MTH150 1.2 Properties of Negatives and Technical Communication

Summary

This educational video script delves into the properties of positive and negative numbers, focusing on how they interact through operations like addition, subtraction, multiplication, and division. It refreshes key concepts such as the outcome of multiplying or dividing numbers with the same or opposite signs. The script also explores notational shortcuts like expressing 'negative a' as '-1 * a' and the flexibility of viewing subtraction as addition of the opposite sign. Practical examples are used to illustrate these properties, emphasizing the importance of precise mathematical language and reasoning in constructing technical arguments about the positivity or negativity of expressions.

Takeaways

• 🔢 The lesson focuses on understanding the interaction between negative and positive numbers, emphasizing their properties and how to use them to determine the sign of expressions.
• 🤔 If two numbers, a and b, have the same sign (both positive or both negative), their product (a * b) and their quotient (a / b) will be positive.
• 🤓 When a and b have opposite signs, their product and quotient will be negative, illustrating the impact of sign differences in arithmetic operations.
• ➕ If a and b are both positive or both negative, their sum (a + b) will maintain the same sign, providing a rule for adding numbers with like signs.
• 🔄 Negative a is equivalent to negative 1 times a, which is a key concept for understanding how to manipulate expressions involving negatives.
• 🧩 Subtraction can be rewritten as addition by using the property a - b = a + (-1 * b), which simplifies understanding of subtraction in terms of addition.
• 📚 The lesson introduces several properties that rely on the distributive, commutative, and associative properties of multiplication, expanding the mathematical toolkit.
• 🔄 Negatives can be manipulated flexibly, such as converting negative a times negative b to a times b, showcasing the power of algebraic manipulation.
• 💡 The ability to think about expressions in different ways is crucial for solving problems efficiently, as demonstrated by the various methods to compute products and sums.
• 📈 The lesson aims to equip students with the tools to construct technical arguments using precise mathematical language, enhancing their problem-solving skills.

Q & A

What are the properties of negatives and positives when they are both the same sign?

When a and b have the same sign (both positive or both negative), the operations a over b and a times b result in a positive value.

What happens when you divide or multiply two numbers with opposite signs?

If a and b have opposite signs, the operations a over b and a times b result in a negative value.

How does the sign of a plus b change if a and b have the same sign?

If a and b have the same sign, the sum a plus b maintains that sign, meaning if both are positive, the result is positive, and if both are negative, the result is negative.

What is the meaning of negative a in terms of multiplication?

Negative a can be expressed as negative 1 times a, which has the opposite sign of a.

How can subtraction be thought of in terms of addition?

Subtraction can be thought of as addition of the opposite sign, so a minus b is the same as a plus negative b.

What is the result of a minus b over a b when a and b have opposite signs?

When a and b have opposite signs, a minus b over a b results in a negative value because the numerator (a minus b) is positive and the denominator (a times b) is negative, and positive divided by negative equals negative.

How can the expression a over b plus b over a be evaluated when a and b have opposite signs?

When a and b have opposite signs, both a over b and b over a are negative, and adding two negatives results in a negative value.

What is the significance of the property that allows us to rewrite negative a times negative b as a times b?

This property simplifies calculations by allowing us to ignore the negatives temporarily and just multiply the absolute values, then apply the negative sign at the end.

Why is it useful to have multiple ways of writing expressions involving negatives?

Having multiple ways to write expressions involving negatives allows for more flexibility in problem-solving and can simplify calculations by choosing the most convenient form for a given situation.

How can the distributive property be used to rewrite negative a plus b?

Using the distributive property, negative a plus b can be rewritten as negative a plus negative b, which then simplifies to negative (a minus b).

What is the significance of the property that negative negative a equals a?

This property shows that two negatives cancel each other out, which is a fundamental concept in algebra and arithmetic operations involving negatives.

Outlines

00:00

🔢 Introduction to Negatives and Positives

The video script begins with an introduction to the concepts of negatives and positives in mathematics. It aims to refresh the viewer's understanding of these fundamental properties and how they interact. The instructor discusses the outcomes of multiplying or dividing numbers with the same sign (both positive or both negative), which result in a positive number, and the significance of zero in this context. The script also touches on the concept of opposite signs and their impact on multiplication and division, leading to negative results. The instructor emphasizes the importance of using precise mathematical language and introduces the idea of converting subtraction into addition by using the concept of opposite signs.

05:00

📚 Constructing Arguments with Negatives and Positives

In this section, the script transitions into using the properties of negatives and positives to construct mathematical arguments. The instructor provides examples to demonstrate how to determine the positivity or negativity of expressions involving variables a and b, which have opposite signs. The discussion includes breaking down expressions into their numerators and denominators and applying the learned properties to determine the overall sign of the expressions. The script also introduces the concept of rewriting subtraction as the addition of the opposite sign, which simplifies the process of constructing technical arguments in mathematics.

10:03

📉 Examples of Applying Properties to Expressions

The script continues with more examples to solidify the understanding of how to apply the properties of negatives and positives. The instructor works through specific cases, such as adding and dividing expressions with variables a and b, which are assumed to have opposite signs. The process involves determining the sign of individual terms and then combining them to find the overall sign of the expression. The instructor also acknowledges a mistake and corrects it, demonstrating the importance of careful reasoning in mathematical arguments. The section ends with a discussion on the importance of showing that an expression's sign can depend on the values of the variables involved.

15:04

🔄 More Properties and Their Applications

This part of the script introduces additional properties related to negatives and positives, such as the distributive, commutative, and associative properties of multiplication. The instructor explains how these properties can be used to manipulate and simplify expressions involving negatives. Examples are given to show how these properties can be applied to rewrite expressions in different but equivalent forms. The script emphasizes the utility of being able to think flexibly about mathematical expressions and how this can be advantageous in solving problems and constructing arguments.

20:06

🎯 Wrapping Up and Reflecting on the Properties

The final section of the script wraps up the discussion by summarizing the properties of negatives and positives that have been covered. The instructor reflects on the technical arguments that have been constructed throughout the script and how they demonstrate the application of these properties. The focus is on the practical use of these properties in mathematical reasoning and problem-solving. The script concludes by encouraging viewers to use the tools and techniques discussed to enhance their own mathematical arguments and understanding.

Keywords

💡Negatives and Positives

Negatives and positives refer to the two types of numbers in the set of integers, where negatives are less than zero and positives are greater than zero. In the video, these terms are fundamental to understanding the properties and interactions of numbers. For instance, when discussing the multiplication and division of numbers with the same sign (both positive or both negative), the result is positive, which is a key concept in the script.

💡Sign

The term 'sign' is used to describe whether a number is positive or negative. It is crucial in the video for determining the outcome of mathematical operations. The script explains that if two numbers have the same sign, their product and quotient are positive, which is a fundamental rule in arithmetic.

💡Opposite Signs

Opposite signs indicate that two numbers have different signs, one being positive and the other negative. The video script uses this concept to explain that when multiplying or dividing numbers with opposite signs, the result is negative. This is a key property used in constructing mathematical arguments within the lesson.

💡Addition

Addition is one of the basic arithmetic operations discussed in the video. The script explains that when adding numbers with the same sign, the sum retains that sign. For example, adding two positive numbers results in a positive number, which is a straightforward application of the rules of addition in the context of signs.

💡Subtraction

Subtraction is another fundamental arithmetic operation. The video script suggests thinking of subtraction as addition of the opposite, which is a strategy to simplify understanding and calculations. This is exemplified when explaining that 'a minus b' can be rewritten as 'a plus negative b', aligning with the property that subtraction is addition of the additive inverse.

💡Distributive Property

The distributive property is a key mathematical property that allows for the simplification of expressions involving multiplication and addition or subtraction. In the script, this property is alluded to as a method for rewriting expressions in different but equivalent forms, which can be useful for easier computation and understanding.

💡Commutative Property

The commutative property of multiplication states that the order of factors does not change the product. Although not explicitly detailed in the script, this property is implicitly used when discussing the multiplication of numbers with the same or opposite signs, as the outcome remains the same regardless of the order of the numbers.

💡Associative Property

The associative property of multiplication, while not directly discussed in the script, is a mathematical rule that allows for the grouping of numbers in multiplication to be changed without affecting the result. This property could be relevant when multiplying more than two numbers, a concept that might be expanded upon in further lessons.

💡Additive Inverse

The additive inverse of a number is the number that, when added to the original number, yields zero. In the video, the concept is used to explain how to handle negative numbers in expressions, such as 'negative a' being equivalent to '-1 times a'. This is a crucial concept for understanding how to manipulate and simplify expressions involving negatives.

💡Technical Arguments

Technical arguments refer to the logical and mathematical reasoning used to explain why a quantity or expression is positive, negative, or dependent on certain conditions. The script emphasizes constructing such arguments using the properties of negatives and positives. For example, using the properties to argue that 'a minus b over a times b' is negative when 'a' and 'b' have opposite signs.

Highlights

The lesson focuses on understanding the interaction between negative and positive numbers.

Property 1: If 'a' and 'b' have the same sign, then 'a/b' and 'a*b' are positive.

Zero is neither positive nor negative, making it a special case in mathematics.

Property 2: If 'a' and 'b' have opposite signs, then 'a/b' and 'a*b' are negative.

Property 3: The sum of numbers with the same sign retains that sign.

Negative 'a' equals negative 1 times 'a', which has the opposite sign of 'a'.

Subtraction can be thought of as addition of the opposite sign.

Using precise mathematical language like 'numerator' and 'denominator' is emphasized.

An example is provided to determine if 'a - b / a * b' is positive or negative.

The numerator of 'a - b / a * b' is positive when 'a' and 'b' have opposite signs.

The denominator of 'a - b / a * b' is negative when 'a' and 'b' have opposite signs.

The overall expression 'a - b / a * b' is negative when 'a' and 'b' have opposite signs.

Property 6: Negative 'a' times negative 'b' is the same as 'a' times 'b'.

Negative 'a' times 'b' can be thought of as negative 1 times 'a' times 'b'.

Property 7: Negative negative 'a' equals 'a'.

Substitution of subtraction with addition can simplify problem-solving.

The lesson provides examples to illustrate the properties and their applications.

The importance of flexible thinking in mathematical problem-solving is highlighted.

The lesson concludes with a discussion on the practical applications of these properties.

Transcripts

okay so this uh lesson is about

uh negatives and positives and how those

two

um types of numbers interact with each

other so we're just gonna do a quick

refresher

on properties of negatives and positives

and their interaction

and then we're going to use those

properties to construct arguments

um explaining why a given quantity

or a given expression is positive or

negative

or maybe it maybe depends um so maybe

let's just do a quick list of

properties so list of

properties

so the first property is if

a and b

have the same sign

meaning they're either both positive or

they're both negative

then a over b

and a times b

are positive

yeah maybe just a quick quick reminder

a greater than 0 or 0 greater than

or 0 less than a means

a is positive so those are both saying

the same thing

they will use b so b less than 0

0 greater than b this means

b is negative that's

that's how we might uh

might say those things uh so if they're

both so if a and b

are either both positive or both

negative then when you multiply them

or divide them you're going to end up

with a positive

positive number and of course right so

here

b is not 0 and a

and b are probably not 0 either because

zero is

is special in the sense that it's not

positive or negative

okay um so if so now what happens if

they have opposite signs so if a

and b have

opposite signs

then the opposite thing happens then a

over b

and a times b are

negative

okay another one

if a

and b have the same sign

then a plus b

so what can we say about a plus b maybe

pause here for a second and

try to figure that out

then a plus b maintains

the sign

meaning if a and b were both positive

then a plus b is positive

if a and b were both negative then a

plus b is

also negative this one's kind of a

notational

thing so but it's important

so negative a equals negative 1

times a and this

has the opposite sign of a

right so we see this whenever i see a

negative

and i used to not do this but i do know

that i've

learned more mathematics i just

automatically change it into a negative

1

times whatever that thing is so if a was

positive

then negative a is negative if a was

negative then negative a is positive

and then 5 this is

another another notational thing so a

minus b

that's the same as a plus negative b

and if we want to write that once more

that's a plus negative 1

times b so this is really breaking down

um that's that we can think of

subtraction as addition and this is

another thing that i also

have started to do only really only once

i started

teaching is just converting all

subtraction into plus the opposite sign

of

of that original thing and this just

this just makes

ends up making things easier in the long

run it might see a little

seem a little cumbersome now but it will

make

constructing technical arguments a

little bit easier

okay now what we're going to do is we're

going to use these properties to

construct

some arguments so let's do

some examples let's say

if a is greater than

zero and b is less than zero

determine

whether

we're not the following

oh follow me i was going to write

expressions they are expressions

yeah sure let's write expressions

following expressions

are positive

negative

or if it depends

and we're going to be using these these

properties um

so let's let's try that uh and maybe

actually

so to to shorten to make our lives a

little bit easier

um maybe let's so here

we can write these as negative times

negative

is positive uh negative

times positive is negative

and positive times

positive is positive

and here this one can be rewritten

as negative plus negative

equals negative and positive plus

positive equals

positive and that should be that should

make

that should make our arguments much much

shorter so let's let's try that or let's

try one

so the first one let's

label the number one

a minus b over a b

and what i would suggest here is to talk

about the numerator and the denominator

separately and then put it all together

and make sure we're using

precise mathematical language like

numerator and denominator

and not top and bottom so we would say

the numerator

is let's see what is the numerator a a

positive

minus a negative is positive

because if you know if you claim

something you should give a reason

claims need reasons the number is

positive because a minus b

equals a plus negative 1 times b

and that's using this property here

sorry this property 5 up here

okay and then we can see that

in then and there maybe we could do

right and there are lots of ways to do

these we could have said this is

a plus negative b as well

and then um this turns into in any event

this is a positive plus a positive

which is positive

plus positive

equals positive

okay so that's why the numerator is

positive because when it boils down to

it we're adding

two positive numbers together okay now

what about the denominator

the denominator denominator

is negative

because

a times b equals let's see

that's a positive times a negative

which is a negative and maybe we could

have done that up here

right we could have said a plus negative

b that's really a

positive plus a positive equals

a positive

right so you can see this is a mix of

mix of

sentence sentences and mathematical

notation

okay so then now overall

a minus b over a b is

what is it over is negative

uh because or since so because

um positive

divided by negative equals negative and

i guess that's something

uh something i didn't do up here so i

did the

the multiplication versions but we also

could have done

um and

negative over negative is positive

positive over positive

is positive and then

negative over positive equals negative

and positive over negative equals

negative

right and if you don't like writing out

these equations you could just say all

this in words

as well

all right so there's the first example

okay let's try another

uh we'll do we'll do two more examples

so

let's do a over b plus b over a

um and maybe we just start with

so a over b and b

over a are both

are they negative

since or because sure let's do since

since a and b

have opposite signs

and that's if we if we look up here if

they have opposite signs then a over b

or b over a are negative

okay so both of those individual terms

so the first term and the second term

are negative um

and then we add two negatives and then

a over b plus b over a

is a negative plus a negative

equals a negative or we could have

written this last part out

um as a sentence and let's

i guess finalize thus overall

the expression

is negative

and like i was saying right you could

have written this out in words as well

okay let's do one more let's do uh a

plus b over a b

uh and here um let's try to use

some specific values let's say that we

didn't we didn't really know

what was going on here so we're like

well let's just pick some numbers so

let's let a

equal 3 and

b equal 4

so then 3 plus 4 over

3 times 4 7 over 12. so that's positive

so we say

so now we're saying okay overall it's

positive so

positive or maybe let's just look at the

numerator and denominator

um right so numerator

is positive

since positive

plus positive is positive

denominator

is positive since

um positive times positive

is positive so then overall

uh the expression

is positive since

positive over positive

equals positive so here i made a mistake

and the mistake is that b is supposed to

be negative

all right okay so all right let's back

up

i guess we can kind of keep that format

so b suppose oh b

is supposed to be negative so let's

change this to negative

negative negative so negative

four so we get

oops this is kind of getting scrunched

up

so here b is supposed to be negative so

let's say b

is negative 4 so then

we have 3 plus negative four over three

times negative four

so that's negative one over negative

twelve positive

over okay it's still positive but we

have to do some

some other reasoning so numerator is

negative

since positive plus negative

is negative uh denominator

is negative

since uh positive times

negative equals negative

that's overall the expression

is positive

since uh negative over negative

equals positive

great but this is this is also wrong

okay so think about why it's wrong

okay uh so if you haven't figured it out

this is one of the cases where um

it's going to depend right it depends on

um well yeah it depends on a and b

really so here so

so what do we do if it depends well we

delete all this stuff

or erase all this stuff

and we say

we say it depends so if it depends

then you have to show me one example one

specific example where it's positive and

one a specific example where it's

negative

it depends so uh

case i guess i

if a equals three and

b equals negative four then

a plus b over a b

equals 3 plus negative 4

over 3 times negative 4 equals negative

1 over negative 12

equals 1 12 is greater than 0.

and the second case so see if you can

think of a situation to make this

negative okay

so if you haven't so if a equals 3

and b equals negative 2

then a plus b over a b

equals 3 plus negative 2

over 3 times negative 2 equals 1

over negative 6 which is less than 0.

there you go okay a couple more things

there are a couple more properties and

these are more

maybe we'll talk about these a little

bit more later um

but what property were we on five

so property six

okay and some of these rely on the

distributive property

uh the commutative property of

multiplication the associative property

of multiplication we'll talk about that

soon enough

um but one uh maybe well here's one of

them

if we do negative a times negative

b that's the same as

a times b another one here is that

negative a times b the idea here is we

can we can think of negative a

as negative 1 times a and then we can

move that negative 1

around however we want that's the same

as a times negative b

which is also the same as negative

a times b uh another one

this kind of you know comes from uh

these previous two but we can write it

down anyway negative

negative a so basically you can see how

um right you can kind of get this

property from six

well what's the end result that just

equals a

um because you could think of this as

negative one times negative one times a

and then the negative one and the

negative one just gives you

one um another property

similar to i think it was five

yeah five is about you know how how can

we

think about um subtraction

in different ways so here we can turn

uh subtraction into addition uh

9 and i think 10 are also going to help

us

think about subtraction in different

ways so if we do negative

a plus b um that turns into again we

think of this negative

uh negative as a negative one and then

we can distribute that negative one to

both of those pieces

and we get negative a minus b or

negative a

plus negative b

and then the last one here that we're

going to talk about is negative a minus

b

again right if we applying

these properties and then applying like

the distributive property

we get negative a plus

b and maybe as a as an intermediate step

maybe i'll do an intermediate step uh

negative b

negative a minus negative b so that

turns into negative a

plus negative negative b

right so you can see how we're using

property five

five yeah five from above to turn this

minus into a plus and then that stacks

these two negatives

and then those two negatives

turn into a plus now maybe you might be

asking yourself

why do i need you know why do i need to

know

like three different ways of writing

this

a negative negative a times b why do i

need to know

all these different ways of writing this

expression and the answer is

you know being able to think about

different expressions

kind of flexibly it can be really useful

i mean as a quick easy example i would i

would say

you know maybe you use this property or

maybe even this property

more often than the other ones you know

if i ask you to compute

negative 30 times negative

seven um right

what i do anyway is i just multiply

um i i just forget about the negatives i

say oh

negative and negative negative those

just

those are going to go away and i'm just

left with 30 times 7 which is 210

okay or if i asked what's you know 5

times

negative 15 i forget about the negative

and i just do 5 times 15

first get 75 and then i say oh yeah

originally there was that negative there

so my

overall product is going to be a

negative

so those are just some some simple

examples of how

how being able to think about problems

flexibly

uh can be can give us an advantage we

should take advantage of that pet

flexibility

okay so what did we do here um hopefully

we

handled our objectives refresh

properties of negative so there are lots

of them

and again right these these ones are a

little more technical

and they they pretty much say these ones

uh the one two and three here

are already kind of taken care of down

here but i

this is a little bit easier uh for me to

think about

so we refreshed all these properties of

negatives

and we constructed some technical

arguments hopefully that gives you some

some some tools to construct your own

technical arguments