By taking away the taking away, what's left is putting things back. The square of 1, i.e. 1 multiplied by 1, equals 1. For example, -3 - 9 = 27. We also have a clock, and so we can measure time. Remark $\ $ More generally the Law of Signs holds for any odd functions > + times = , If you remove weight (neg.) rev2022.12.7.43083. 2 -3 = ? If one is positive and the other negative, the answer will be negative. Take one element from each of these sets, thus forming a set of $a$ elements. So, instead of subtracting a negative, you are adding a positive. Both the number of groups and the direction of each group are to the right. Can you really warm up a drink by removing ice cubes from it? Well if I were to explain this in an intuitive way to someone (or at least try), I would like to think of an analogy with walking over the real line, by agreeing that walking left will be walking in the negative direction and walking right in the positive direction. When a number is multiplied by a negative number, the result is the additive inverse of the answer if there had been no negative. is linear hence odd (viewing the ring in Cayley-style For latex paint, dampen your roller cover with water from your spray bottle or the faucet. Then how many times you will walk that distance? of negative numbers, and I asked my father to explain this peculiar tive times negative. there are no negative signs in front of them. Rule 2: A positive number times a negative number gives you a negative number. 2 +1 Answers. I believe it's the most natural (yet totally mathematical) explanation of a basic notion like multiplication of negative numbers. -2 x 3 has one flip, so you start with 2x3 = 6 but with one flip so it becomes -6 instead. Adding something negative or taking away something positive makes the situation worse (down the number line we go).. Have you shown him a sequence, to see a pattern? At the time, however, This happens whether the "something" is positive or negative. Now, so you are already starting to feel better about this part right over here negative times a positive, or a positive times a negative is going to give you a negative. I can find many cool ways to show addition and subtraction with + and - numbers, but I can't find anything meaningful or cute to explain why negative x . e.g. Suppose you have two numbers - 20 and -4 and wish to divide the first integer by the other. This proof of the Law of Signs uses well-known laws of positive integers (esp. We will have, -20 -4 = 20 4 = 5. Since I also like to show students that 9 x 9 (which they know is 81) is the same as (10-1)(10-1). For instance, the orientation of $1*(-1)$ is the same as $(-1)*1$ because both require a clockwise rotation from the first vector to the second: $(0,1)\mapsto(0,-1)$ for the former and $(-1,0)\mapsto(0,1)$ for the latter. if you added 2 negative #s you'd get the same result. Multiplying two negative numbers together produces a positive number. I get your point, but I find this example works well enough for pedagogical purposes.. Similarly for $-4P, -5P$, and so on. ), the moment (neg. $$(-a)b + ab = b(a-a) \begin{array}{r} There are different possible answers to this question, depending on the standard of proof one needs and the background knowledge one brings to the question. Explain the definition of negative numbers. From these axioms, we can prove that a negative times a negative is a positive. Maybe you could think of the negative sign in the second factor to imply that you change direction, that is, it makes you turn around and start walking the specified distance. 3 times 4 equals 12. @PaulSinclair Indeed! Well, here's a cute one that a friend once told me. Best wishes, $\mathcal H$akim. When you multiply a negative number to a positive number, your answer is a negative number. This approach has applications with Complex numbers. So, in a sense, the distributive law is a keystone of the ring structure. I get it that a negative * a positive = 0. Why is this immoral? With the interpretation in terms of oriented areas we would have $xy=-yx$ which is kind of problematic. \textbf{0} & & & 0 & 0 & 0\\ \hline \textbf{-2} &4& 2& 0& -2& -4\\ \hline Wish the answer was that easy. I doubt the child is ready for a talk about specific heat. of the rule of signs, and I have come to the following conclusion. & (\text{Answer} = -6 )\\ (2) Every number has exactly one additive inverse. Applying this twice along with the identity $-(-a)=a$, $(-x)\cdot(-y)=-(-x)\cdot y=-(-(x\cdot y))=x\cdot y$. .hide-if-no-js { These fundamental laws of "numbers" are axiomatized by the algebraic structure known as a ring, and various specializations thereof. The result is a big amount of potential cancelling. Is investigation an art so mechanical, that it may be conducted by certain manual operations? It doesnt matter which order the positive and negative numbers are in that you are multiplying, the answer is always a negative number. Example 1: This is the kind of multiplication you've been doing for years, positive. \end{align*}$$. I have since been offered a moderately convincing explanation that features a film of a Is the menstrual cycle positive or negative feedback? Extend reals to the complex plane. Contrary to the deductive theories of my father and Descartes, as a To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The real problem is trying to explain what a group of negatives is. Two times X minus plus five equals 10. \end{array}\end{equation*}$$ What happens when a number is multiplied by a negative number? By. \hline \underline {\color{#c00}{a}}}\underline{\ + \color{#0a0}{-a}}\ $ in $\,2\,$ ways (note over/underlined terms $ = 0)$. number. Well,I wouldn't exactly call that a SIMPLE answer. Ukraine's 10 is X Plus four. Easily the best and most satisfying answer here. Similar to before, we distribute -5 through both terms. \end{array}$$ 3 1 = ? I teach a group of very low level math students. Greenrune113. Consider the number line. This primes the roller cover to soak up as much paint as possible, Jessica explains. Advertisement. Can you use paint thinner on a paint roller? instead. Okay, so we can use language to better cultivate our understanding of negative numbers. This is a feasible problem, and it helps children Is negative 1 squared positive? $$*(-a)(-b) = ab $$ What is the best varnish for kitchen cabinets? Case 2 - The quotient of a positive and negative integer is a negative integer and its absolute value is equal to the quotient of the corresponding absolute values of the integers. From Dr. Alex Eustis, we have this algebraic proof that a negative times a negative is a positive. on the left side of the lever (neg. First, we prove that a = (a). Solving for distance we get: $$\rm Velocity\times{\rm Time}={\rm Distance}.$$ It is not possible definite orientation (rotation from one vector to another can be \textbf{2} &-4 & -2& 0& 2&4\\ \hline Justify _that_! Result: A big positive gain. Multiplication by $-1$ is a rotation by $\pi$ radians. My memory of how I learned this is somewhat uncertain, because I was only 8 or 9 years old at the time, but I think this is very similar to how Isaac Asimov explained multiplication of negative numbers in either. Here's a proof. The difference between instrumental and relational understanding. If a negative times a negative equals a positive, why does a negative squared equal a negative? Prerequisite knowledge: One has to know what these symbols mean, what is meant by finding one number times another, and how negative numbers work in terms of counting down and subtraction. Since you have to add. }. distance) is clockwise (pos.) to explain to algebraists that their axiomatic method is mostly Imagine we represent multiplication as jumps on a number line. I have never been able to leam by heart what is not properly Like, the first column is counting upwards by $2$ when we move up - it goes $-4$ then $-2$ then $0$ so we should continue counting this way to $2$ then $4$. One thing that must be understood is that this law cannot be proven in the same way that the laws of positive rational and integral arithmetic can be. Given that the goal of an argument that something is true is to leave the other person convinced of the truth of the argument, whenever anyone uses any justification, representation, or proof, it behooves one to check that ones audience is left convinced. The equation: (-a)b + ab = (-a)b + ab Likewise, the additive inverse of -N is N. Imagine now that you have a video of the above scene, and time is positive if you play the video normally but is negative if you play it backwards. The answer is -2 x 4 = -8. a are the same number. Another thesis of Descartes theory and methods of education is even 2 & \times & -3 & = & ? Prepping your roller ensures faster painting and saves paint. A negative divided by a negative equals a positive. Subtracting $x\cdot y$ from both sides, $x\cdot(-y)=-(x\cdot y)$. \textbf{1} & -2&-1 &0 &1 &2 \\ \hline In this way $-1 \times -1$ is the same as doing nothing, or is the same as multiplying by $1$. Negative times negative equals positive? This is my logic. So for consistency the next product in the pattern must be $0 + 2 = 2$. Not a proof, but for an example in realia from the physical world of levers: If you add weight (pos.) +1. "Huh??? Now, we multiply the first equation by (-1) and use the distributive property to get (-1)(-1)+(-1)(1)=(-1)(0). equation of time, which takes into account a correction for the Finally, you'll have to try to picture what could $(-2)*(-3)$ mean. with reality should be a part of science (why should results I went ahead and gave them a proof by contradiction like so: Then divide both sides by $(-x)$ and you get $(-y) = y$. Multiplying integers is fairly simple if you remember the following rule: If both integers are either positive or negative, the total will always be a positive number. clockwise or anti-clockwise). Roll the roller through the paint thinner to clean it. We also know the following: $$\rm Velocity=\dfrac{\rm Distance}{\rm Time}.$$ Rule 2: A negative number times a positive number equals a negative number. Here, the loss per day is one negative and going backwards in time is another. $$\begin{array}{|c|c|c|c|c|c|} If you film a man running forwards ($+$) and then play the film forward ($+$) he is still running forward ($+$). The number line is unchanged. Similarly, we can prove that a negative times a negative is a positive. The reason for this is that negatives lack any "external" (external to mathematics, ie. Not a proof though! If you want to write negative two squared you need parentheses, (-2) More answers below Lee Veinot Math Teacher 2 y Its not. When you apply the two flips it gets you back to where you started because you flip to negative and then flip back. How did you think to use it here? It doesn't matter which order the positive and negative numbers are in that you are multiplying, the answer is always a negative number. If you annihilite the annihitation the result is a net gain: Result: positive. At this stage, many people will notice the answers are 3 smaller each time and the number being multiplied by 3 is one smaller each time, so they continue that pattern to answer the following questions. How much more money did they have 5 days ago? But what about a physical example? For 3 3, we draw 3 groups of 3 moving to the right. This video is an excerpt from the Mathemagic workshop in which you learn a lot of maths tricks. So if we take a number and multiply the number times -1 it is represented below: (-1)x = -x. me was to pretend to agree that negative times nega- > + times + = +, If you remove weight (neg.) It doesn't matter which order the positive and negative numbers are in that you are multiplying, the answer is always a negative number. I teach engineers and for many years this has been the assumption. Is there precedent for Supreme Court justices recusing themselves from cases when they have strong ties to groups with strong opinions on the case? Well, you're at the right of the origin so you are in the positive section. How to negotiate a raise, if they want me to get an offer letter? What factors led to Disney retconning Star Wars Legends in favor of the new Disney Canon? + is the positive sign, ? is the negative sign. -(f(-g)) \iff\,&\rm \ f(-g)\ = -(f(g))\\ \phantom{\times9}2\\ What does it mean when negative times are positive? @DavidK No, we should suppose that the scenario is grossly simplified. Which is better adding something positive or taking something negative? This is two times the square root of by. Well. this always made sense to me (but I've found it doesn't for others). overcome my strong sense that multiplying intensifies something, and thus two negative numbers -196 + (-71) = -267. principles of axiomatic science: a definition is chosen such that the Notice in each case, as we reduce the first multiplier by 1, the product is being increased by 2. This value should be positive since it results in you receiving money. That's from the reddit ELI5 from a few days ago, $\overbrace{\bf\ Law\ of\ Signs}^{\rm\Large {(-x)(-y)}\ =\ xy} $, $\rm\,\ (-x)(-y) = (-x)(-y) + \color{#c00}x(\overbrace{\color{#c00}{-y} + y}^{\Large =\,0}) = (\overbrace{-x+\color{#c00}x}^{\Large =\,0})(\color{#c00}{-y}) + xy = xy$, $\rm\,\ \overline{(-x)(-y)\ +\ } \overline{ \underline {\color{#c00}{x(-y)}}}\underline{\phantom{(}\! Multiplying and dividing When multiplying (or dividing) two numbers with the same signs, the resulting answer is positive. there are no negative signs in front of them. \overline{\phantom{+}\! been plus, this law would be broken. Second, to establish that a negative times a negative is positive: we now know that $3 \times (-2) = -6, 2 \times (-2) = -4, 1 \times (-2) = -2, 0 \times (-2) = 0$. Altitude - above sea level is positive, below sea level is negative. For example: 3 x 2 = 6. 2 & \times & -1 & = & ? Ideally, the second negative should change the sign of our original number (which is also negative). I don't know if this will help, but it's the only way I can think of this in some intuitive sense. This last proof though is unlikely to justify that a negative times a negative is a positive for any students though. Each number will now be superimposed over its negative: $-1$ will be where $+1$ was; $+2$ will be where $-2$ was. In fact, for many students, mathematics stopped making sense somewhere along the way. $$(-1)a\cdot (-1)b$$ Getting the right answer, (-1)(-1)=1, uses a couple more steps: First, you must agree that (1)+(-1)=0, (1)(-1)=-1, and (0)(-1)=0. So you tell your son that your best friend $\color{red}{\fbox{took-away}}$ seven $\color{red}{\fbox{debts}}$ of $5$ ($\color{red}{-7}\times\color{red}{-5}$) and this equals a gain of $\color{blue}{35}$. What's the translation of "record-tying" in French? Lets talk about signs. \underline {\color{#c00}{a}}}\underline{\ + \color{#0a0}{-a}}\ $. @AndreasBlass Or in another form: "negative multiplication is reverse intensification". The material you quoted contains the interesting statement "Whatever explanations they offered could not overcome my strong sense that multiplying intensifies something". Why is negative times negative equal positive? An integer n is divisible by a nonzero integer m if there exists an integer k such that =.This is written as . Electrons have an electric charge of 11, which is equal but opposite to the charge of a proton, which is +1+1. the video is played backwards, you'll see that the car moves along the $+$ direction! \hline What's the logical thinking skills or tactic used to jump in one step from. But, you need to learn to control your emotions. I wonder whether anyone tried the very short answer "yes, but negative intensification is attenuation"? I take three \$20 notes from you: -3 * +20 = -\$60 for you \phantom{\times9}4\\ The axiomophile Rene Descartes stated that neither experimental tests Adding $-2$, two times, yields the diagram in (2). The cancel out positive numbers. If multiplication by a negative is a reflection across 0 on the number line, and we think of negative numbers as being reflections across 0 of the number line, then multiplication of a negative number times a negative number is a double-reflection. That's what the symbology above says: multiplying by $-a$ is the same as multiplying by $a$ then by $-1$ and similarly for $-b$. negative). If you play the film backward ($-$) he appears to be running backwards ($-$) so the result of multiplying a positive and a negative is negative. If you look at it on the number line, walking backwards while facing in the negative direction, you move in the positive direction. $$(-a)(-b) + (-ab) = (-a)(b-b) $$ But in the same way you can play this idea with a negative times a positive. Then I will try to convey the idea that if you are multiplying two numbers (let's suppose they are integers to make things easier to picture) then a product as $2*3$ would just mean that you have to walk right (in the positive direction) a distance of $2$ (say miles for instance) three times, that is, first you walk $2$ miles, then another $2$ miles and finally another $2$ miles to the right. \end{array} $$\begin{array}{|c|c|c|c|c|c|} is pointing out that logical negation works the same way as multiplying negative numbers (two negatives make a positive), not belittling your question. $$-3 \times -5 = 15$$. What is positive message and negative message? The simplest way to prove why is by using the group axioms and thier consequences. Once I Hey look, we know these two things are true, therefore this third thing must also be true.. \textbf{2} & & & 0& 2&4\\ \hline Rule 2: A negative number times a positive number equals a negative number. This aims not at the algebraic or arithmetic properties of numbers but more at the oppositeness of negative numbers. $$\begin{equation*}\begin{array}{c} They try to learn rules for operations with positive and negative numbers. Addition of a positive number and a negative number: While adding a positive and a negative number, we take the difference of the absolute values of both the numbers and attach the sign of the greater number with the answer. Why a negative times a negative is a positive | Pre-Algebra | Khan Academy | log Posted on 07/12/2022 by Michael W. Fanning rings of polynomials, power series, matrices, differential operators, etc. Separates us from the nerds who live their lives without intuition. Here the important part comes, if the car is moving in the $+$ direction and the time the video is played is positive, i.e. Add and subtract positive and negative numbers When adding and subtracting numbers it's important to be consistent with positive and negative values. Required fields are marked *, Enter in the following * When you multiply a negative number to a positive number, your answer is a negative number. I only have a vague intuitive notion that I probably can't explain well, but I sometimes think of a negative number like $-5$ as being "$5$ in the other direction", and so multiplying by $-5$ means "multiply by $5$ and switch direction", i.e., sign. Multiplying by a negative is repeated subtraction. Mathematical consistency and mathematical properties. Result: a loss; negative. By what definition of $inverse$ are they inverses? first time I used this good site! $$ 3 \times -1 = -3$$, Removing two of the ice cubes will raise the temperature by $2$ degrees, or Films could make sure they get the representations by making a diversity charter that tells you what you are going to do. This is a good answer, because the rate at which the individual is running in the video multiplied by the rate of the playback equals the apparent rate at which they run in the playback. Can you explain this answer? the next topic, and the only practical course open to Why is negative times negative = positive? Yes, or the paint will run down your wall surface. . Is it not an easier way to explain why Apply the paint to the surface in long, light, even strokes. Here's how the reasoning goes: (1) Zero times anything equals zero. The paint will lay on much better when dry; if the roller is wet and mixed in with the paint and you start painting your walls (it will run, plus leave a type of water mark look on the walls). distributive identity a(b+c)=ab+ac holds. We know that this is the same as -5 times 0, so this has a value of 0. Since there is one positive and one negative number, the product is negative 12. McNeill, the turning point was clearly defined. = But what if we distribute 5 through both terms first? But dont go too crazyJessica suggests removing excess moisture with a paper towel and a good shake of the roller so its just slightly damp. The Difference Between Formative Assessments and Formative Assessment. together as we normally would, and then put a negative sign in front of our answer. a + (a) = 0a (b) + (a) (b) = 0 (b)ab + (a) (b) = 0. Khurshed Batliwala who put together this workshop holds a Masters Degree in Mathematics from the very prestigious IIT Bombay. on the right side of the lever (pos. People also need to make sure they know what diversity they are trying to represent. Youll waste a good gallon of paint and have to let the wall dry and start over again with a new dry roller. So)i) positive x positive: add a bunch of positive numbers a positive number of times. understood by the age of thirty when it would be possible to read and Therefore -15 and -5 -3 are opposites since they add to 0, so -5 -3 must be positive. That is, multiplication by a negative is the same as two steps: multiplying by the thing as if it had no negative, then applying the negative sign. Before using a paint brush, it should be pre-wetted with water if a latex paint is being used, or mineral spirits for an oil base paint. Without the distributive law a ring degenerates to a set with two completely unrelated additive and multiplicative structures. On a case-by-case basis, it's not obvious that a negative times a negative should be positive. Simple Answer: Rule 2: A negative number times a positive number equals a negative number. This would read negative two minus negative 4. If the bag contains two 6s, among other things, we can easily taking that "minus two lots of six from the bag is the same as minus 12 from the total". (-5) + 7 = 2. And, if we trace back the steps that we used to generate this correct table, we can recover $(-1)\times (-1)=1$ as follows: Firstly, we note that one times something leaves that thing unchanged. Point out that the definition of $-x$ implies that $-(-x) = x$. I've become convinced that my education cheated me on how deep an idea negative numbers are, and I expect to remain puzzled by them for many years. You cant use water or wash it down your sink. -1 * (-1) then simply flips it back from the negative to the positive side. correspond to reality if the initial principles do not correspond to out. Copyright Warner Bros. 1988 First: (4) Now, we are forced to accept a new law, that negative times positive equals negative. In particular, we want the distributive property to apply. The anti-clockwise rotation is 3Dampen the roller cover. The answers to these problems are below but I really do recommend taking the time to solve the problems above on your own first, so you get the sense of how students might think through this set of problems. Subtract another $P$ and you get $-3P$, which is still negative. student can be taught as successfully as the most gifted one, while pre-axiomatic, intuitive, conceptuel, empirical, physical, etc.) -2 x -4 are both negative, so we know the answer is going to be positive. The equation: So -3 x -2 = 6. If you have (-3) (-3) then there are two minus signs being multiplied, and the answer is positive. \textbf{-1} &2& 1& 0& -1& -2\\ \hline Net result is that i get Bachelors level students that can not do anything with a computer, not even frigging graph a function. The best answers are voted up and rise to the top, Not the answer you're looking for? So far, the claim "This is intuitive . Having established these definitions,.." hanging around "This is intuitive". It's a negative. Distances are more useful than number of things in this case. From this, we can show that ab and ab have opposite signs and therefore that a positive times a negative is a negative. For example, we can define rolling a 6 on a die as a success, and rolling any other number as a failure . Hint: starting from the very reasonable axioms $a(b-c)=ab-ac\ ,\ a - (b - c) = a - b + c$, consider the product $(a-b)(c-d)$. It doesn't matter which order the positive and negative numbers are in that you are multiplying, the answer is always a negative number. And therefore: $$\color{grey}{\boxed{\color{white}{\underline{\overline{\color{black}{\displaystyle\rm\, negative\times negative=positive.\,}}}}}}$$ As you have seen, it takes only a little bit of imagination for it to make sense. Two Signs: The Rules "Two like signs make a positive sign, two unlike signs make a negative sign" Example: (2) (+5) The signs are and + (a negative sign and a positive sign), so they are unlike signs (they are different to each other) So the result must be negative: (2) (+5) = 10 Example: (4) (3) distance) is counterclockwise (neg.). This is because we can use the distributive law on an expression like 2* (3 + (-3)). Similarly, we can prove that a negative times a negative is a positive. 3 3 = ? ten year old, I started thinking about a naturally-scientific sense A negative times a negative equals Multiplying Negative Numbers When multiplying negative numbers, we have a certain rule that allows us to know what the result will look like. When you multiply a negative number to a positive number, your answer is a negative number. ciety were clearly more powerful than I was. Rule 3: Subtracting a negative number from a negative number a minus sign followed by a negative sign, turns the two signs into a plus sign. It might be easiest to explain using whole numbers. (adsbygoogle = window.adsbygoogle || []).push({}); Copyright 2022 Find what come to your mind. Same idea for positive times negative. @DavidK Just use the word "pretend." 3 -1 = ? document.getElementById( "ak_js_1" ).setAttribute( "value", ( new Date() ).getTime() ); This site uses Akismet to reduce spam. Using them, one can come up with a correct system of definitions in no time, if they have to. \textbf{2} &-4 & -2& 0& 2&4\\ \hline representing each ring element $\rm\ r\ $ by the linear map $\rm\ x \to r\ x,\ $ The illustration of a negative times a positive is easier to understand. The opposite of being billed would be billing someone else. 1. \textbf{0} & 0 & 0 & 0 & 0 & 0\\ \hline You have a bank account. Delete faces inside generated meshes on surface. When you multiply two negative numbers or two positive numbers then the product is always positive. $$\begin{equation*}\begin{array}{c} Quite a good explanation is that one wants the distributive law to work in general with positive quantities when you add (smaller) negative ones: A great scene from the movie Stand and Deliver. you can't multiply numbers and letters!". That is and therefore 4 + 4 +4 = 12, therefore, because - (-4) - (-4) - (-4) = 12. & (\text{Answer} = 6 )\\ &= (-1 \times -1) \times (a \times b) \\ One should ask children: at what time will high tide be tomorrow if Distribute the negative two. \end{array}$$ In it he states -3 *-3 = 9. We know that 4 and 3 are both positive because. negative two times zero, anything times zero, needs to be equal to zero, but then once again, we can distribute negative two times six so we get negative two times six, then plus negative two times negative six plus negative two times negative six, then once again all of this is going to be equal to zero, now based on the five experiment we just $$ The point isn't to teach physics, but to paint a mental image of what opposites mean in terms of multiplication. $$(-a)(-b) + (-ab) = 0 $$ Now do the rotation twice. This is an excerpt from Arnold's wonderful memoir "Yesterday and Long Ago" (3d ed., available in English from Springer), full of world history, drama and ingenious storytelling. other is along a perpendicular axis in the plane). also thanks for this post, it would have been so helpful when we were initially looking, but maybe i wouldnt have tried to reason it out like this if i had seen this. They are anti-numbers. :). Rule 2: A negative number times a positive number equals a negative number. @KGhatak the question explicitly asks for an intuitive explanation. iii) negative x positive: take a bunch of positive numbers and take them away. One way to think of this is to think of taking 3 groups of the number away. or I was never good at it. The elementary intuition behind the product of two negatives can be thought of as follows. could suddenly see how a proof was going to come On the number line, opposite numbers are mirrored in their distances from zero, which provides a nice visual aid as well. @MichaelS You are, of course, correct. time I have disliked the axiomatic method with its non-motivated @PaulSinclair The children of today are used to using computers, they can easily be taught to think of $x$ as storing some arbitrary number. You could of course make it bigger to make the patterns clearer. For example. In my opinion, negatives are ultimately best understood as purely abstract objects. While I can understand the sentiment, I'm not sure this is the kind of morality we should be teaching 8-year-old kids. Prepping your roller ensures faster painting and saves paint. For example -6 is the opposite of 6, so if you said "find the opposite of 6" you'd get -6, but if you said "find the opposite of -7," or "find the opposite of the opposite of 11 . So, when we get to $(-1)\times (-1)$ we have to be one more than $0$ since $-1$ is one less than $0$. it?). The most commonsense of What happens when a negative times a negative? The 2007 translation into English, I believe, is not of best quality, but it's the only one so far. Its the kind of thing which is a required level of justification for a mathematician interested in rigorous proof who would likely consider the other justifications patterning and not sufficient. You see the person walking forwards, because negative*negative=positive! In this lesson, you will learn how to prove that a negative times a negative equals a positive by using the distributive property. Thus, the rule of signs is not an axiom taken out of the blue, but \textbf{-1} & & & & &\\ \hline In this video Khurshed Batliwala, fondly called as Bawa, explains why multiplication of two negative numbers equals a positive. \textbf{1} & & &0 &1 &2 \\ \hline In general adding $-(-P)$ to itself $Q$ times gives $(-Q)(-P)$, which is therefore positive as well. A negative plus a negative equals a negative. law (x+y)z=xz+yz holds. $$(-a)b + ab = 0 $$ I faced a real difficulty with school mathematics several years after (-2) x (-8) = 16. There are so many analogies that. First, for all $x$, $x\cdot 0=x\cdot(0+0)=x\cdot 0 +x\cdot 0$. $$ I'll reproduce Dr. Eustis's proof below and include the reference to the axioms used. I reformat the most upvoted answer (also my favorite) with MathJax, from Reddit: I give you three \$20 notes: $+3 +20 =$ you gain $60, I give you three \$20 debts: $+3 -20 =$ you lose $60, I take three \$20 notes from you: $-3 +20 =$ you lose $60, I take three \$20 debts from you: $-3 -20 =$ you gain $60. (-} Negative Plus+ (+} Positive = (Sign of Biggest#) the Difference MULTIPLICATION/ DIVISION: Negative numbers times (or divided by) a positive number equals a negative number (-} Negative Times/Divided by (+} Positive = (-} Negative Bad Guy . $$(-a)(-b) + (-ab) = (-a)(-b) + (-a)b $$ 4 is positive, 3 is positive, thus, 12 is positive. &\textbf{-2}& \textbf{-1} & \textbf{0} & \textbf{1} & \textbf{2} \\ \hline the video is played normally, then you'll see that the car moves along the $+$ direction and you'll calculate that it moves "a positive distance". 5 groups of the inverse of 3? For latex paint, dampen your roller cover with water from your spray bottle or the faucet. -1 & \times & -3 & = & ? correspond to these numbers (one vector is along one axis and the So in this case the $-2$ tells you to walk left a distance of $2$ miles but the $-3$ tells you to first turn around, and then walk $3$ times the $2$ miles in the other direction, so you'll end up walking right and end in the point that is $6$ miles to the right of the origin, so you'll be in the positive section, and $(-2)*(-3) = 6$. 2 & \times & 3 & = & 6\\ Now, we decrease the first number in the pattern by 3 and one has to make some deductions about what the answer should be. several weeks, trying to get a sensible explanation Posted by 11 months ago. \textbf{-2} & & & & &\\ \hline 0 & \times & -3 & = & 0\\ Multiplying two negative quantities results in reversing direction twice, QED. Notice in each case, as we reduce the second factor by 1, the product is being reduced by 3. ), the moment (pos. Secondly, looking at the table again, we see that multiplying by $(-1)$ "reverses" the order of our usual counting - that is $(-1)\times 2$ is $-2$ then $(-1)\times 1$ is one more at $-1$ and $(-1)\times 0 =0$. Then with the axiom that "+ and - provide opposite direction in a number line" will not be sufficient to deduce "- followed by - will change direction". Positive times negative We can show that these facts imply what multiplication of negative numbers has to look like, in two steps. How to fight an unemployment tax bill that I do not owe in NY? For example, 5 times 3 is 15. We can use the term to describe arithmetic operations: The opposite of three times five is the opposite of 15. Similarly, we can prove that a negative times a negative is a positive. The level to which you speed up the rewind doesn't matter ( 3x or 4x) these results hold true. We know from before that -5 3 is -15 so we can substitute that value for -5 3 in the left-hand side of the equation. Using the fact multiplication is commutative, a negative times a positive is also negative. 2 & \times & 2 & = & 4\\ This equals 2* (0), which is zero. 2 & \times & -2 & = & ? for Electrical Engineering (EE) 2022 is part of Electrical Engineering (EE) preparation. & (\text{Answer} = -2 )\\ Im reading Neil de Grasse Tysons Origens. @user500668 Good question. Or is truth so easily discovered, that intelligence is not necessary to give success to our researches? Prerequisite knowledge: The prerequisite knowledge for this proof is much less than the other one, but it does assume a fair bit of fluency with manipulation of algebraic structures. e.g. Your proof implicitly uses the fact that $-xy=(-x)y$, and assumes that there are only two possibilities, $xy$ or $-xy$, then shows that the latter is impossible. Oh, you could probably come up with one involving opposite "directions", and notions of symmetry, but it would be quite artificial and not at all obviously "the best" definition. Now we have two negative numbers, so the result is positive. So $-2(-P)$, which is $-(-P)$ added to itself, is still positive. Gurvich) who treated an ignorant interlocutor with full respect and Solve for (-1)(-1), and you get (-1)(-1)=1. What is the meaning of professional behaviour? As a consequence, a product of two negative numbers is positive. Then $-P$ is negative. What do you see if you play a film backwards of someone walking backwards? Subtracting $x\cdot0$ from each side, $x\cdot0=0$. For example, 5 + (3) is the same as 5 3, and equals 2. Taking away a negative number from another is the same as adding the positive number with the same numerals. tried to explain non-trivial ideas and facts of various sciences such Using the fact multiplication is commutative, a negative times a positive is also negative. These seem like plausible assumptions, but I tried to be very careful in my proof above (thus using $-(x\cdot y)$ rather than simply $-xy$ to not be confused with $(-x)\cdot y$). I think about addition when multiplying. understood. So you either have to make some complicated "averaging" argument or you have to rely on a false physical intuition to be "unlearned" later. Lets look at a problem that we can do in more than one way, borrowed from the Khan Academy. > times = +, so a couple years ago my son asked me why a negative times a negative is a positive, i didnt have a good answer so we looked online for a proof, and i didnt find anything i found super compelling, so i started trying to think about what we are literally saying. It's not perfect, but it introduces the notion of the number line having directions at least. But even that explanation doesn't altogether satisfy me. and Dedekind) started explaining to his eleven-year-old son the from my teacher, my classmates, my parents, any- Let's fill this in: Probably it was for this reason that by this time I got a + (a) = 0a b + (a) b = 0 bab + (ab) = 0. \underline{\times\phantom{1}-2}\\ how can I prove negative times negative is positive. 5 comments. Does he understand why a negative times a positive is negative? Can LEGO City Powered Up trains be automated? In this video Khurshed Batliwala, fondly called as Bawa, explains why multiplication of two negative numbers equals a positive. The fact that a negative times a negative equals a positive can be proven mathematically using algebraic manipulation. 2022 The Reflective Educator. &= a \times b. Whatever explanations they offered could not Does a negative times a positive equal a positive? As to what negative times negative is positive actually means I dont have a clue. Now, I say that it cannot be the sign -: for -a by +b gives -ab, and -a by -b cannot produce the same result as -a by +b With no disrespect to Euler (especially consdiering this was intended as an introductory textbook), I think we can agree that this is a pretty philosophically dubious argument. 3 2 = ? Your email address will not be published. Just as before $3$ times and in the end you'll be $6$ miles to the left of the origin so you'll be in the negative section. Usually the process is gradual, but for Ruth Rule 2: A negative number times a positive number equals a negative number. In an $$ Multiplication and Division If two positive numbers are multiplied together or divided, the answer is positive. weight times pos. 2 x -3 is the same, only one flip, so it's 2x3 = 6 but flipped to -6. That is, - * + = - So, our original negative sign is changed into a positive sign when a negative is multiplied to it. You pay 3 bills for 40 dollars each, $3 \cdot (-40) = -120$ is added to your account. Same goes for if you film a man running backwards ($-$) and play it normally ($+$) he appears to be still running backwards ($-$). customarily considered positive and the clockwise rotation is then Was this reference in Starship Troopers a real one? @Mathemagician1234 you know, I agree with Arnold on the idea that explanations or intuition like this are much much more powerful than any proofs or definitions, because they are real and without artificial restrictions of formal systems. Now, play the film back, but in reverse (another negative rate). 2 & \times & 0 & = & 0\\ Rule 2: A negative number times a positive number equals a negative number. I also needed to not lose sight of the overall goal and to be able to recognize the structure of each part of the argument and match that structure to the axioms. iv) negative x negative: take a bunch of anti-numbers and take them away. becomes a natural property of orientation which is easily verified x is a letter! i.e. Also, what would be an intuitive way to explain the negation concept, if there is one? Now we have 3 groups of the number still, but the number is negative. Rule 1: A positive number times a positive number gives you a positive number. and thank you for trying to make it clear, So, our original negative sign is changed into a positive sign when a negative is multiplied to it. negative result. I think a lot of answers are either too simple or stray away from mathematics too much. This $(-a)(-b)=ab$ stuff started when we gave up on distinguishing multiplier and multiplied in a multiplication. Mathemagic has been designed to be a thoroughly entertaining 2 hour learning experience; a delightful romp through some of the basics of some fairly advanced maths.To know more about this workshop please write to info@bawandinesh.inVideo Created byAbhiram ViswanathanGowrishankar VenkatramanConcept and Overall Interference by Bawa and Dinesh (BnD)Share, Support, Subscribe!! Funny - this is the second argument that I typically present to elementary teachers and their students after my cops and robbers story! Should you wet your paint roller before painting? &= -1 \times -1 \times a \times b \\ The full set of axioms required is below. Definition. This algebraic explanation was not able to shake either my hearty $$. Which rules of arithmetic are worth keeping? Required fields are marked *. Thus: $$\rm negative\times positive=negative.$$, If however, the car moves along the $+$ direction but the time the video is played is negative, i.e. Solution: We can calculate the result of negative time's positive number by using the following steps: Step 1: First we can identify the given problem. Here, not in axioms, is laid true A negative plus a positive depends on which number is bigger for the sign. Similar informal (but entirely convincing, reasonable, and I would say irrefutable) reasoning can be used to demonstrate the rules for manipulating positive fractions, say. In the above case of rings, distributivity implies that multiplication Any dimension cannot be negative in real sense. For one over square root of two, times the square root times integration from 0 to infinite. $$ With kids of that age, a common response is that $x$ means 24. Times you will walk that distance wall dry and start over again with new... * -3 = 9 your point, but for Ruth rule 2 a... Trying to represent have opposite signs and therefore that a negative is a net gain result. Specific heat clockwise rotation is then Was this reference in Starship Troopers a real one ].push... Of Descartes theory and methods of education is even 2 & = & this example works well enough for purposes. So ) I ) positive x positive: take a bunch of positive numbers a positive number of in. As jumps on a case-by-case basis, it & # x27 ; s the. Groups and the only one so far equals a positive ) Every number exactly! Is also negative ) in time is another, which is kind of problematic in! Something positive or negative x 4 = -8. a are the same signs and... And -4 and wish to divide the first integer by the other and. A consequence, a product of two negative numbers, so the negative times positive equals is a negative equals a negative having! Have two numbers with the same as 5 3, we can prove that a divided... Use water or wash it down your sink positive, below sea level is negative and... Suppose that the definition of $ inverse $ are they inverses are a... Groups of the number is bigger for the sign of negative numbers has to look like, a! Is negative of three times five is the menstrual cycle positive or taking something negative interesting statement Whatever. Discovered, that it may be conducted by certain manual operations get an offer?. Soak up as much paint as possible, Jessica explains simple or away! Mathematics, ie additive and multiplicative structures intuitive '' = x $ Wars in! 3 groups of 3 moving to the top, negative times positive equals the answer is -2 x -4 are both negative the. Is kind of morality we should be positive since it results in you money. Interesting statement `` Whatever explanations they offered could not overcome my strong sense that multiplying something... Taking something negative ties to groups with strong opinions on the right side of the number,... Of definitions in no time, if they want me to get a sensible explanation Posted 11. Neil de Grasse Tysons Origens x\cdot ( -y ) =- ( x\cdot y ) $ have! ) 2022 is part of Electrical Engineering ( EE ) preparation taking 3 groups of 3 moving to positive... Very low level math students x\cdot ( -y ) =- ( x\cdot y ) $, the. Produces a positive is also negative negative times positive equals not overcome my strong sense that intensifies. The origin so you are adding a positive $ what is the kind of morality we should that. The menstrual cycle positive or taking something negative rolling any other number a... ( external to mathematics, ie \times -5 = 15 $ $ what happens when a negative equals negative. Altitude - above sea level is negative 1 squared positive the opposite of negative times positive equals to mind... Number away explanation that features a film of a basic notion like multiplication of negative numbers together a! In that you are multiplying, the resulting answer is always positive convincing explanation that features a film of. You play a film of a basic notion like multiplication of negative numbers has to look like, two... As purely abstract objects $ * ( -a ) ( -b ) =ab $ stuff started we... Is $ - ( -P ) negative times positive equals added to your account ) then simply flips gets. Zero times anything equals zero law a ring degenerates to a positive 1. Start with 2x3 = 6 but flipped to -6 positive depends on which number is bigger the! The other anyone tried the very prestigious IIT Bombay friend once told me negative s. Electrical Engineering ( EE ) 2022 is part of Electrical Engineering ( EE ).. Is two times the square root of by no negative signs in front them. Negatives can be thought of as follows are multiplied together or divided, answer! Film back, but it 's 2x3 = 6 but flipped to -6 much as... A basic notion like multiplication of negative numbers, and so we can measure time s how the goes! Adding a positive = 0, fondly called as Bawa, explains why multiplication of two times... Hearty $ $ now do the rotation twice as to what negative times a negative equals a.. To be positive since it results in you receiving money as Bawa, why! Maths tricks and thier consequences commonsense of what happens when a number is multiplied by a nonzero m. Ring, and so on our answer || [ ] ).push ( { } ) Copyright! Opposite to the following conclusion \times -5 = 15 $ $ what is the opposite being! Math students fundamental laws of positive numbers are multiplied together or divided, the loss day... Positive or negative feedback $ inverse $ are they inverses for others ) for latex paint dampen! And their students after my cops and robbers story positive, why does a negative is a positive without. Course open to why is negative times a negative, the product negative times positive equals being reduced 3..., we have this algebraic proof that a simple answer: rule 2: a negative times. Times 0, so we know that this is intuitive '' x\cdot ( -y ) =- ( x\cdot y $. To groups with strong opinions on the left side of the number of groups and the answer -2! Workshop in which you learn a lot of answers are voted up and rise to the top, not answer. The charge of a basic notion like multiplication of two negative numbers positive. 3 3, we can show that ab and ab have opposite and... Prove why is by using the group axioms and thier consequences window.adsbygoogle || [ ] ).push {! Pretend. multiplied together or divided, the loss per day is one positive the. A number is bigger for the sign surface in long, light, even strokes how much more money they. Numbers a positive that it may be conducted by certain manual operations this! The opposite of being billed would be billing someone else from 0 to infinite natural ( yet totally )... Of paint and have to let the wall dry and start over again with a new dry roller learn! Added 2 negative # s you & # x27 ; s 10 x! The rotation twice structure known as a ring degenerates to a positive number times a positive is also negative.! Zero times anything equals zero back to where you started because you flip to negative and going in! In French for Electrical Engineering ( EE ) preparation keystone of the number line directions! Negative is positive that multiplying intensifies something '' a die as a failure being billed would be billing else. Is intuitive '' these definitions,.. '' hanging around `` this is the opposite of being billed be! Reality if the initial principles do not owe in NY ( -40 ) = 0 one... A correct system of definitions in no time, however, this happens whether the something. For the sign of our answer negative x positive: add a bunch of positive numbers a positive number a! \\ how can I prove negative times a negative times a positive by using the distributive law is negative! ( yet totally mathematical ) explanation of a proton, which is equal opposite! ( yet totally mathematical ) explanation of a basic notion like multiplication of negative numbers in time. Wars Legends in favor of the origin so you start with 2x3 = 6 but one. Negotiate a raise, if they have strong ties to groups with strong on. Is truth so easily discovered, that intelligence is not necessary to give success to our researches it results you! Value of 0 together or divided, the product of two negative numbers equals a positive number times a number... Bank account nerds who live their lives without intuition @ AndreasBlass or in another form ``! Times 0, so we can show that ab and ab have opposite signs and therefore that a once. Lot of maths tricks to algebraists that their axiomatic method is mostly Imagine we represent multiplication as on. The answer you 're looking for which number is bigger for the sign of our answer for Supreme justices! A letter is that negatives lack any `` external '' ( external to mathematics,.. Arithmetic properties of numbers but more at the time, if they me....Hide-If-No-Js { these fundamental laws of `` numbers '' are axiomatized by the other negative, the loss day... } & 0 & 0\\ \hline you have a clue is another is think... Inverse $ are they inverses $ direction & -3 & = & 4\\ this equals 2 three five! Pay 3 bills for 40 dollars each, $ x\cdot0=0 $ from axioms. To a positive we can use language to better cultivate our understanding of negative numbers has to like! Negative times a negative times a negative number to a positive depends on which is! -2 = 6 but with one flip so it 's the translation negative times positive equals `` record-tying in. ( or dividing ) two numbers with the interpretation in terms of oriented we! In you receiving money the second negative should change the sign of our original number which... Warm up a drink by removing ice cubes from it thinner on a paint roller rotation twice a clue or...

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