically. When you have two negative signs, one turns over, and they add together to make a positive. If you have a positive and a negative, there is one dash left over, and the answer is negative. This is a simple and visual aide-mémoire, despite not necessarily being satisfying to those who want to understand the rule.
Negative signs can look a bit daunting, but the rules that govern their use are simple and straightforward. Keep these in mind, and you will have no problems.
Negative numbers are numbers with a minus sign (-), such as -1, -2, -3. It reads like: minus one, minus two, minus three.
An example of the use of negative numbers is a thermometer that indicates the temperature of the body, air, soil or water. In winter, when it is very cold outside, the temperature is negative (or, as the people say, «minus»).
For example, -10 degrees cold:
The usual numbers that we considered earlier such as 1, 2, 3 are called positive. Positive numbers are numbers with a plus sign (+).
When writing positive numbers, the + sign is not written, so we see the usual numbers for us 1, 2, 3. But it should be borne in mind that these positive numbers look like this: +1, +2, +3.
Coordinate line is a straight line on which all numbers are located: both negative and positive. Looks like this:
Only numbers from -5 to 5 are shown here. In fact, the coordinate line is infinite. The figure shows only a small fragment of it.
Numbers on the coordinate line are marked as dots. In the figure, the bold black dot is the starting point. The countdown starts from zero. To the left of the reference point, negative numbers are marked, and to the right, positive ones.
The coordinate line continues indefinitely on both sides. Infinity in mathematics is denoted by the symbol ∞. The negative direction will be denoted by the symbol −∞, and the positive one by the symbol +∞. Then we can say that all numbers from minus infinity to plus infinity are located on the coordinate line:
Each point on the coordinate line has its own name and coordinate. The name is any Latin letter. Coordinate is a number that indicates the position of a point on this line. Simply put, the coordinate is the same number that we want to mark on the coordinate line.
For example, point A (2) is read as «point A with coordinate 2 » and will be indicated on the coordinate line as follows:
Here A is the name of the point, 2 is the coordinate of the point A.
Example 2. Point B (4) is read as 900 05 «point B with coordinate 4 » and
Here B is the point name, 4 is the coordinate of the point B.
Example 3. Point M ( −3) is read as «point M with coordinate minus three» and will be denoted on the coordinate line like this:
Here M is the name of the point, −3 is the coordinate of the point M .
Points can be marked with any letters. But it is generally accepted to designate them with capital Latin letters. Moreover, the beginning of the report, which is otherwise called origin , is usually denoted by a capital Latin letter O
It is easy to see that negative numbers lie to the left of the origin, and positive numbers to the right.
There are such phrases as «the more to the left, the less» and «the more to the right, the more» . You probably already guessed what we are talking about. With each step to the left, the number will decrease downwards. And with each step to the right, the number will increase. The arrow pointing to the right indicates the positive direction of counting.
Compare negative and positive numbers
Rule 1. Any negative number is less than any positive number.
For example, let’s compare two numbers: -5 and 3. Minus five is less than than three, despite the fact that the five catches the eye in the first place, as a larger number than three.
This is because −5 is negative and 3 is positive. On the coordinate line, you can see where the numbers −5 and 3 are located
It can be seen that −5 lies to the left, and 3 to the right. And we said that «the more to the left, the less» . And the rule says that any negative number is less than any positive number. Hence it follows that
−5 < 3
“Minus five is less than three”
Rule 2. Of the two negative numbers, the one located to the left of the coordinate line is less.
For example, let’s compare the numbers −4 and −1. Minus four is less than than minus one.
This is again due to the fact that on the coordinate line -4 is located more to the left than -1
It can be seen that -4 lies to the left, and -1 to the right. And we said that «the further to the left, the smaller» . And the rule says that of two negative numbers, the one that is located to the left on the coordinate line is less. It follows that
−4 < −1
Minus four is less than minus one
Rule 3. Zero is greater than any negative number.
For example, let’s compare 0 and −3. Zero is greater than than minus three. This is due to the fact that on the coordinate line 0 is located to the right than −3
It can be seen that 0 lies to the right, and −3 to the left. And we said that «the more to the right, the more» . And the rule says that zero is greater than any negative number. It follows that
0 > −3
Zero is greater than minus three
Rule 4. Zero is less than any positive number.
For example, let’s compare 0 and 4. Zero is less than 4 . In principle, this is clear and true. But we will try to see it with our own eyes, again on the coordinate line:
It can be seen that on the coordinate line 0 is located to the left, and 4 to the right. And we said that «the more to the left, the less» . And the rule says that zero is less than any positive number. It follows that
0 < 4
Zero is less than four
Tasks for independent solution
Task 1. Compare the numbers −2 and 1
−2 < 1
Task 2 Compare the numbers −5 and −2
−5 < −2
Show the solution
Task 3. Compare the numbers −5 and −16
−5 > −16
Show the solution
Task 4. Compare the numbers 15 and 20
90 002 15 < 20
Show the solution
Task 5. Compare the numbers −7 and 0
−7 < 0
Show the solution
Task 6. Compare the numbers 5 and 0
Show the solution
Task 7. Compare the numbers 5 and 7
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Why does a minus times a minus give a plus?
«The enemy of my enemy is my friend.»
The easiest answer is: «Because these are the rules for dealing with negative numbers.» The rules we learn in school and apply throughout our lives. However, the textbooks do not explain why the rules are the way they are. We will first try to understand this from the history of the development of arithmetic, and then we will answer this question from the point of view of modern mathematics.
A long time ago, people knew only natural numbers: 1, 2, 3, … They were used to count utensils, booty, enemies, etc. But the numbers themselves are rather useless — you need to be able to handle them. Addition is clear and understandable, besides, the sum of two natural numbers is also a natural number (a mathematician would say that the set of natural numbers is closed under the operation of addition). Multiplication is, in fact, the same addition if we are talking about natural numbers. In life, we often perform actions related to these two operations (for example, when shopping, we add and multiply), and it is strange to think that our ancestors encountered them less often — addition and multiplication were mastered by mankind a very long time ago. Often it is necessary to divide one quantity by another, but here the result is not always expressed by a natural number — this is how fractional numbers appeared.
Subtraction, of course, is also indispensable. But in practice, we tend to subtract the smaller number from the larger number, and there is no need to use negative numbers. (If I have 5 candies and I give 3 to my sister, then I will have 5 — 3 = 2 candies, but I can’t give her 7 candies with all my desire. ) This can explain why people did not use negative numbers for a long time.
Negative numbers appear in Indian documents from the 7th century AD; the Chinese, apparently, began to use them a little earlier. They were used to account for debts or in intermediate calculations to simplify the solution of equations — it was only a tool to get a positive answer. The fact that negative numbers, unlike positive ones, do not express the presence of any entity, aroused strong distrust. People literally avoided negative numbers: if a task got a negative answer, they believed that there was no answer at all. This distrust persisted for a very long time, and even Descartes — one of the «founders» of modern mathematics — called them «false» (in the 17th century!).
Consider, for example, equation 7x – 17 = 2x – 2 . It can be solved like this: move the terms with the unknown to the left side, and the rest to the right, you get 7x — 2x = 17 — 2 , 5x = 15 , x = 3 . With this solution, we did not even meet negative numbers.
But it could be done in a different way: move the terms with the unknown to the right side and get 2 – 17 = 2x – 7x , (–15) = (–5)x . To find the unknown, you need to divide one negative number by another: x = (-15)/(-5) . But the correct answer is known, and it remains to be concluded that (–15)/(–5) = 3 .
What does this simple example demonstrate? Firstly, the logic that defines the rules for actions on negative numbers becomes clear: the results of these actions must match the answers that are obtained in a different way, without negative numbers . Secondly, by allowing the use of negative numbers, we get rid of the tedious (if the equation turns out to be more complicated, with a large number of terms) search for the solution path in which all actions are performed only on natural numbers. Moreover, we can no longer think every time about the meaningfulness of the quantities being converted — and this is already a step towards turning mathematics into an abstract science.
Rules for actions on negative numbers were not formed immediately, but became a generalization of numerous examples that arose when solving applied problems. In general, the development of mathematics can be conditionally divided into stages: each next stage differs from the previous one by a new level of abstraction in the study of objects. So, in the 19th century, mathematicians realized that integers and polynomials, for all their outward dissimilarity, have much in common: both can be added, subtracted, and multiplied. These operations obey the same laws, both in the case of numbers and in the case of polynomials. But the division of integers by each other, so that the result is again integers, is not always possible. The same is true for polynomials.
Then other collections of mathematical objects were discovered, on which such operations can be performed: formal power series, continuous functions … Finally, the understanding came that if you study the properties of the operations themselves, then the results can be applied to all these collections of objects (such approach is characteristic of all modern mathematics).
As a result, a new concept appeared: ring . It’s just a bunch of elements plus actions that can be performed on them. The fundamental rules here are just the rules (they are called axioms ), which are subject to actions, and not the nature of the elements of the set (here it is, a new level of abstraction!). Wishing to emphasize that it is the structure that arises after the introduction of axioms that is important, mathematicians say: the ring of integers, the ring of polynomials, etc. Starting from the axioms, one can derive other properties of rings.
We will formulate the axioms of the ring (which, of course, are similar to the rules for operations with integers), and then we will prove that in any ring, multiplying a minus by a minus results in a plus.
A ring is a set with two binary operations (that is, two elements of the ring are involved in each operation), which are traditionally called addition and multiplication, and the following axioms:
- the addition of the elements of the ring is subject to commutative ( A + B = B + A for any elements A and B ) and associative ( A + (B + C) = (A + B) + C ) laws; there is a special element 0 (neutral in addition) in the ring such that A + 0 = A , and for any element of A there is an opposite element (denoted (–A) ) that A + (–A) = 0 ;
- multiplication obeys the combination law: A (B C) = (A B) C ;
- addition and multiplication are related by the following bracket expansion rules: (A + B) C = A C + B C and A (B + C) = A B + A C .
Note that rings, in the most general construction, do not require multiplication to be permutable, nor is it invertible (that is, division is not always possible), nor does it require the existence of a unit, a neutral element with respect to multiplication. If these axioms are introduced, then other algebraic structures are obtained, but all the theorems proved for rings will be true in them.
Now we prove that for any elements A and B of an arbitrary ring, firstly, (–A) B = –(A B) , and secondly (–(–A )) = A . From this, statements about units easily follow: (–1) 1 = –(1 1) = –1 and (–1) (–1) = –((–1) 1) = –( -1) = 1 .
For this we need to establish some facts. First we prove that each element can have only one opposite. Indeed, let the element have A there are two opposite ones: B and C . That is A + B = 0 = A + C . Consider the sum A + B + C . Using the associative and commutative laws and the property of zero, we get that, on the one hand, the sum is B : B = B + 0 = B + (A + C) = A + B + C , and on the other hand, it is equal to C : A + B + C = (A + B) + C = 0 + C = C .