Subtraction

Subtraction is the mathematical process of taking a quantity away from another to find the difference between two numbers. It’s written as $$ a - b = c $$ where "a" is the number you’re subtracting from (the minuend), "b" is what you’re subtracting (the subtrahend), and "c" is the result (the difference).
$subtraction$
In other words, the difference (c) between two numbers is a natural number which, when added to the subtrahend (b), equals the minuend (a). $$ a+c=b $$

Subtraction is all around us: when you check your bank balance after shopping, when you take books off a shelf, or when you calculate how much time is left to finish a task.

Let me give you an example.

Imagine you have 10 cookies. Now comes the subtraction: you decide to eat 3.

The question is: how many do you have left? It’s simple: 10 minus 3 leaves 7. Voilà, you’ve just done a subtraction!

$$ 10 - 3 = 7 $$

In this example, 10 is the minuend, 3 is the subtrahend, and 7 is the difference.

Want to see another example?

Let’s say you have 20 euros, and you spend 6 euros on a book and another 8 euros on pizza. How much money do you have left? First, subtract the cost of the book:

$$ 20 - 6 = 14 $$

Then subtract the cost of the pizza:

$$ 14 - 8 = 6 $$

In the end, you have 6 euros left. See how each subtraction gradually reduces what you started with?

Now, here’s something interesting to consider.

What happens if you try to subtract a bigger number from a smaller one?
Subtraction as the inverse of addition
Subtraction of Signed Numbers
Properties of subtraction

What happens if you try to subtract a bigger number from a smaller one?

This is where negative numbers come into play!

Let’s say you have 5 cookies, but for some reason, you promised a friend 8 cookies. Now you’re in trouble:

$$ 5 - 8 = -3 $$

What does this negative 3 mean? It simply means you owe your friend 3 cookies.

This concept also applies to things like bank accounts: if you spend more money than you have, you end up in the negative, or in debt.

Subtraction as the inverse of addition

Another useful way to think about subtraction is as the inverse of addition.

For example, if 7 minus 3 equals 4:

$$ 7 - 3 = 4 $$

You can think of it in reverse: if you add 3 back to that 4, you get 7 again.

$$ 4 + 3 = 7 $$

Once you understand this reversal with addition, subtraction becomes a really handy tool!

Subtraction of Signed Numbers

The difference between two signed integers is calculated by adding the first number (the minuend) to the opposite of the second number (the subtrahend).

$$ a - b = a + (-b) $$

This method turns subtraction into addition, keeping things simple without introducing extra rules.

Unlike with natural numbers, subtraction is an operation that stays within the set of integers and, more broadly, within the set of real numbers.

This means that the difference between two integers is always another integer.

$$ 3 - 5 = 3 + (-5) = -2 $$

More generally, subtraction is closed within the set of real numbers as well.

Properties of subtraction

Now things get a little more interesting because subtraction, unlike addition, has some quirks. It doesn’t always behave as expected.

But don’t worry, there are just a few key points to keep in mind:

Invariant Property of Subtraction
The difference between two numbers $ a - b $ stays the same if the same number $ c $ is either added to or subtracted from both. $$ a-b = (a-c)-(b-c) $$ $$ a+b = (a+c)-(b+c) $$
Example. The difference between 10 and 6 is 4. If you add 3 to both the minuend and the subtrahend, the difference remains the same: $$ (10+3)-(6+3)=13-9= 4 $$. Likewise, if you subtract 3 from both, the difference still doesn’t change: $$ (10-3)-(6-3)=7-3= 4 $$
It’s not commutative
Commutativity works perfectly with addition: if you say $5 + 3 = 8$, then $3 + 5 = 8$ as well. The order doesn’t matter. It’s like the numbers saying: "You go first, or I’ll go first—it doesn’t change where we end up." But subtraction doesn’t work that way! Try subtracting 5 from 3: $$ 3 - 5 = -2 $$ Now reverse the order and subtract 3 from 5: $$ 5 - 3 = 2 $$ See the difference? The results are completely different! This means subtraction is not commutative. You can’t just switch the numbers around because it changes everything.
It’s not associative
With addition, you can group numbers however you like, thanks to its associative property: $$ (2 + 3) + 4 = 2 + (3 + 4) $$ $$ 5+4 = 2+7 $$ $$ 9 = 9 $$ But subtraction is different. Look at this: $$ (10 - 5) - 2 $$ $$ 5 - 2 = 3 $$ But if you group them differently: $$ 10 - (5 - 2) $$ $$ 10 - 3 = 7 $$ So, subtraction is not associative. You can’t just move the parentheses around like with addition, because it changes the result.
The identity element
Every operation has an "identity element," a number that doesn’t change anything. In addition, it’s zero, and in subtraction, zero plays a similar role. If you subtract zero from a number, nothing changes. $$ a - 0 = a $$ If you have 10 apples and no one takes any, you still have 10 apples. Simple, right?
But watch out! If you subtract a number from itself, you get zero: $a - a = 0$. It’s like saying, "I have 5 apples, and I take all 5 apples away." What’s left? Nothing, zero! Remember, subtraction isn’t commutative, as we mentioned earlier.
Subtracting negative numbers
Here’s where subtraction gets a bit tricky. What happens when you subtract a negative number? You might have heard the saying, "Two negatives make a positive." In math, that’s absolutely true! If you have $$ 5 - (-3) $$ This is like saying, "I’m removing a debt." And what happens when you remove a debt? You end up with more! So, $$ 5 - (-3) = 5 + 3 = 8 $$ In other words, subtracting a negative number is the same as adding it.
Tip: When you see subtraction with a negative subtrahend, like $$ 5 - (-3) $$ just imagine there’s an implied $ 1 \times $ between the minus sign and the parentheses: $$ 5 - 1 \times (-3) $$ In algebra, multiplying two negatives gives a positive. So multiplying -1 by -3 gives you 3. $$ 5 + 3 $$ Now you’ve turned it into addition, and the result is clearer: $$ 5 + 3 = 8 $$
It’s not a closed operation with natural numbers
Here’s another interesting point. When you subtract two natural numbers (like 5 - 3 = 2), you stay in the realm of natural numbers—the numbers we use for counting (0,1,2,3,...). But if you subtract a bigger number from a smaller one (like 3 - 5 = -2), you enter the world of negative numbers, which aren’t considered "natural."
So, subtraction can take you out of the set of natural numbers and into the world of negative numbers. You won’t always stay in the same "set" of numbers, depending on what values you subtract. That’s why we say subtraction isn’t "closed" with natural numbers.

In summary, subtraction is kind of a "rule-breaker" in math. It doesn’t follow the same easy rules as addition; it’s a bit more complicated, and you have to pay attention to the order and placement of the numbers.

But once you get the hang of it, subtraction becomes a crucial tool that opens up more advanced concepts, like negative numbers, which are incredibly useful for explaining many real-world situations.