Addition and subtraction
We can count forwards: 1, 2, 3, 4, …and the question is: if you can go in one way, can you go the opposite way? The answer is: negative numbers
Now we can go forwards and backwards as far as we want.
+ is the positive sign
– Is the negative sign
No sign means positive
What you see above is called the number line. The negative numbers fall to the left of 0. (We say that a negative number is less than 0.) The positive numbers fall to the right. (We say that a positive number is greater than 0.)
We imagine every number to be on the number line. And so the fraction ½ will fall between 0 and 1; the fraction −½ is between 0 and −1; and so on.
IN ARITHMETICS we cannot subtract a larger number from a smaller:
2 − 3.
But in algebra we can. And to do it, we invent “negative” numbers.
2 − 3 = −1 (“Minus 1” or “Negative 1”).
What are the two parts of a signed number?
Its algebraic sign, + or − , and its absolute value, which is simply the arithmetical value, that is, the number without its sign.
The algebraic sign of +3 (“plus 3” or “positive 3”) is + , and its absolute value is 3.
The algebraic sign of −3 (“negative 3” or “minus 3”) is − . The absolute value of −3 is also 3.
If a number has no written sign it means that is positive.
Positive numbers start from zero and go to the right, negative numbers start from zero and go to the left.
Adding positive numbers
In algebra we speak of “adding,” even though there are minus signs.
Considering that positive numbers start at zero and go to the right and negative numbers start at zero and go to the left, if you want to add 5+(-9) you start from zero and go five “steps” to the right, from that point you jump nine steps to the left.
Look at the number line below:
The answer is -4.
Adding negative numbers works like a “regular” addition, but in the opposite direction: instead of going to the right you jump to the left.
Look at the number line to see how -2+(-3) is calculated:
The answer is -5.
- If the terms have the same sign, add their absolute values, and keep that same sign.
2 + 3 = 5. −2 + (−3) = −5. −2 − 3 = −5.
- If the terms have opposite signs, subtract the smaller in absolute value from the larger, and keep the sign of the larger.
2 + (−3) = −1. −2 + 3 = 1.
Algebra, after all, imitates arithmetic, and it is easy to justify these rules by considering money coming in or going out. For example, if you borrow $10 and then pay back $4, we express that algebraically as
−10 + 4 = −6.
You now owe $6.
Subtracting positive numbers
What sense can we make of
2 − (−5) ?
Remember that every number has a negative. For example the negative of 5 is -5…..and the negative of -5 is –(-5) that is 5. This rule will be true for any number.
The first term does not change. Changes only the second!
For example:+6-(+3)=+3 but careful:
So Positive minus Positive gives you:
POSITIVE if the first number is bigger, but NEGATIVE if the first is smaller!
Here is a fundamental rule for 0:
Adding 0 to any term does not change it.
- a) 0 + 6 = 6 b) 0 + (−6) = −6
- c) 0 − 6 = −6 d) −6 + 0 = −6
Positive and negative together
Two like signs become a positive sign
Two unlike signs become a negative sign
Exercises (solutions at the end)