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Positive and Negative Numbers - I
Arithmetic and fractions, positive and negative numbers. In this video, we will discuss how to add and subtract positive and negative numbers. Now this a very basic topic. Again, if you are proficient in this topic, do not feel compelled to watch this entire video.
This is a video designed to get people comfortable with this topic if it is unfamiliar or they have some uncertainty with the topic. So wherever you are starting, I will assume that you are proficient with two basic cases, how to add two positive integers or how to subtract two positive integers when they are in the form (bigger) minus (smaller).
First thing I’ll say is if you need practice with this, practice every day. It’s very important to be proficient in two digit addition and subtraction. Be able to do that as mental math that will make the test much smoother. Now the good news is, if you can do these two things, you can do anything else. This entire topic is very easy if you know these two things. There are many ways to discuss this material.
Let’s begin with subtraction. Some mathematicians would say subtraction doesn’t really exist. What does that mean? Well, subtraction of any number can be rewritten as the addition of a number of the opposite sign. And so, some mathematician would say that this addition is actually the true form.
So, let’s make sure we understand this. Subtraction of any number can be rewritten as the addition of a number of the opposite sign. Here are four different instances of subtraction. We have a positive minus a positive, a negative minus a positive, a positive minus a negative, and a negative minus a negative.
In all four cases we could rewrite that subtraction as addition of a number of the opposite sign. In the cases where we are subtracting a positive that’s the same as adding a negative in the cases where we are subtracting a negative that’s the same as adding a positive.
Well, notice that we get some simplification, but it’s not a simplification in every case. For example in the first one, it looks like we are better off where we started, we are better off without changing it to addition. In the third one, it looks like we clearly made things better off by changing it to the addition of two positive numbers.
In that fourth one, notice that now its addition, its commutative so we can switch the order around and when we switch the order around, we can rewrite it as subtraction and that is much easier. So sometimes this is really an important move for simplification but not always. You don’t always have to rewrite subtraction as addition but it can be a very good simplifying trick to have up your sleeve.
Notice in particular for the case positive minus negative, this trick will always simplify it. It will always become positive plus positive, which is one of those fundamental things that I assume you know how to do already. Now let’s look at that tricky double negative case, which could appear in the form negative minus positive, or in the form negative plus negative.
The big idea is we can always factor out a negative sign. Now what does this mean exactly? Let’s look at that first one. Negative 46 minus 37. We can factor out a negative sign, if we factor out a negative sign everything inside becomes positive.
So, it just becomes 46 plus 37. Addition of two positive numbers. So you perform that addition and then just stick a negative in front of the sum. You might want to try these others on the page. Pause the video here. Try these others and then you can compare your answers to mine.
Here are the answers. One other case folks find tricky is the case small positive minus big positive, which also shows up as small positive plus big negative. Here, the big idea is factoring out a negative sign reverses the order of subtraction. So what is this mean exactly, suppose I have 23 minus 64 by factor out a negative sign then what I get is subtraction in the reverse order, 64 minus 23.
Well, now that’s bigger minus smaller that we can do that’s one of the fundamental skills. So do that subtraction, and then just stick a negative sign in front of it. Let’s look at another one, 26 minus 63, factor out the negative, and we get a negative in front of the reversed order of subtraction, 63 minus 26. Perform the subtraction and stick a negative sign in front of it.
Here’s some more. You might wanna pause the video here and practice these on your own. Here are the answers I get. Very important that you are able to do things like this, and it’s very good practice for mental math to be able to do this in your head. These ideas allow you to change any addition or subtraction to either the sum of two positives or the difference of larger minus smaller.
Here, I just discussed integers for simplicity, but all these same ideas would also be true for positive and negative decimals and fractions. The core skills are addition of two positives or larger positive minus smaller negative, smaller positive. From here if we’re doing positive minus negative, we can change that to positive plus positive.
If we have the double negative case, we can factor out a negative sign. And whenever we have smaller minus bigger, we can factor out the negative sign, reverse the order of subtraction and then what’s inside bigger minus smaller that’s something we can do.
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