INSTRUCTORS Carleen Eaton Grant Fraser Eric Smith

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Operations on Numbers

• To add numbers that are the same sign, add the absolute value of each number together. The answer will have the same sign as the sign of the two numbers.
• Adding numbers that are different in sign can be done using subtraction. To subtract numbers, find the difference of the absolute value of the two numbers. The answer will have the same sign as the number that was larger in absolute value.
• When multiplying and dividing, if the numbers have the same sign the result will be positive. If the numbers have a different sign, the result will be negative.
• When raising a number to a power of a natural number, we can think of it as repeated multiplication
• When taking the square root of a number, we want the positive number that when multiplied by itself would give use the number under the square root.
• Square roots can be split up over multiplication and division. Be careful not to split up a square root over addition or subtraction.

Operations on Numbers

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Objectives 0:06
• Operations on Numbers 0:25
• Subtraction
• Multiplication & Division
• Exponents
• Bases
• Square Roots
• Principle Square Roots
• Perfect Squares
• Simplifying and Combining Roots
• Example 1 8:16
• Example 2 12:30
• Example 3 14:02
• Example 4 16:27

Transcription: Operations on Numbers

Welcome back to www.educator.com.0000

In this lesson we are going to take a look at operations on numbers.0002

When you hear me use that word operations, I'm talking about ways that we can combine together numbers or do something to a number.0007

It gets very familiar things such as adding, subtracting, multiplying and dividing.0015

As well as exponents and square roots.0020

To combine numbers together, we use a lot of familiar operations in order to do so.0028

Again, adding, subtracting, stuff like that.0033

Note that depending on the types of numbers being used, certain rules applies that will help us put them together.0036

The rules that I will focus on are the ones that involve positive and negative numbers.0042

How should we deal with those negative signs? Let us see what we can do with addition.0048

When you add numbers that have the same sign then you are looking at adding their absolute values together.0055

The results if they do have the same sign or have the same sign as the original numbers.0062

In other words, if you are adding the other two positive numbers its result will be positive.0068

For adding together two negative numbers, then your result will be negative.0073

Now if you happen to have two numbers that you are adding together and they are different in sign,0079

then we will actually handle these using subtraction.0084

Wait for the rules on what to do with positive and negative on my next slide.0087

When dealing with subtraction and what we want to do is subtract the number that is smaller in absolute value0095

from the number that is larger in absolute value and looking at each of their absolute values taking the smaller one from the larger one.0102

When looking at the final result, the answer will have the same sign as whichever number was larger in absolute value.0111

If your larger number was or the larger number in absolute value is negative, then the result will be negative.0119

Remember that we will also think of adding negative numbers using subtraction in this way.0128

In different way we are writing it but instead we are really using subtraction.0134

The other two familiar operations that we have are multiplication and division.0142

The way we handle a lot of signs with these ones are by thinking of this.0147

If we multiply a negative × a negative then the result will be positive.0153

A negative times a negative results positive.0158

If we take two things that are different in sign such as negative times a positive, then the result will be negative.0164

Now these are the same two rules that we end up using for division.0175

If we divide a negative by a negative, you will get a positive.0180

Negative divided by negative result positive.0184

If they are different in sign such as negative divided by positive, then the result is negative.0189

Keep these in mind when working with that your signs, multiplication and division.0198

When we have an exponent, we can think of it as taking a number and multiplying it by itself in a certain number of times.0210

For example, maybe I have 37, 7 would be our exponent.0218

I can interpret that as multiplying that number by itself 7 times or some other key vocabulary you want to pick up on.0224

The exponent is that number raised next of the 3, exponent.0236

In the base of the number is the number that we are actually being repeatedly multiplied, base.0244

Once we see what number that is and how many times we need to multiply it, usually we can go ahead and simplify from there.0252

There are some common exponents that we usually give other names.0258

If I'm taking 5 and I'm raising it to the power of 2, 5 × 5, this is often said 52.0262

Another good common one would be something like 23, 2 × 2 × 2 that would be 8.0278

I could also say that this is 23.0290

Key on these special words for some of these other powers.0294

Another operation that we can do with numbers is taking the root of a number.0303

The principal square root of a number is the non negative number (n)0309

that you know if I were to say multiply it by itself, I would end up giving that (n).0313

This seems a little funny special worry but let us see if I can describe it using (n) as an example.0319

Let us say I wanted to find out the square root of 25.0326

What I'm looking for is what number would multiply it by itself in order to get a 25? That has to be 5.0330

It is what I'm talking about here, positive number such that when it is multiplied by itself you get that number underneath the root.0340

For this reason we have a little bit of a problem with our negatives underneath the root.0353

After all, what number would you multiply it by itself in order to get -16?0360

We learned from our rules of positive and negative numbers that -4 and +4 would work to get 160364

but unfortunately they are different in sign and we need them to be exactly the same, that is not going to work.0373

We are dealing with the principal square root because we are only interested in the positive numbers0382

that when multiplied by themselves would give us that number.0387

A perfect square is a number that is the square of a whole number and this one is usually reduced very nicely.0393

For example the square root of 9 would be the example of a perfect square, as it reduces down to a nice whole number, 3.0401

If you have a (a, b) being non negative real numbers,0416

then there are a few different ways that you can say combine or rip apart those roots.0421

3 multiplied underneath the same root and you can apply the root to each of those pieces.0426

If you are dividing and you have a root then you can put it over each of its pieces in the numerator and in the denominator.0432

You can use these rules in two different ways to simplify or combine words together.0442

We will see that a lot in some future lessons.0446

I'm pointing this out now so you do not make a common mistake.0449

Do not try and split up your roots over addition or subtraction.0453

We do not have a rule to do that yet or a good way to handle it.0458

In fact, in this example I have written below you can see that the two are not equal by simply evaluating each side.0461

9 + 16 would be a 25 and the square root of 25 is 5.0469

Looking at the right side square root of 9 is 3, square root of 16 is 4.0475

I'm putting those together I will get 7 and you can see that these things are not the same.0482

Be very careful when working with your roots.0492

Now that we know a few things about combining these, let us go through some examples and just practice with them.0498

The first one I want to add together a -3 and a -6.0504

I'm adding together two numbers that have exactly the same sign.0510

I will look at their absolute value and add those together.0514

The absolute value of 3 is 3, the absolute value of -6 is 6, if I combine those together I will get 9.0519

Since I'm adding together numbers that have exactly the same sign, the result will also have the same sign.0531

Adding two negative numbers my result is negative, -3+ -6 is a -9.0537

Moving on, 19 + a -12, I want to think of this as a subtraction problem since their signs are different.0545

How do I handle subtraction? Again I will look at their absolute value.0558

The absolute value of 19 and the absolute value of a-12, 19, 12.0562

I will subtract the smaller number from the larger number 19-12, what will that give me? I will get7.0571

I have to determine what sign this should be.0584

In subtraction we take the same sign as the larger number.0588

19 was larger in absolute value, it was positive, I know my result is 7.0594

If we do get a positive result as our answer, we do not write that positive sign up there.0604

We are just having 7, we will assume that that is a positive 7.0609

-8-11, an interesting way we can look at the problem, we could look at this as adding a -11.0614

The reason why I have looked at it in that way is that I could use my rules for addition.0626

If I'm adding things that have the same sign, I have looked at the absolute value of each of them, 8, 11.0631

Then I can simply add up those two values and get 19.0640

Since I'm adding two negative numbers, I know my result will also be negative, -19.0645

Let us try another one, 8 - -13.0651

When you subtract the negative this is another good situation that you could end up rewriting in a much simpler form.0657

When you subtract the negative, you can change it into addition.0662

This is 8 + 13 and now I'm adding together two positive numbers.0667

You made a positive 21 and one more, negative the absolute value of a -4 + 9.0673

Let us start inside those absolute values and see what we can do.0683

I'm adding together two things but they have different signs.0687

Let us look at the absolute value of a -4 and the absolute value of 9, 4, 9.0692

We want to subtract the smaller value from the larger value, 9-4 and that result would be 5.0699

I know what sign should that have, or the number that is larger in absolute value is 9.0708

And that was a positive value over here, I'm looking at a 5.0716

There are a lot of other things I have left out here so far,0721

those would be the absolute values and that leading negative sign.0724

Let us go ahead and put those in there now.0727

I want to do the absolute value of 5, all that is simply 5.0731

I still have that negative sign hanging out front and it is been there since the very beginning.0740

I can see that after done evaluating this one all the way, my answer is actually a 5.0743

Let us work on adding, multiplying and dividing the following numbers.0752

We only have one rule to take care of that is when we are multiplying together two negative numbers, we get a positive.0757

If we are multiplying together two numbers that are different in sign, the result should be negative.0766

4 × -7, I just want to think of 4 × 7 that would be 28.0775

Since they are different in sign, I know that this will be a -28.0784

Moving on, -6 × -5, 6 × 5 would give me 30 and now here I have a - × - I know the result will be positive.0789

But again we usually do not write that positive sign in there so just leave this as 30.0801

-12 ÷ 3, 12 ÷ 3 would be a 4, negative ÷ positive would be negative, our result is a -4.0806

One last one, -60 ÷ a -5, 60 ÷ 5 goes in there 12 times and negative ÷ negative is positive, my result is a 12.0820

You need to be very careful with these rules for multiplication, make sure you have these memorized.0835

Let us do a few involving our exponents.0843

Remember we can think of these as repeated multiplication.0846

2/34, it can be the same as 2/3 × 2/3 × 2/3 × 2/3, we are doing it 4 times.0850

I simply multiply it across the top and across the bottom 2 × 2 × 2 × 2 would be 16.0862

Then I have 3 × 3 × 3 × 3 = 81.0871

All the numbers here are positive so I know my result is positive.0876

Here is a tricky one, -5 ×-5 × -5, let us take this two other times so we can what is going on here.0881

Here I have a -5 × -5 the result of taking 5 × 5 would be 25.0896

Taking a negative × negative I would know that this would be 25.0904

That looks good, let us go ahead and work in this last value of -5.0910

We want to multiply 25 × -5, the result there would be 125.0918

Since I'm multiplying a positive × a negative result is -125, -125 would be my answer.0929

The next one looks very similar but it is actually very different.0939

That one is 23 and that negative sign is just out front of all that.0943

We recognize that the 2 is the base and that the negative is not included in that base0949

since there is no parenthesis given to group it in that way.0956

We want to figure this one as 2 × 2 × 2, I have multiplied it out three times.0960

And as for what to do with that negative sign, if we have not put it up front it is a long pretty ride.0966

2 × 2 × 2 that would give me an 8, all of those numbers are positive, 8.0973

Of course let us put our negative sign out front since it was out front at the very beginning.0979

We can see that our result is -8.0984

Now on to some square roots, these ones we simply just want to break them down and simplify them as much as possible.0990

The first one I have the square root of 64.0998

Think for yourself what number would you have to multiply it by itself in order to get 64?1001

I have a couple of options that could be 8 and 8, that would have given me 64.1008

Or it could be -8 × -8, that would also give me 64.1013

We are only interested in the positive values that do so, let us not worry about those -8.1018

We will say that our answer for the square of 64 is 8.1024

Moving on, the square root of 169 ÷ 81.1031

This one we can use one of our rules and break up the root over the top and over the bottom.1037

Now we can end up taking the root of each of these individually.1045

What number multiplied by itself would give us a 169? That have to be a 13.1048

A number that would be multiply by itself to get an 81? 9.1055

The answer in this one is 13/9.1060

Continuing on, this one has a negative sign out front but that is not underneath the root.1064

I would not worry about it just yet instead let us just focus on the square root of 36.1071

That would be 6, of course we will go ahead and put our negative sign and see that our final result is a -6.1079

One more, this last one involves the square root.1091

What two numbers when multiplied together would give us a -49 remember they must be the same.1095

We got a few problems, do not we? If I try and use 7 and 7 that would give me 49, that does not work.1102

If I try and use a couple of-7, that does not work, that still gives me 49.1109

I can not use one positive and one negative even though those give me a -49.1114

Those are not the same sign, one is positive and one is negative.1121

What is going on here, if you remember about your types of numbers, these are imaginary numbers.1125

I will leave that one just as it is until we learn about simplifying imaginary numbers in some later lessons.1137

Some various different ways that you can go ahead and combine numbers using some very familiar operations.1144

Remember that most of the rules that I covered will give you some tips on what to do when they are different in signs.1150

Positive, negative, negative, positive and all those will be handy in figuring out the overall sign of your answer.1155

Thank you for watching www.educator.com.1166