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Absolute Value | Calculus And Analytical Geometry | MTH101_Lecture02

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This content provides a comprehensive introduction to the concept of absolute value for real numbers, covering its definition, properties, and applications in solving equations and inequalities, as well as its geometric interpretation as distance.

abstractly speaking is said

what is the order of a real number

inequalities we also saw what is a what

it's very important stuff introductory

what exactly it is that we'll be talking

about in this lecture um

absolute values okay what is an absolute

value of a number

absolute value um

you want to relate the positive square

root of a real number with the absolute

value of that real number is

important theorem uh mathematics

the triangle inequality that will be the

third thing or the last thing we'll see

at the end of this this lecture introduction

absolute value absolute value of a real

definitions

what is the absolute value of a real

number definition is the absolute value

is defined as

a the absolute value of the real number

a is equal to a if a is bigger than

equal to 0 yeah if a is greater than

equal to 0 or in other words

it's a positive number

positive number definition you have over

here the absolute value of a is equal to

minus of a or minus of the real number a

if a happens to be a number less than

zero any a a negative number so a

I think it will become very obviously definition

mathematics something looks difficult

unless you look at one example and

they might look difficult but once you

try them you know you'll see that

they're very easy

example what this definition means in

terms of evaluating the absolute value

basically I want to Define this or you

in other words if a real number a

happens to be greater than or equal to 0.

a is greater than zero

absolute value of five equals five

absolute value of negative 4 over 7 is

equal to the negative of negative 4 over

7 which happens to be four over seven or

three is an example has zero equals

to evaluate the absolute value of five

we should probably work with the

definition right which we just stated definition

is it a number greater than equal to

zero or less than zero obviously it's

greater than or equal to zero by

definition the absolute value of 5 will

be just that number itself five that's

how we defined it

next example absolute value negative

four over seven

by definition

negative four over seven is less than zero

we should multiply that number by a

negative one that's exactly what the

yeah negative of negative four over

seven turns out to be positive four over

seven Okay negative one multiplied by

negative is always uh positive that's

how we get the second example or these

three example three the absolute value

absolute value of a was defined to be a

so we just get absolute value of zero

equals zero so nothing big no big deal

basically nothing important or nothing

fancy happening just trivial stuff foreign

uh just by using the definition

an absolute value takes a real number

if you take a real number the negative

of that real number a is less than equal

to the absolute value of a and the

absolute value a of a is less than equal

to positive a uh it's a cute statement

inequality I'll let you prove it

yourself is

depends on what the value is

if I say a is a real number don't assume

automatically that it's a positive

number just because there's not a

negative sign alongside with that a uh four

so what I really mean is plus 4 positive

four uh it has a a positive number

minus three so obviously there's a sign

alongside with that so that will be a

negative number so really what I'm

saying when I say negative 3 I'm saying

positive parenthesis minus three

so that's how you basically distinguish

between a positive and a real number uh

when you're given an abstract variable

for that number so again

a is a real number it could be positive

or negative any don't be tempted to say

that a is negative if I say a negative a

uh it may be that the final result is

positive because if a is negative then

minus a will be a positive number and if

a was if I gave you just a by itself and

here a happens to be a negative number

then the final result will be a minus a

or negative number so just a little

technical point we need to keep in mind

this is a subtle thing hopefully we'll

keep it in mind when we talk about

um absolutely further showed us a point

in your coffee

problems so we'll keep that in mind

Let's do an example example

involving absolute values

find the

um or solve the following equation

absolute value of x plus 3 is equal to 4.

X plus 3 equals 4.

or I would have subtracted three from

both sides and the result would have

been x equals one like when you have

absolute value involved so how do we

solve this uh looks intimidating like in

as long as you remember the definition

first course in calculus so for a long

time I could not come to terms with

absolute value definition

absolute value of a number is always

positive but when you solve examples you

absolute value basically gives a

positive number no matter what the input

is you should also keep in mind that

that's an easy way to work with absolute

value sometimes but when you solve

equations involving or inequalities

involving absolute value we have to work

with the definition of absolute value so

let's do that so question is solve x

plus three absolute value equals four so

let's start solving that here are some steps

remember that x minus 3 will play the

role of that real number a in the

definition so when we use that

definition we get x minus 3 absolute

value will equal to just x minus 3 if x

minus 3 is greater than equal to 0

and it'll equal to minus of the quantity

x minus 3 when is

because the minus goes is multiplying

the whole quantity x minus 3. and that

will happen if x minus 3 is less than 0.

basically by definition of absolute

and note that this is exactly what will

happen let's move on

just by the definition of absolute value

of a real number what can we say now

well just by that definition we can say

that what we have written down on the

screen and now let's try to solve it we

have two equations to solve in the first

part we have that absolute value of x

minus 3 is equal to quantity x minus 3.

x minus 3 will equal to 4 in the first

case or in the second case it will be

minus of the quantity x minus 3 will

absolute value of x

minus 3 should equal 4.

so working with that let's see if we can

solve this

it's very easy in the first case we have

x minus 3 equals 4 very simple add 3 to

both sides and you get x equals 7.

in the second case we have minus of the

quantity x minus 3 equals 4.

you multiply throughout the equation

that's given to you by a negative one to

simplify things a little bit

x minus three equals minus four

and now you just add 3 to both sides

just like we did before

x equals

4 minus 4 plus 3 which is negative 1.

equation absolute value equation

x equals seven

and x equals minus one so you don't know

uh numbers seven or minus one original inequality

inequality

result I will get a true statement

basically that was a very good example I

think let's move on and see if we can do

example here

the absolute value of 3x minus 2 is

equal to the absolute value of 5x Plus 4.

interesting example equations

equations

by itself no absolute value now you have

to uh absolute value quantities equating

each other equal to each other so how do

we tackle this um

you take the first part which is 3x

minus 2

and take three x minus 2 and the

absolute value and apply the definition

of absolute value to this quantity or

then expand it and do all the algebra

and you'll get it all worked out nicely

I will let you do that as an exercise

the student should always have something

to do so I'll let you do that I'm sure

you are enjoying uh all this extra work

I give you uh like in uh I'll make it

easy for you actually I will show you

how to do it in a simpler way

mathematics you want to take shortcuts

when I was in America I made a

differential equations he used to say

that a good mathematician is a lazy

mathematician so I hope you take that to

heart and you know use it to good

purpose we shouldn't get lazy for uh you

know for wrong purposes but whenever we

using that let's see if we can actually

uh I can show you a quick way to do this

exam this problem uh you know key JK if

you have uh two numbers uh in the

absolute value

and they're equal to each other if the

absolute value of two numbers is equal

to each other then those two numbers

inside the absolute value have to differ

only in sign

or they must be equal and if for example

absolute value of 4 is equal to the

absolute value of 4 of course and also

notice that absolute value of 4 is equal

just by this example also I think it

to the right hand side quantity in the

basically what I'm saying is K on the

screen you'll see right now that

3x minus 2 is equal to 5x plus 4

or 3x minus 2 is equal to minus of the

quantity 5x Plus 4. so this is just the

same uh you know rule that I followed

that we just saw earlier with the 4 and

negative 4.

we don't have to use the definition of

absolute value and I'm being a good

mathematician being a lazy mathematician

3x minus 5x in the first case first

so we get 3x minus 5x is equal to four

plus two uh

do the algebra you get or the arithmetic

you get minus 2X equals six

of a simply divide both sides by the

appropriate numbers to get x equals

similarly next part because I'll leave

it up to you as an exercise simple

algebra and I think you can do it like

in pointy we can appreciate the the idea

mathematician like can you notice that

being in being lazy you're actually

being uh very ingenious and you have to

do a lot of thinking so I hope you keep

that in mind when you try to be lazy in

between the absolute value of a real

number and the positive square root of a

real number

what is a square root and what is the

positive square root note if I give you

then the question is K what should B be

in order to satisfy this above equation

we have to be careful yeah

which is minus three

and that also works because if you

square minus 3 you also get 9. so so

basically the point is that every

positive real number has two square roots

symbolically we'll say a square root of

a is

square root of positive square root of a

and the negative square root of a um

the square root of a square

would I be right in saying that would I

be correct to say that uh

let's look at the screen and see what we

are saying

square root of a square is a so the

statement basically is saying that the

positive square root of the square of a

number is equal to that number

but is that correct

so let's take an example what happens if

a equals negative 4 well if we take that

a equals negative 4 and plug it into the

original equation we have on the screen

we will get square root the positive

square root of minus 4 quantity squared

is equal to the square positive square

root of 16 which is equal to 4 but that

is not equal to a remember a was

negative 4. so this is certainly not a

true statement I mean there must be

something wrong with this statement we

just wrote down which is that the

positive square root of a square equals a

maybe we can modify it somehow and see

if we can get some more information out

of it so going back to the screen foreign

for any real number a the positive

square root of a square is equal to not

just that number but the absolute value

of that number a that is for every real

number a the absolute value of a is

equal to the positive square root of a square

square

root of negative 4

the positive square root of negative 4

quantity squared is equal to the

positive square root of 16 which is

equal to 4 which is also equal to the

absolute value of minus 4 by definition

of absolute value

that's a true statement

again it's a technical Point

um but which is a very important

technical point it's not just something

we can you know ignore

powerful theorem it actually helps us in

proving many things later on in calculus

so we'll use this result a lot to find

the derivative of some function and

stuff like that is

and I'll just throw them at you very

frequently so I hope you don't mind so

here's another one on the screen

theorem 1.2.3

if a and b are real numbers then

first statement the absolute value of

negative a is equal to the absolute

value of a

H basically they're saying that a number

and its negative have the same absolute

value well obviously that's true just by

you know what we know intuitively about

absolute value statement number B says

the absolute value of the product of two

numbers say we have two numbers a times

B then we take the absolute value of

that it's equal to the product of the

individual absolute values of A and B so

basically it's saying the absolute value

of a product is the product of the

absolute value absolute values and

statement C says that the if you have

two real numbers a and b and you divide

them you take their quotient or they

take their ratio

then the absolute value of that ratio is

going to be the ratio of the individual

absolute values

before we try to prove this thing let's

look at a few examples to understand

so here's an example of the first statement

statement

uh absolute value of negative 4

is the same thing as the absolutely F4

we use this result remember to prove

something else earlier simply to to

actually solve the equation we earlier

let's take the example 2 times -3 is

this the result of that is minus six and

if I take the absolute value of that I

get 6 but that is exactly the same thing

as the product of absolute value of 2

multiplied applied by the absolute value

of negative three

which is 2 times 3 again and the result

is 6.

and for part number c let's look at the

example of the absolute value of 5 over 4.

well the absolute value of 5 over 4 is

first of all just 5 over 4 right but

remember 5 is the same thing as absolute

value of 5 and 4 is the same thing as

absolute value of 4. so we get 5 over 4

equals absolutely of 5 divided by

absolute value of 4 which is again just

equal to five or four so that's also a

true statement and by example we have

seen you've gotten a flavor of what that

okay the absolute value of negative a is

equal to

the positive square root of negative a

squared which is the same as the

positive square root of a squared but

that's just equal to the absolute value

and why we actually use it many times in

proving other proofs proof of Part B is

on the screen and I will let you look at

it and convince yourself of that um

division by zero is not allowed and if B

was equal to zero then we'll be in

trouble so we'll ignore that uh the case

we'll talk about b or division by zero

under absolute value so I will get the

absolute value of a subscript 1

multiplied by a sub 2 all the way to a

sub n will equal to the absolute value

of course it's a true fact I'll let you

prove it maybe you can use

Part B of the previous theorem as

reference and prove this or a core result

result

the absolute value of a raised to the

power n is the same as the absolute

value of a raised to the power n again

I'll let you have fun with this and

prove it yourself

definition theoretical abstract level

let's see if we can make it concrete we

want to look at applications also so uh

so the absolute value comes in naturally

absolute value of a number is always

positive really that's what the absolute

the absolute value of a number is always

positive positive and so is the distance

so when I travel from Lahore I travel

300 kilometers approximately

on the car reads 300 kilometers once I

get to Lahore when I start moving back

uh towards from Lahore odometer

odometer

so again my odometer reads 300.

I would have my odometer would have read

600 kilometers by the time I reached Islamabad

Islamabad

when I moved back to Islamabad distance

is measured in positive values always

there's no such concept of negative and

there's no reference in terms of

so again distance is always positive Sim

similarly so is absolute value and we'll

see how we can Define distance in terms

of absolute values so let's see how

let's define it as say we have two

points capital A and capital B let's

call it a and b and the coordinates on

those two points are the numbers A and B

small a and small B uh because distance

is non-negative which is well we said

it's positive but really it's not

negative because it may be zero I mean

if the same if we measure the distance

between the same point A and A you just

get the distance as zero so we'll allow

the fact that distance could be zero and

so by that keeping that in mind we have

that the distance between A and B is

defined as or is going to be B minus a

if a is less than b it's going to be a

minus B if a is greater than b

and it's going to be 0 if the two points

equal uh to each other which is exactly

what I said a while ago in the first

case B minus a is positive so B minus a

is equal to the absolute value of B

minus a

H just by definition right we all know

that's if you have a positive number the

absolute value is always positive

similarly in the second case uh if we

have where we have a is greater than B B

minus a would be negative because a is

greater so a minus B will equal minus of

B minus a and that just happens to be

the absolute value again of B minus a

thus in all cases we have the following result

result

theorem 1.2.4 and we call it the

distance formula we will write it down

if a and b are points on a coordinate

line with coordinates A and B small A

and B

respectively then the distance between D

between A and B is defined as D equals

the absolute value of B minus a

H this formula all which is actually

given to you on the screen

provides a useful geometric

interpretation of some common

geometric interpretations being absolute

value ski your distance is

convince yourself of these facts and do

some examples that might be helpful in

let's do some examples of these let's

see if we can solve them so here's some

x minus 3 absolute value less than 4. is

inequality as uh negative 4 is less than

x minus 3 quantity and that is less than

4. we solved an inequality like this in

if we add three throughout the

inequality we will be left with X is

between negative 1 and 7. or that will

be your solution I mean that exactly is

going to be your solution to the

inequality so the numbers between

negative 1 and 7 are the numbers which

if you plug into the original inequality

will satisfy that inequality or make it

true and you can try that actually and

the result can also be written as

interval notation as parentheses

negative one comma seven results

I know you're enjoying it so far right

let's do another example and see uh just

to make it you know this whole idea much

my practice is very essential so the

more you do the more you learn and the

better you get at math example

example

and hopefully you'll get a better grasp

of the whole idea that we're trying to

solve the absolute value of x plus 4 is

first of all X plus 4 is less than or

equal to negative 2 or X plus 4 is

inequalities both the top and the bottom

one then I will get the result X is less

than or equal to negative 6 or X is

greater than or equal to minus 2. foreign

as absolute value of x minus the

quantity minus 4

is greater than equal to two

so is

the the solution of this inequality

consists of all X whose distance from

so this is exactly what this inequality

but you know it's good to remind

the triangle inequality very important

fact very important theorem uh

Heisenberg's uncertainty principle

physics make here quantum physics

so it exactly it's actually a direct

the absolute value of two real numbers a

plus b is the same as the absolute value

of the individual numbers added together

for example a equals to a b equals minus

one absolute value of j e which is equal

any absolute value of a plus the

absolute value of B so result at the

absolute value of 2 plus the absolute

value of minus 3 which is equal to 2

absolute value of a plus b is equal to

absolute value of the sum individually

in the absolute value of a plus the

absolute value of B in generally it's

not always true some cases it may be but

in in general it's not true but what is

true is the triangle inequality uh

you know superficially and I'll let you

work out the details

uh inequality theorem 1.2.5 triangle

inequality if a and b are any real

numbers then the absolute value of a

plus b

is less than equal to the absolute value

of a plus the absolute value of B so is

this is not always equal it may be equal

sometimes but in general is less than

we saw this a lot earlier in this

lecture that the any given real number a

is between the negative of its absolute

value and the positive of its absolute value

value

and I hope you convince this of yourself

of this and you can prove this also if

you want to uh

let's consider that we know that the

negative of the absolute value of a is

between is less than equal to a and that

is less than equal to the absolute value

of a

and the negative value of a negative

absolute value of B is less than equal

to B and that's less than equal to the

absolute value of B positive 1. if we

add these two together we get the

following result it's on the screen I

will let you work out the details

since A and B were real numbers adding

them will also result in a real number

obviously right well there are two types

of real numbers what are they

well either they are bigger than equal

to zero or they're less than 0. so that

is why we have that the apps the the sum

of a plus of A and B A plus b is greater

than equal to 0 or a plus b is less than 0.

I hope you can do that and convince

yourself of this uh

so the result is that in gen in general

absolute value of a plus b is less than

equal to the absolute value of a plus

the absolute value of B

so I hope you like this triangle

inequality I know you have methodology

it's a well established result it's an

old proof very important thing so it

must not be wrong but to convince

yourself again this is your part as a

student that you see the result the

proof and you go over it through it and

convince yourself by examples or in

and I hope you appreciated the the nice

thing about that maybe you don't do it

right now but eventually you will when

you see higher and better results in mathematics

so let's hope that my odometer on my car

reads 300 actually if I set it to 0 or

600 total okay

I'll probably find myself in kohika see

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