# Subtraction of a negative number

Why subtraction of a negative number from a positive number is addition? Eg: $a - (-b) = a + b$ When looking through the number scale I am unable to relate this, please kindly clarify ?

• Have you looked at sites like mathsisfun.com/positive-negative-integers.html? They have some explanations that may be worth reading first. – JB King Jun 30 '14 at 7:57
• @JBKing, I went through the example of balloon and weight but not satisfactory with that example. In the number scale, positive numbers come to the right of zero and negative numbers go to the let of zero. If subtraction of negative number is explained using the number scale it would be very helpful. – wcs Jun 30 '14 at 8:29
• This is because the enemy of your enemy is your friend :). – Surb Jun 30 '14 at 8:29
• @Surb but what about zero? :-) :-) – Carl Witthoft Jun 30 '14 at 12:15
• What grade are you in? Are you just now being introduced to this concept? – David Jun 30 '14 at 16:10

Two ways to explain this.

1. One way to understand what subtraction is: $a-b = a+(-b)$. When you subtract $b$ from $a$, you are actually adding $-b$ to $a$. This is the way $a-b$ is defined. With this definition, you know that $a-(-b)$ equals $a+ (-(-b))$. Now, what is the number $-(-b)$? Well, by definition, for any $c$, $-c$ is the number that satisfies the equation $c+(-c) = 0$. This means that the number $x=-(-b)$ is the number that solves the equation $(-b) + x = 0$. Of course, it is obvious that, since $(-b) + b = 0$, this means that $-(-b) = b$, and that means that $$a-(-b) = a+(-(-b)) = a+b.$$

2. In the real line, you can see the number $a+b$ as the number you get when you add the lines $a$ (line from $0$ to $a$) and the line from $0$ to $b$ and put them one after the other (end to end). Subtraction in this sense means that calculating $a-b$ is taking the line from $0$ to $a$ and the line from $b$ to $0$ and putting them end to end. But since the line from $-b$ to $0$ is the same as the line from $0$ to $b$ (just shifted), this also means subtracting $-b$ is the same as adding $b$.

To calculate $a+b$, you take the line from $0$ to $a$ (directed to the right) and the line from $0$ to $b$ and but the end of one to the beginning of the other, so you have something that looks like

:------>:------------>
a          b
:-------------------->
a+b


However, if you are subtracting $b$, you must reverse the direction of the line which belongs to the number you are subtracting, so you have

:------------------>
a       <------:
b
:----------->
a-b


So, the point is that to add $b$ to $a$, you add the line from $0$ to $b$ to the line from $0$ to $a$. If you are subtracting it, you have to flip the direction of the line you are subtracting. The main idea is that if you flip the direction of $b$ and add it to $a$, it's the same thing as if you took the line for $-b$.

• thanks for your explanation. I understood clearly the section one explanation. I am not able to follow the section 2 explanation (using real line). Could you please explain me more on it. I am not able to follow "Subtraction in this sense means that calculating a−b is taking the line from 0 to a and the line from b to 0 and putting them end to end. But since the line from −b to 0 is the same as the line from 0 to b (just shifted), this also means subtracting −b is the same as adding b." – wcs Jun 30 '14 at 10:53
• @wcs I hope my edit helps. – 5xum Jun 30 '14 at 11:08
• thanks a lot for your explanation. I understood now clearly – wcs Jul 1 '14 at 3:29

\begin{align} a - (-b) &= a +\color{red}{0} - (-b) \\ &= a + \color{red}{\left(\; b + (- b) \;\right)} - (-b) \\ &= a + b + \color{blue}{\left(\;(-b)-(-b)\;\right)} \\ &= a + b + \color{blue}{0} \\ &= a + b \end{align}

The idea is this: We "can't" subtract $(-b)$ from an expression ---in this case, $a$--- that doesn't have a $(-b)$ in it, so we cleverly manipulate the expression into an equivalent form that does have a $(-b)$ in it. The easiest way to do this is to add zero (which changes the expression's value not at all), and immediately interpret that as the sum of $b$ and its negator, $(-b)$. At that point, we have a $(-b)$ in our expression, from which we can subtract $(-b)$, achieving our goal. We find ourself with another $0$, which we can ignore.

I'll note that a similar strategy works to explain the rule for "dividing by a reciprocal": \begin{align} a \div \frac{1}{b} &= a \cdot \color{red}{1} \div \frac{1}{b} \\ &= a \cdot \color{red}{\left(b \cdot \frac{1}{b}\right)} \div \frac{1}{b} \\ &= a \cdot b \cdot \color{blue}{\left(\frac{1}{b}\div\frac{1}{b}\right)} \\ &= a\cdot b \cdot \color{blue}{1} \\ &= a b \end{align}

consider what would happen if this were not the case, then 1-(-1) = 0 (I assume this is what you would think it would be otherwise) so we add (-1) to both sides giving 1=(-1) which is absurd! without the use of standard a-(-b)=a+b we run into contradictions

• Not sure why this got downvoted, I think it cuts to the chase rather neatly. – Jack Aidley Jun 30 '14 at 10:44
• @JackAidley I didn't downvote, but I guess the problem is that It only explains why $a-(-b)=a-b$ is not the case (from the example $1-(-1)=0$). It does not explain why $a-(-b)=a+b$ should hold. And $\neg(a-(-b)=a+b)$ is not the same as $a-(-b)=a-b$. – 5xum Jul 1 '14 at 13:27
• my aim was to try to get out of the posters mind that a-(-b) was a-b, it wasn't meant as any kind of rigorous proof – Mmm Jul 3 '14 at 17:08

When it comes to negative numbers, don't bother too much with physical intuition. Ultimately, negative numbers are abstract constructs that we use because they make algebra more elegant. The rules are like the rules of a board game: they're artificial. Asking why such and such rule is true is a bit like asking why the bishop moves diagonally in chess. The answer is "Because we say so".

1. Once you have negative numbers, one possibility is to define $a-b$ as $a+(-b)$, in which case this rule is true because the negative of a negative is a positive, which in turn is true because the definition of $-a$ is "that which, when added to $a$, produces $0$".

2. Or, you could continue to have subtraction as a separate operation from addition. One definition for subtraction is that $a-b$ means "that which, when added to $b$, produces $a$". You can then prove that there's exactly one number satisfying that criterion, and that's $a+(-b)$.

And finally, you can also think of it as "subtracting a debt means someone cancelled the debt, so I gained money", but I still advise against thinking about negative numbers as debts. Thinking about debts can help to illustrate certain things, but ultimately negative numbers are just abstract objects. An example of why this is a bad idea is because it breaks down when you try to explain multiplication. Why is $(-a)(-b)=ab$? What does it mean to multiply two debts together?

(This is a bit like how the ancient Greeks used to think of numbers as lengths, and of multiplying two numbers as calculating an area. Multiplying three numbers was thought of as volume, but what about multiplying together four numbers? Some medieval mathematicians even considered it absurd to multiply together more than three numbers, which shows how limiting it is to tie yourself down with physical intuition.)

Addition and subtraction are operators whereas positive and negative are properties of numbers. The difference can be understood by reducing the definition of these operators to the unary operators increase and decrease.

Let's assume:

• The absolute value of a number is a property independent of its positive or negative property, representing the distance from zero
• We understand the standard ordering relations for positive and negative numbers (i.e. the "number line", where zero is a positive number with absolute value 0, and where values increase to the right of zero and decrease to the left of zero)

Then we can define the following operators.

Increase and decrease are unary operators that shift the absolute value of a number using the standard ordering relations with respect to zero and the number line, to the right and to the left, respectively.

Addition (x + y = z) is an operation that combines the values of a pair of operands (x and y) into a resulting number (z) by the following rules:

1. Assign to z the absolute value of zero
2. If x is positive, increase the absolute value of z by the absolute value of x
3. Else if x is negative, decrease the absolute value of z by the absolute value of x
4. If y is positive, increase the absolute value of z by the absolute value of y
5. Else if y is negative, decrease the absolute value of z by the absolute value of y

Subtraction (x - y = z) is the inverse operation of addition with respect to the increase and decrease operators for the y operand:

1. Assign to z the absolute value of zero
2. If x is positive, increase the absolute value of z by the absolute value of x
3. Else if x is negative, decrease the absolute value of z by the absolute value of x
4. If y is positive, decrease the absolute value of z by the absolute value of y
5. Else if y is negative, increase the absolute value of z by the absolute value of y

In other words, the reason we define subtraction is to provide a convenient operator to generate the opposite result of addition. The brutally laborious definition above serves to disambiguate the differences between properties of numbers and operators. The standard +/- (plus, minus) notation can cause confusion in these rules because - (minus) is used both for the subtraction operator and the negative property.

Instead of using the minus sign for negative numbers, we can use the ~ (tilde) as the notation for negative numbers. We can use our new notation to demonstrate the above operators.

3 + 4 = 7

z = 0 (rule 1)
z = 3 (rule 2, increase positive x)
z = 7 (rule 4, increase positive y)


3 + ~4 = ~1

z = 0 (rule 1)
z = 3 (rule 2, increase positive x)
z = ~1 (rule 5, decrease negative y)


3 - 4 = ~1

subtraction:
z = 0 (rule 1)
z = 3 (rule 2, increase positive x)
z = ~1 (rule 4, decrease positive y)


To answer the original question:

3 - ~4 = 7

subtraction:
z = 0 (rule 1)
z = 3 (rule 2, increase positive x)
z = 7 (rule 5, increase negative y)


Final note: does intuition serve us better in this case? Probably yes. But I like this approach because I find value in different ways of thinking about problems.

I always understood the minus sign as meaning two things:

1. Change direction
2. Subtract

So let's say you have 4. This would be on the right side of 0. -4 would be on the other side of zero, in other words, negative. -(-4) says to take 4, put it on the left side of zero, then on the right side of zero, ergo +4. Of course, in your mind you do this faster, as in -4 you probably already think of as a negative number.

Take the addition 5 + 10 = 15. If you add one less, the result is one less: 5 + 9 = 14. Again, if you add one less, the result is one less: 5 + 8 = 13. Repeat this a few times, until you get 5 + 2 = 7, 5 + 1 = 6, 5 + 0 = 5. Now try to add one less. That means you add 5 + (-1). The result is one less than the previous addition 5 + 0 = 5, so 5 + (-1) = 4. And adding one less again gives 5 + (-2) = 3 and so on.

Since $-a$ is $-1\cdot a$ we see that “$-$” is the abbreviation for “$-1\cdot{}$” (such as “$+$” is for “$+1\cdot{}$”). From here exploiting that $-1\cdot-1=+1$ we arrive in $$a-(-b)=a-1\cdot(-1\cdot b)=a-1\cdot(-1)\cdot b=a+1\cdot b=a+b.$$