OPERATIONS ON INTEGERS

Summary

In this educational video from Field Korean TV's Math Corner, viewers are introduced to the fundamentals of integer operations. The video explains the concept of integers, including positive, negative, and zero. It then delves into addition and subtraction rules for integers, emphasizing the importance of absolute values and the signs of the numbers involved. The tutorial also covers multiplication and division, illustrating how to determine the sign of the result based on the signs of the operands. Each operation is accompanied by clear examples, making the lesson accessible for learners.

Takeaways

• 🔢 Integers are whole numbers that can be positive, negative, or zero, and include both counting numbers and their negative counterparts.
• ➕ When adding integers with the same sign, simply add the numbers and keep the common sign.
• ➖ For integers with different signs, subtract the smaller absolute value from the larger and keep the sign of the integer with the larger absolute value.
• 📝 Positive numbers are written without a sign, as it's understood that they are positive by default.
• 🔁 Subtracting integers involves changing the subtraction sign to addition and then applying the rules for adding integers.
• 🚫 When the subtrahend is greater than the minuend and both are positive, the result is negative, and vice versa for negative numbers.
• ✖️ In multiplication, multiply the absolute values of the integers and then apply the sign rule: same signs result in a positive product, different signs result in a negative product.
• ➗ For division, divide the absolute values and apply the same sign rule as in multiplication: same signs give a positive quotient, different signs give a negative quotient.
• 📉 Dividing by a smaller number results in a decimal or fraction, and the sign is determined by the original signs of the dividend and divisor.
• 🎓 The video provides a comprehensive guide to performing basic arithmetic operations with integers, emphasizing the importance of understanding integer properties and the rules for handling different signs.

Q & A

What are integers?

Integers are whole numbers that can be positive, negative, or zero. They include counting numbers like 1, 2, 3, etc., and their negative counterparts like -1, -2, -3, etc., as well as zero.

What is the rule for adding two positive integers?

When adding two positive integers, you simply add the numbers together without the need to write a plus sign before the first number, as it is understood to be positive.

How do you add two negative integers?

To add two negative integers, you add the numbers and then apply a negative sign to the result.

What is the process for adding integers with different signs?

When adding integers with different signs, you subtract the number with the smaller absolute value from the one with the larger absolute value and then apply the sign of the integer with the larger absolute value to the result.

What is the meaning of absolute value in the context of adding integers?

The absolute value of a number is the distance of that number from zero, disregarding its sign. For example, the absolute value of both 6 and -6 is 6.

How do you subtract a positive integer from a negative integer?

To subtract a positive integer from a negative integer, you change the subtraction sign to addition, take the opposite sign of the subtrahend, and then apply the rules for adding integers with different signs.

What is the shortcut for subtracting a smaller positive integer from a larger one?

The shortcut for subtracting a smaller positive integer from a larger one is to simply subtract the smaller number from the larger one and then apply a negative sign to the result.

What is the rule for multiplying integers?

When multiplying integers, you first multiply their absolute values and then apply the following rules to determine the sign: if the signs are the same, the result is positive; if the signs are different, the result is negative.

How do you divide two integers with the same sign?

When dividing two integers with the same sign, the result is positive. You divide the absolute values of the numbers.

What is the outcome when dividing two integers with different signs?

When dividing two integers with different signs, the result is negative. You divide the absolute values of the numbers and then apply a negative sign to the result.

Outlines

00:00

📚 Introduction to Integers and Their Operations

This paragraph introduces the concept of integers, defining them as whole numbers that can be positive, negative, or zero. Examples of integers are given, such as 35, -20, -100, 7, and 0. The paragraph emphasizes that integers include all counting numbers and their negative counterparts, as well as zero. It then transitions into explaining how to add integers, starting with the addition of two positive integers (e.g., 6 + 9 = 15), followed by the addition of two negative integers (e.g., -6 + (-9) = -15). The rules for adding integers with different signs are also outlined, where the integer with the larger absolute value determines the sign of the sum after subtracting the smaller absolute value from the larger (e.g., 6 + (-9) = -3 and -6 + 9 = 3). The absolute value is defined as the distance of a number from zero, disregarding its sign.

05:00

🔢 Subtracting Integers: Steps and Examples

This paragraph explains the process of subtracting integers, outlining a four-step method: keeping the first number (minuend), changing the subtraction sign to addition, getting the opposite sign of the second number (subtrahend), and applying the rules of integer addition. Examples are provided to illustrate the process, including subtracting both positive and negative integers. The technique is also discussed for when the subtrahend is greater than the minuend, simplifying the process by directly subtracting the smaller number from the larger and applying the appropriate sign. The paragraph concludes with examples of subtracting a negative number from a positive number and vice versa, demonstrating how to determine the sign of the result based on the absolute values of the integers involved.

10:03

📈 Multiplication and Division of Integers

The rules for multiplying and dividing integers are discussed in this paragraph. The process involves first multiplying or dividing the absolute values of the integers and then determining the sign of the result based on the signs of the original numbers: same signs yield a positive result, while different signs yield a negative result. Several examples are provided to illustrate these rules, including multiplying positive and negative integers, as well as dividing them. The paragraph covers various scenarios, such as multiplying negative six by positive four (resulting in -24) and dividing negative eight by positive two (resulting in -4). The importance of considering the absolute values and the signs of the integers when performing these operations is emphasized.

15:05

📉 Final Examples and Conclusion

This final paragraph presents additional examples of dividing integers, including dividing negative two by positive eight, which results in a negative 0.25 or -25 hundredths. The paragraph concludes with a summary of the key points covered in the video, highlighting the rules for integer operations and expressing a hope that viewers have gained a better understanding of the topic. The video ends with a sign-off and a blessing, accompanied by closing music.

Keywords

💡Integers

Integers are whole numbers that can be positive, negative, or zero. They are the backbone of the arithmetic operations discussed in the video, as they include all counting numbers and their negative counterparts. In the script, integers are used to demonstrate addition, subtraction, multiplication, and division, highlighting their fundamental role in mathematics. For instance, the video explains how to add positive integers (6 + 9 = 15) and negative integers (-6 + (-9) = -15), showcasing the rules specific to integer operations.

💡Addition of Integers

Addition of integers is a fundamental arithmetic operation covered in the video. It involves combining two integers to find their sum. The video script explains that when adding integers with the same sign, you simply add their values and keep the common sign. For example, adding -6 and -9 results in -15. When adding integers with different signs, you subtract the absolute value of the smaller number from the larger and keep the sign of the number with the larger absolute value, as shown when adding 6 and -9, which equals -3.

💡Subtraction of Integers

Subtraction of integers is another basic arithmetic operation that the video script elaborates on. The process involves changing the subtraction sign to addition and then determining the sign of the result based on the absolute values of the integers involved. For example, when subtracting a smaller positive integer from a larger one (like 2 - 7), the result is negative, as per the rules explained in the video. The script also covers the case where a negative integer is subtracted from a positive one, such as -2 - 7, which results in -9.

💡Absolute Value

The absolute value of a number is the distance of that number from zero on the number line, disregarding its sign. This concept is crucial when adding or subtracting integers with different signs. The video uses absolute value to determine which number to subtract from when the integers have different signs, and it also helps in understanding how to handle operations where the result's sign needs to be determined based on the numbers' magnitudes.

💡Multiplication of Integers

Multiplication of integers is the process of scaling one integer by another. The video script explains that when multiplying, you first multiply the absolute values of the integers and then apply a rule to determine the sign of the product: if the integers have the same sign, the result is positive; if they have different signs, the result is negative. For example, multiplying -6 by 4 yields -24 because the absolute values (6 and 4) are multiplied to get 24, and the different signs result in a negative product.

💡Division of Integers

Division of integers is the process of splitting one integer into equal parts determined by another integer. Similar to multiplication, the video script explains that you divide the absolute values and then determine the sign of the quotient based on the integers' signs: same signs result in a positive quotient, and different signs result in a negative quotient. An example from the script is dividing -8 by 2, which equals -4, reflecting the rule that division of two negative integers results in a positive quotient.

💡Minuend

The minuend is the number from which another number (the subtrahend) is to be subtracted. In the context of the video, the minuend is the first number in a subtraction operation. The script explains how to handle subtraction by changing the operation to addition and considering the opposite sign of the subtrahend, as seen in examples like 7 - (-2) which simplifies to 7 + 2.

💡Subtrahend

The subtrahend is the number that is to be subtracted from another number (the minuend). In the video, the subtrahend's sign is changed to its opposite when the subtraction operation is converted to an addition operation. The script uses this term to explain how to perform subtraction with integers, such as in the example of 2 - 7, where the subtrahend 7 is converted to -7 before performing the addition.

💡Positive Integers

Positive integers are whole numbers greater than zero. In the video, positive integers are used to demonstrate basic arithmetic operations like addition and multiplication. The script clarifies that when adding or multiplying positive integers, the result is positive, as seen in the addition of 6 + 9 = 15 and the multiplication of 6 x 4 = 24.

💡Negative Integers

Negative integers are whole numbers less than zero. The video script uses negative integers to illustrate how arithmetic operations change when dealing with numbers that have a negative sign. For instance, when adding negative integers, the sum retains the negative sign, as shown in the example of -6 + (-9) = -15, and when multiplying, the product's sign depends on the rules of integer multiplication.

Highlights

Integers are whole numbers that can be positive, negative, or zero.

Positive and negative integers represent counting numbers in their respective directions from zero.

The addition of two positive integers is performed by simply adding their values together.

When adding two negative integers, the numbers are added and the result retains the negative sign.

Adding integers with different signs involves subtracting the absolute values and taking the sign of the integer with the larger absolute value.

The absolute value of a number is the distance from zero, disregarding the sign.

Subtraction of integers is converted to addition by changing the sign of the second integer and applying addition rules.

When subtracting a smaller positive integer from a larger one, the result is negative.

Subtracting a larger negative integer from a smaller positive integer results in a positive number.

Multiplication of integers follows rules based on the signs of the numbers involved: same signs result in positive, different signs result in negative.

Division of integers also follows sign rules similar to multiplication: same signs are positive, different signs are negative.

When multiplying or dividing, first calculate the absolute values and then apply the sign rules.

Parentheses in mathematics indicate multiplication.

Integer multiplication and division rules apply to both positive and negative numbers.

The video provides practical examples to illustrate the rules of integer operations.

The video concludes with a summary of the rules for integer operations and a sign-off.

Transcripts

hi welcome to field korean tv

math corner in this video

we will tackle the operations on

integers

first we must know the meaning of

integers

when we say integer it is a whole number

not a fractional number that can be

positive

negative or zero examples

35 negative 20

negative 100 7

and 0. just keep in mind

that integers are all counting numbers

1 2 3 4 5 and so on

and also the negative counting numbers

negative 1 negative 2

negative 3 and so on as well as

zero now let's proceed

to addition of integers

okay let's add first both positive

integers

example 6 plus

9 the numbers are both

positive so we will just add 6 plus 9

equals

15 so positive 6 plus positive 9

equals positive 15 but in writing

positive numbers

we don't need to put the plus sign

because it is understood that a number

without a sign

is a positive number

next let's add both negative

integers okay let's add negative

six plus negative nine

to add integers with the same negative

signs we just add the numbers

then we copy the negative sign in the

answer so negative 6 plus negative 9

equals

negative 15. this time

we will add integers with unlike or

different

signs let's get the sum

of positive six plus negative

nine to add integers

with unlike or different signs just

subtract

the number with a smaller absolute value

from the number with higher absolute

value and copy the sign of the number

with higher absolute value

when we say absolute value it is at the

distance of a number from

0 regardless of its sign

examples the absolute value of six

and negative six is six

the absolute value of nine and negative

nine

is nine as you can see

positive 6 and negative 6 have the same

absolute

value that means that when we get the

absolute value of

a number we disregard the sign

okay let's go back to the problem

okay let's add now so the rule is we

will subtract this

number with smaller absolute value from

the

number with higher absolute value so we

will subtract six from

9 9 minus 6 equals 3

and then to put the sign in our answer

we need to copy the sign of the number

with higher

absolute value so since negative 9 has a

higher absolute

value so we will copy its sign and it is

negative therefore positive 6

plus negative 9 equals negative

3. how about if we add

negative six plus positive nine

the signs of the addends are different

so the rule is to subtract we will

subtract the smaller number

to a higher number in terms of absolute

value

so we will subtract nine minus

six equals three

then to put the sign in the sum we need

to copy the

sign of the number with a higher

absolute

value and since 9 is higher than

6 so we will copy the sign of 9 and it

is

positive so negative

6 plus positive nine equals positive

three as i've said we don't need

to put the sign in writing the answer

when it is positive

let's proceed to subtraction of integers

okay let's know first the steps in

subtracting integers

first keep the first number or

minuend second change the subtraction

sign

into addition sign so that means

minus will become plus third

get the opposite sign of the second

number or

subtrahend and fourth

proceed to the rules in adding integers

to get the answer okay

let's follow these steps in subtracting

integers

so let's have our first example positive

seven minus positive two or

seven minus two it's very easy

so we will just subtract seven minus two

equals

5 in this example

we can get immediately the answer

without changing

the subtraction sign into addition sign

because we only subtract whole numbers

or natural numbers

now let's solve another problem this one

positive two minus positive

seven okay let's follow the steps in

subtracting

integers so we will keep

the first number or the minuend

the two then we will change the

subtraction sign

into a addition sign then we will get

the

opposite sign of the subtrahend so the

subject is positive 7

so then the opposite is negative

7 the last step is to apply

the rules in adding integers as you can

see

the problem becomes positive 2

plus negative 7 the signs

are different so we need to subtract

the smaller number to a higher number in

terms of

absolute value so that means that we

will subtract two

from seven okay let's do it

seven minus two equals

5 then to put the sign in the

answer we need to copy the sign of

the number with a higher absolute value

and that is 7 the sine of 7

is negative so we will copy the negative

sign so the answer for

positive 2 minus positive 7

or 2 minus 7 equals negative

5 but there is a technique in

subtracting

both positive integers in which the

subtrahend

is greater than the minuend

so the technique is just subtract the

smaller number from the higher number

and then put the negative sign in the

answer

so that's it you don't need to do

step by step to save time

but for both negative numbers

and the subtrahend is greater than in

the minuend in terms of absolute value

the answer is positive

ok let's have another example positive

seven

minus negative two okay let's solve this

step by step

first keep the minuend seven

then change the subtraction sign into

addition

so minus will become plus and then get

the opposite

sign of the subtrahend so negative 2

will become

positive 2 and then

apply the rules in adding integers

so the problem becomes 7 plus

2 or positive 7 plus positive 2

so the answer is 9 we will just add

because the numbers

are both positive so

seven minus negative two equals

positive nine

last example negative two

minus positive seven

okay so first step keep the minuend

negative two second change

minus two plus third

get the opposite of the subtrahend

positive seven so will be it will become

negative seven okay

the problem now becomes negative two

plus negative seven

so we will apply the rules in adding

integers

with both negative signs

okay so we will just add the numbers and

then

copy the negative sign 2 plus 7 equals 9

and copy the negative sign so negative

two minus

positive seven the answer is negative

nine

now let's proceed to multiplication

and division of integers

the rules in multiplying and dividing

integers

are the same first multiply

or divide the absolute values of

the numbers next apply the following

rules

in determining the sign of the product

or

quotient so positive

times or divide positive the answer is

always positive negative

times or divide negative the answer is

always positive and when we

multiply or divide integers with

different or unlike signs the answer is

always negative okay

let's solve some problems for you to

better understand the rules

okay let's multiply 6

times 4 or positive 6 times positive 4

okay both positive so we will just

multiply the numbers

and the answer is always positive so 6

times 4

equals 24

next example negative 6

times negative 4 in this problem

i use parenthesis because in mathematics

parenthesis means multiplication

okay so we will multiply negative six by

negative

four the absolute value of negative six

is six

the absolute value of negative 4 is 4

so multiply the absolute value 6 times 4

equals

24 same sign so the answer is

always positive another example

negative six times positive four

the signs here are different so the

answer

must be negative so we will multiply the

absolute value of negative

six and positive 4. so

just multiply 6 times 4 the answer

is 24 and since the

signs are different so the answer must

be

negative so the answer is negative

twenty-four same with if we multiply

positive six times negative four

different signs so the answer is

negative twenty-four

this time we will divide integers

first example 8 divided by

2 or positive 8 divided by

positive 2 the signs are the same

so the answer must be positive we will

just divide the numbers

eight divided by two equals four

okay next problem negative

8 divided by negative 2

again the signs are the same

both negative so the answer must be

positive we will just divide the

absolute value of

the numbers the absolute value of

negative eight is eight

and absolute value of negative two is

two so eight divided by two equals

four next example

negative eight divided by positive two

okay different signs so the answer must

be

negative we will just divide the

absolute value of

negative eight and two so eight divided

by two

equals four and the sign must be

negative so the answer is negative

four last example

negative 2 divided by positive 8

okay so we will divide the absolute

value

of negative 2 which is 2 by the absolute

value of 8 which is 8. okay let's divide

2 divided by

8 so since 2 is less than 8 we will add

0 it will become 20

20 divided by 8 equals 2

two times eight equals sixteen

then subtract twenty minus sixteen

equals

four okay let's add another zero

it will become forty forty divided by

eight equals five

five times eight equals forty

forty minus forty equals zero the answer

is

twenty five hundredths or point twenty

five

but since the signs are different

so we need to use the negative sign

therefore negative 2 divided by positive

8 equals negative 25

hundredths or negative 0.25

that's all for this topic i hope

you learned in this video see you next

time

god bless

[Music]