Multiplying Integers Rules

Multiplying integers is just like the multiplication of whole numbers, except that with integers, you have to keep tract of your signs.



Recall that 6 + 6 + 6 = 6 × 3

Instead of adding 6 three times, you can multiply 6 by 3 and get 18, the same answer.

Similarly,

6 + 6 + 6 + 6 + 6 + 6 + 6 = 6 × 7 = 42

Still by the same token,

2 + 2 + 2 + 2 = 2 × 4

In algebra, 2 × 4 can be written as (2)(4)

You can think of this as four groups of 2

In general,when multiplying integers, remember the followings:

Positive × Positive = Positive

For example,

7 × 6 = 42

2 × 5 = 10

3 × 10 = 30

8 × 2 = 16

Now, try adding -3 to -3

- 3 + -3 = -3 × 2

The reasoning is the same; Instead of adding -3 two times, you can just multiply -3 by 2.

To model this on the number line, just start at 0 and put 2 groups of -3 of the number line. You end up at -6 and -6 is negative.

In general,when multiplying integers

Positive × Negative = Negative

For example:

8 × -5 = - 40

2 × -10 = -20

3 × - 6 = - 18

5 × -5 = - 25

The last case we need to cover is:

Negative × Negative = ?


In general,when multiplying integers,

Negative × Negative = Positive

For example:

-9 × -5 = 45

-4 × -2 = 8

-1 × -1 = 1

-2 × - 6 = 12

Source: tes.com

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Set notation

I was taught set notation when I was in sixth grade. The teacher started by giving us the definition of a set.
Basically, he said that anything with a collection of objects is a set. Having said that, we felt afterward that we could easily make up our own sets.

Set notation

You could be surprised what a set could be made of. Basically, any collections of things that make sense.

Examples of sets:

1. The set of all letters in the modern English alphabet

What is included in that set? a, b, c, d, e, f, etc...

2. The set of all great mathematicians in the past

What is included in that set? We could mention with Carl Gauss, Isaac Newton, Einstein, Blaise Pascal, Euclid, Pierre de Fermat, etc..Sorry, I am not silly to list them all here!

3. The set of all positive numbers less than 10

What is included in that set? 1, 2, 3, 4, 5, 6, 7, 8, 9

4. The set of all types of sausages

What is included in that set? I am not sure I am knowledgeable enough in this area. I know Italian sausage and ???. What else? Did I miss something?

5. The set of all states in the United States

Oh Boy.Geography! You know what.I think you got the point. Let us move on to something else



Finite and infinite set:

A set is finite if you can list all its elements and infinite otherwise.

All sets described above are finite because you can list or count all their elements

However, among the five sets, one set can be turned into an infinite set with one small change. That set is set number 3

If I get rid of the word positive and say instead "the set of all numbers less than 10" the set in now infinite because you cannot count all those numbers less than 10.

Ways to define a set:

a. With a verbal description: All sets above are described verbally when we say, " The set of all bla bla bla "

b. A listing of all members separated by commas with braces ({ and }):

A listing of set 1 is written as: {a, b,c,d,e,f,....,z}

c. Set-builder notation:

Set 1 and set 4 can be written as { x / x is a letter of the modern English alphabet} and { x / x is a type of sausage}

{ x / x is a letter of the modern English alphabet} is read, " The set of all x such that x is a letter in the modern English alphabet

Set-builder is an important concept in set notation. You must understand it.

We use capital letters such as A, B, and so forth to denote sets

For example, you could let A be the set of all positive numbers less than 10.

We use the symbol Î to indicate that an object belongs to a set and the symbol Ï to indicate that an object does not belong to a set

For example, if A is the set of all positive numbers less than 10, then 2 ∈ A, but 12 ∉ A

A set that has no element is called empty set and is denoted by { } or ∅

For example, {x / x is a human being who have lived 10,000 years} is an empty set because it is impossible to find at least one human being who have lived so long

Two sets are equal if they have exactly the same element

For example, { x / x is a number between bigger than 1 and less than 5} and { 2, 3, 4} are equal sets


Subtleties with set notation:

Two sets are still equal even if the same element is listed twice

{ 2, 3, 4} and { 2, 3, 3, 4} are equal

The order of elements in sets does not matter

{ 2, 3, 4} = { 2, 3, 3, 4} = { 4, 3, 2}

This concludes the lesson about set notation


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Operations on Sets

Now that we have an idea about what a set is, we can move on to understand how to work with them. Is there a way to merge two sets together? What is the intersection of sets and how do we find the intersection of two sets? Should we even be allowed to add two sets together? Here we will find out all the answers!

Operations on Sets

Operations on Sets

Unlike the real world operations, mathematical operations do not require a separate no-contamination room, surgical gloves, and masks. But certainly, expertise to solve the problem, special tools, techniques, and tricks as well as knowledge of all the basic concepts are required to obtain a solution. Following are some of the operations that are performed on the sets: –

Union
Intersection
Difference
Complement
Let’s deal with them one by one.

Union of Sets
Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12}. Then, A U B is represented as the set containing all the elements that belong to both the sets individually. Mathematically,

A U B = {x : x ϵ A or x ϵ B}

So, A U B = {2, 4, 6, 8, 10, 12},

here the common elements are not repeated.

Properties of (A U B)
Commutative law holds true as (A U B) = (B U A)
Associative law also holds true as (A U B) U {C} = {A} U (B U C)
Let A = {1, 2} B = {3, 4} and C = {5, 6}
A U B = {1, 2, 3, 4} and (A U B) U C = {1, 2, 3, 4, 5, 6}
B U C = {3, 4, 5, 6} and A U (B U C) = {1, 2, 3, 4, 5, 6}
Thus, the law holds true and is verified.

A U φ = A (Law of identity element)
Idempotent Law – A U A = A
Law of the Universal set (U): (A U U) = U


Intersection of Sets
An intersection is the collection of all the elements that are common to all the sets under consideration. Let A = {2, 4, 6, 8} and B = {6, 8, 10, 12} then A ∩ B or “A intersection B” is given by:

“A intersection B” or A ∩ B = {6, 8}

Mathematically, A ∩ B = {x : x ϵ A and x ϵ B}

Properties of the Intersection – A ∩ B
The intersection of the sets has the following properties:

Commutative law – A ∩ B = B∩ A
Associative law – (A ∩ B)∩ C = A ∩ (B∩ C)
φ ∩ A = φ
U ∩ A = A
A∩ A = A; Idempotent law.
Distributive law – A ∩ (BU C) = (A ∩ B) U (A ∩ C)
Difference of Sets
The difference of set A and B is represented as:

A – B = {x : x ϵ A and x ϵ B}

Conversely, B – A = {x : x ϵ A and x ϵ B}

Let, A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8} then A – B = {1, 3, 5} and B – A = {8}. The sets (A – B), (B – A) and (A ∩ B) are mutually disjoint sets; it means that there is NO element common to any of the three sets and the intersection of any of the two or all the three sets will result in a null or void or empty set.


Complement of Sets
If U represents the Universal set and any set A is the subset of A then the complement of set A (represented as A’) will contain ALL the elements which belong to the Universal set U but NOT to set A.

Mathematically, A’ = U – A

Alternatively, the complement of a set A, A’ is the difference between the universal set U and the set A.

Properties of Complement Sets
A U A’ = U
A ∩ A’ = φ
De Morgan’s Law – (A U B)’ = A’ ∩ B’ OR (A ∩ B)’ = A’ U B’
Law of double complementation : (A’)’ = A
φ’ = U
U’ = φ
Hence, these are the basic concepts and operations on Sets.



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INTEGER

An integer (pronounced IN-tuh-jer) is a whole number (not a fractional number) that can be positive, negative, or zero.

INTEGER

Examples of integers are: -5, 1, 5, 8, 97, and 3,043.

Examples of numbers that are not integers are: -1.43, 1 3/4, 3.14, .09, and 5,643.1.

The set of integers, denoted Z, is formally defined as follows:

Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

In mathematical equations, unknown or unspecified integers are represented by lowercase, italicized letters from the "late middle" of the alphabet. The most common are p, q, r, and s.

The set Z is a denumerable set. Denumerability refers to the fact that, even though there might be an infinite number of elements in a set, those elements can be denoted by a list that implies the identity of every element in the set. For example, it is intuitive from the list {..., -3, -2, -1, 0, 1, 2, 3, ...} that 356,804,251 and -67,332 are integers, but 356,804,251.5, -67,332.89, -4/3, and 0.232323 ... are not.

The elements of Z can be paired off one-to-one with the elements of N, the set of natural numbers, with no elements being left out of either set. Let N = {1, 2, 3, ...}.

In infinite sets, the existence of a one-to-one correspondence is the litmus test for determining cardinality, or size. The set of natural numbers and the set of rational numbers have the same cardinality as Z. However, the sets of real numbers, imaginary numbers, and complex numbers have cardinality larger than that of Z.





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Operations of Arithmetic

The first thing to note is that in algebra we use letters as well as numbers. But the letters represent numbers. We imitate the rules of arithmetic with letters, because we mean that the rule will be true for any numbers.

Here, for example, is the rule for adding fractions:

a/c + b/c = a/c + b/c

The letters a and b mean: The numbers that are in the numerators. The letter c means: The number in the denominator. The rule means:

"Whatever those numbers are, add the numerators
and write their sum over the common denominator."

Algebra is telling us how to do any problem that looks like that. That is one reason why we use letters.

(The symbols for numbers, after all, are nothing but written marks. And so are lettersexclamation As the student will see, algebra depends only on the patterns that the symbols make.)

The numbers are the numerical symbols, while the letters are called literal symbols.

The four operations of arithmetic

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

1. Addition:  a + b.   The operation sign is + , and is called the plus sign.  Read a + b as "a plus b. For example, if a represents 3, and b represents 4, then a + b represents 7.
2. Subtraction:  a − b.   The operation sign is − , and is called the minus sign.  Read a − b as "a minus b." If a represents 8, for example, and b represents 2, then a − b represents 6.
3. Multiplication:  a· b.  Read a· b as "a times b." The multiplication sign in algebra is a centered dot.  We do not use the multiplication cross ×, because we do not want to confuse it with the letter x. And so if a represents 2, and b represents 5, then a· b = 2· 5 = 10. "2 times 5 equals 10." Do not confuse the centered dot -- 2·5, which in the United States means multiplication -- with the decimal point:  2.5. However, we often omit the multiplication dot and simply write ab.  Read "a, b."  In other words, when there is no operation sign between two letters, or between a letter and a number, it always means multiplication.  2x  means  2 times x.
4. Division:  a/b. Read as "a divided by b." In algebra, we use the horizontal division bar.  If a represents 10, for example and b represents 2, then a/b  =   10/ 2   =  5. "10 divided by 2 is 5."

Note:  In algebra we call  a + b  a "sum" even though we do not name an answer.  As the student will see, we name something in algebra simply by how it looks.  In fact, you will see that you do algebra with your eyes, and then what you write on the paper, follows.

Similarly, we call  a − b  a difference,  ab

 a product, and  a
b  a quotient.
This sign = of course is the equal sign, and we read this --

a = b

-- as "a equals (or is equal to) b."

That means that the number on the left that a represents, is equal to the number on the right that b represents.  If we write

a + b = c,

and if a represents 5, and b represents 6, then c must represent 11.



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https://www.themathpage.com/alg/algebraic-expressions.htm#parentheses

Properties of Integers

We will learn about properties of integers. This is resume of integers properti

properties of integers

Let us now study these properties in detail.

Closure Property

The System of Integers in Addition
It states that addition of two Integers always results in an Integer. For example, 7 + 4 = 11,  the result we get is an integer. Therefore, the system is closed under addition.

The System of Integers under Subtraction
It states that subtraction of two Integers always results in an Integer. For example, 7 – 4 = 3, the result we get is an integer. Also, 2 – 4 = -2. The result is also an integer. Therefore, the system is closed under subtraction.

The System of Integers under Multiplication
It states that multiplication of two integers always results in an integer. For example, 7 × 4 = 28, the result we get is an integer. Therefore, the system is closed under multiplication.

The System of Integers under Division
It states that division of two integers does not always result in an integer. For example, 7 ÷ 4 = 74, the result we get is not an integer. But, 8 ÷ 4 = 2, the result we get is an integer. Therefore, a system is not closed under division.

Commutative Property

It is a property that associates with binary operations or functions like addition, multiplication. Take any two numbers a and b and subtract them. That is a – b, say 5 – (-3). Now subtract a from b. That is b – a or -3 – 5. Are they same? No, they are not equal. So, the commutative property does not hold for subtraction. Similarly, it does not hold for division too.

Again take any two numbers a and b and add them. That is a + b. Now add b and a which comes to be b+ a. Aren’t the same? Yes, they are equal because of commutative property which says that we can swap the numbers and still we get the same answer.

Associative Property

Associative property of integers states that for any three elements(numbers) a, b and c

1) For Addition a + ( b + c ) = ( a + b ) + c
2) For Multiplication a × ( b × c ) = ( a × b ) × c
3) For Subtraction. Associative property does not hold for subtraction a – ( b – c ) != ( a – b ) – c
4) For Division. Associative property does not hold for division a ÷ ( b ÷ c ) != ( a ÷ b ) ÷ c

Multiplicative Identity for Integers
The multiplicative identity of any integer a  is a number b which when multiplied with a, leaves it unchanged, i.e. b is called as the multiplicative identity of any integer a if a× b = a. Now, when we multiply 1 with any of the integers a we get a × 1 = a = 1 × a  So, 1 is the multiplicative identity for integers.

Additive Identity for Integers

The additive identity of any integer a  is a number b which when added with a, leaves it unchanged, i.e. b is called as the additive identity of any integer a if a + b = a. Now, when we add 0 with any of the integers a we get a + 0 = a = 0 + a  So, 0 is the additive identity for integers.

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Square Number

When you multiply a whole number (not a fraction) by itself, the result is a square number. For example 3 x 3 = 9. Nine is the square of three multiplied by itself.
As mathematicians often try to shorten the way things are written, we show that we want to square a number by simply writing that number with a small  '2' to the right of it.

Source: 34auburn.weebly.com

Squaring Negative Numbers

As you may know already, if you multiply a negative number by another negative number, it becomes a positive.

Example:  -3 x -3 would become 9 just the same as it would if both the 3’s were positive!

However, if you are multiplying a negative with a positive, like -3 x 3 it would become negative -9 and then, of course, it wouldn’t be a square number (because -3 is a different number to 3)!

Squaring Decimals

Just like whole numbers (integers), it’s easy to square a decimal number too!

1.23 Squared = 1.23 × 1.23 = 1.5129

Square Root

A square root is a number that’s been multiplied to get the square number. For example the square root of 9 is 3 because 3 x 3 = 9.

Finding the square root of a number is much trickier than calculating the square number in the first place, so many calculators have a square root button. This is the one that looks like a tick √. It’s called the radical.


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