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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 is 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're proficient with two basic cases: how to add two positive integers or how to subtract two positive integers when they're 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 re-written as the addition of a number of the opposite sign. And so, some mathematicians would say that this addition is actually the true form. So, let's make sure we understand this.

Subtraction of any number can be re-written 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 were better off where we started, we were better off without changing it to addition. In the third one, it looks like we've clearly made things better off by changing it to the addition of two positive numbers. In that fourth one, notice that now it's addition, it's commutative, so we can switch the order around, and when we switch the order around we can rewrite it as subtraction and that's 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)- (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. (-46) -37.

We can factor out a negative sign. If we factor out a negative sign, everything inside becomes positive. So it just becomes 46 + 37, addition of two positive numbers. So you perform that addition and then just stick a negative in front of the sum. You might wanna 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 does this mean exactly? Suppose I have 23- 64, if I factor out a negative sign, then what I get is subtraction in the reverse order, 64-23. 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- 63, factor out the negative and we got a negative in front of the reversed order subtraction, 63- 26, perform the subtraction and stick a negative sign in front of it. Here are some more, you might want to pause the video here and practice these on your own.

Here are the answers I get. Very important that you're 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 into either the sum of two positives or the difference of larger minus smaller. Here I just discussed integers with 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 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 a negative sign, reverse the order subtraction. And then, what's inside bigger minus smaller, that's something we can do.

Read full transcriptThis 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're proficient with two basic cases: how to add two positive integers or how to subtract two positive integers when they're 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 re-written as the addition of a number of the opposite sign. And so, some mathematicians would say that this addition is actually the true form. So, let's make sure we understand this.

Subtraction of any number can be re-written 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 were better off where we started, we were better off without changing it to addition. In the third one, it looks like we've clearly made things better off by changing it to the addition of two positive numbers. In that fourth one, notice that now it's addition, it's commutative, so we can switch the order around, and when we switch the order around we can rewrite it as subtraction and that's 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)- (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. (-46) -37.

We can factor out a negative sign. If we factor out a negative sign, everything inside becomes positive. So it just becomes 46 + 37, addition of two positive numbers. So you perform that addition and then just stick a negative in front of the sum. You might wanna 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 does this mean exactly? Suppose I have 23- 64, if I factor out a negative sign, then what I get is subtraction in the reverse order, 64-23. 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- 63, factor out the negative and we got a negative in front of the reversed order subtraction, 63- 26, perform the subtraction and stick a negative sign in front of it. Here are some more, you might want to pause the video here and practice these on your own.

Here are the answers I get. Very important that you're 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 into either the sum of two positives or the difference of larger minus smaller. Here I just discussed integers with 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 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 a negative sign, reverse the order subtraction. And then, what's inside bigger minus smaller, that's something we can do.