Each of these symbols + (plus) and - (minus) has different meanings. First, it can indicate that a number is either positive or negative. Second, it can indicate the operations (addition or subtraction) between numbers.
In section 2.1, we have learned to add two whole numbers. In fact, we can make use of a number line to illustrate the addition of whole numbers. Let us look at a problem for which we know the answer. Suppose we want to add numbers 3 and 4. The problem is written as 3 + 4. To put it on the number line, we read the problem as follows:
Step 1: The number 3 tells us that to "start at the origin and move 3 units in the positive direction."
Step 2: The + sign is read "and then move."
Step 3: The 4 means "4 units in the positive direction."
To summarize, 3 + 4 means startng from the origin, we move 3 units in the positive direction, and then move 4 units more in thre positive direction.
We end up at 7, which is the answer to our problem: 3 + 4=7.
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To add two integers with
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We are now ready to subtract integers. Suppose we use positive integers to indicate money earned and negative integers to indicate money spent. If we earn Php100 and we need to spend Php120, then we are short of Php20. Thus,
100 - 120=100 +(-120)=-20
Also,
-50 - (+100)=-50 + (-100)=-150. -- Taking away (subtract) the money earned is the same as adding the expenditure.
because if we spend Php50 and then spend Php100 more, we are now short of Php150.
What about -100 - (-30)? We claim that
-100 - (-30)=-70
because if we spend Php100 and then subtract (take away) a Php30 expenditure (represented by -30), we save Php30; that is
-100 - (-30)=-100 + 30=-70 -- When we take away (subtract) a Php30 expenditure, we save (add) Php30.
In general, we have the following definition.
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To subtract two integers a and b, the difference a - b=a + (-b). In other words, to subtract b from a, add the opposite or additive inverse of b to a. |
This definition of subtraction is not in conflict with that we already know, but it allows us to do subtraction using negative numbers.
In elementary math, we learned te meaning of multiplication as repeated addition:
For example, 3 x 4=4 + 4 + 4=12
Applying this idea, we have the following:
From the first two examples, we can observe that the product of two positive factors is positive. That is, if a and b are both positive, a x b is positive. In the last two examples, we can observe that the product of a positive and negative number is negative. That is, (+a) x (-b)=-(a x b).
Similarly, we have (-a) x (+b)=-(a x b).
What is the value of (-2) x (-3)?
To answer this question, consider the following:
(-2) x (+5)=-10=-[2 x (+5)]
(-2) x (+6)=-12=-[2 x (+6)]
Following this pattern, we should have
(-2) x (-3)=-[2 x (-3)
=-(-6)
=6.
From the above result, we can see that
(-a) x (-b)=+(a x b).
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In general, for any two integers a and b, (+a) x (+b)=ab |
We have already studied that division of whole numbers is the inverse of multiplication, that is 2 x 3=6 means 6 / 2=3 and 6 / 3=2.
The ruled in multiplication of integers are also applicable to division of integers. Let us consider some examples of division.
Here, we need to find a number which when multiplied by (-2) gives the product +6.
Since (-2) x (-3)=6, therefore 6 / (-2)=-3.
The answer is -3.
Here we need to find a numbe which (-5) should be multiplied so that the product is (-50).
Since (-5) x (+10)=-50, thus, (-50) / (-5)=+10.
The answrr is +10.
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The rules for division of integers are as follows:
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Power of IntegersFor any integer a and any natural numbers n, an can be determined by the following these steps. Step 1: Observe the sign of integer a. Step 2: If a is positive, an would be positive. Step 3:
Step 4: an=(Sign defined by step 2 or step 3) x |a|n. |