Algebra: Consequences for Real Numbers
Consequences for Real Numbers
In this algebra video tutorial, I explain and prove some consequences of the properties of the real numbers. These consequences are rules that are commonly used in algebra and a proper understanding of how they are derived is essential to learning algebra. For this purpose, I have created a reference page where you can analyze the proofs in the video more deeply at your own pace.
Here, I will go through the basic thinking involved in writing one of these proofs from scratch. In this instance, I will look at how to write the proof of the Multiplication by Zero rule: a0 = 0. The steps of the proof are listed at the bottom of the page as a reminder.
To begin, consider what we start with and where we need to go. We start with a0 and we need to end up with 0. So, something must cancel out the a0 term. Using the properties of real numbers, we see that there are two ways to cancel out a term: the inverse or the identity. Since we have the term a0 which is not known be zero, we will need to use its inverse at some point to remove it. That begins our thinking.
Now, in order to get the inverse, we need something to pull it from. The most obvious option we have is to use the additive identity to get a zero term. That's easy. So, we use the additive identity property to get a0 + 0.
We can use the additive inverse to replace the zero and get a0 + (a0 - a0). At this point, we have the -a0 term that we can use to cancel the original a0 term. However, doing this will get us back to where we started. So, we need to think of something else.
This is the most subtle step of the proof. As a rule, whenever something is not obvious in algebra, it probably involves the distributive property. This is really the key to the proof. Once you realize that step, it does not take too much thinking to apply additive associativity and get (a0 + a0) - a0 before you apply the distributive property to get a(0 + 0) - a0.
Once you can get to this point, the rest should be clear. The additive identity cancels one of the zeroes because it says that adding a number to zero just gives you that number or 0 + 0 = 0. So, this gets us to a0 - a0 and the property of additive inverses finishes off the proof to get to 0.