Subtracting Negative Numbers From Positive Numbers
By Kathleen Cantor, 07 Oct 2020
Addition and subtraction can get complex the more numbers, signs and variables are involved. This is why, before we dive into the subtraction of negative numbers from positive numbers, we will revisit the basics of numbers, addition and subtraction first so that the concepts are built on one another.
Numbers can be positive or negative
Numbers can be positive or negative. Think of all the numbers that exist; they are all on a spectrum from negative through zero to positive.
Imagine all these numbers on a line; this will make it easier for you to envision each number as positive or negative. For example:
-9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
All of the numbers to the right of the zero are positive.
Denoting positive integers
When denoting positive numbers or integers, we do not need to add the plus "+" sign before the number; it is assumed that the number is positive when there is no negative '-' sign.
This means that the number '3' is actually '+3'. If we want 3 to be noted as negative, we would have to list it as '-3'.
- Adding a positive number to another positive number
When we are adding two or more positive numbers together, we usually use the plus '+' sign. For example:
5 + 7 = 12
5 and 7 are both positive numbers. 7 is adding to the value of 5 and thus increasing the total value of the sum.
- Subtracting a positive number from another positive number
When we are subtracting a positive number from another positive number, we usually use the minus '-' sign. For example:
8 - 4 = 4
8 and 4 and both positive numbers (note the lack of the minus sign before either) and thus 4 is taking away value from eight.
- Adding a negative number to a positive number or vice versa
When the negative sign '-' and the positive sign are added together, the sign with the larger value will be the sign of the sum total. For example:
3 + (-2) = 1 i.e. (+1) because the positive sign has a larger value than the negative sign.
Another example is 2 + (-4) = (-2) because the negative sign has a higher value than that of the positive sign.
or even (-5) + 7 = 2 i.e. (+2) because the positive sign has the larger value.
Remember that a positive number plus a negative number is always a subtraction, and the sign of the total is determined by which of the signs had a larger magnitude. i.e
4 +( -2) = 4 - 2 = 2
4 + (-6) = 4 - 6 = (-2)
- Subtracting a positive number from a negative number
When subtracting a positive number from a negative number, the difference will always have a negative sign. This is because you are actually adding the numbers on the negative side of the number line.
Think of the number line: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
Now imagine taking away 2 from 5, what would you do? You would take two steps to the left from five, and that would make your answer 3. The same concept is applied when taking away a positive number from a negative number. For example
(-3) - 2 = (-5) that is, you take two steps to the left of negative three on the number line in order to get your difference.
This gives the same effect as adding two positive numbers, only with a negative sign added before the answer. So, next time you are stuck when subtracting a positive number from a negative number, simply add the two numbers together and put the negative sign '-' before the sum total. For example:
-5 - 7 = -5 - (+7) = (-12) but if you are stuck simply change the question to 5 + 7 = 12 and then add the negative sign '-' to your answer.
- Subtracting a negative number from another negative number
When subtracting a negative number from another negative number, the minus sign will become positive. This is due to the fact that a double minus signifies a multiplication of signs, that is, -2 - -2 = -2 + 2.
The negative sign for the number two, together with the minus sign that denotes subtraction yield a positive sign; two negatives equal a positive.
Therefore, when subtracting a negative number from another negative number, remember that the sign turns positive.
-4 -( -5) = 4 +5 = 9
- Subtracting a negative number from a positive number
When subtracting a negative number from a positive number, the sign will also become a positive sign because, as discussed in step number five, a negative times a negative is a positive.
Knowing that this same principle applies, we can now begin subtracting.
5 - (-9) = 5 + 9 = 14
A negative number subtracted from a positive number always yields a positive result.
The key take away is this:
(+) (+) = + [a positive and a positive will yield a positive]
(+) (-) = - [ a positive and a negative will yield a negative]
(-) (+) = - [ a negative and a positive will also yield a negative]
(-) (-) = + [ a negative and a negative will yield a positive]
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