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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https://www.basic-mathematics.com/set-notation.html

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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https://www.toppr.com/guides/maths/sets/operations-on-sets/

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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Source:
https://whatis.techtarget.com/definition/integer

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.



This article was originally published on The Math Page. Read the original article.

Source:

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.

This article was originally published on The toppr. Read the original article.

Source:
https://www.toppr.com/guides/maths/rational-numbers/properties-of-integers/

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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Source:
https://www.edplace.com/blog/what-are-square-numbers-square-numbers-explained/

Finding General Term of Squence

A sequence is a function whose domain is an ordered list of numbers. These numbers are positive integers starting with 1. Sometimes, people mistakenly use the terms series and sequence. A sequence is a set of positive integers while series is the sum of these positive integers. The denotation for the terms in a sequence is:

a1, a2, a3, a4, an, . . .

Finding the nth term of a sequence is easy given a general equation. But doing it the other way around is a struggle. Finding a general equation for a given sequence requires a lot of thinking and practice but, learning the specific rule guides you in discovering the general equation. In this article, you will learn how to induce the patterns of sequences and write the general term when given the first few terms. There is a step-by-step guide for you to follow and understand the process and provide you with clear and correct computations.

An arithmetic series is a series of ordered numbers with a constant difference. In an arithmetic sequence, you will observe that each pair of consecutive terms differs by the same amount. For example, here are the first five terms of the series.

3, 8, 13, 18, 23

Do you notice a special pattern? It is obvious that each number after the first is five more than the preceding term. Meaning, the common difference of the sequence is five. Usually, the formula for the nth term of an arithmetic sequence whose first term is a1 and whose common difference is d is displayed below.

an = a1 + (n - 1) d

Source: teacherspayteachers.com

Steps in Finding the General Formula of Arithmetic and Geometric Sequences
1. Create a table with headings n and an where n denotes the set of consecutive positive integers, and an represents the term corresponding to the positive integers. You may pick only the first five terms of the sequence. For example, tabulate the series 5, 10, 15, 20, 25, . . .

2. Solve the first common difference of a. Consider the solution as a tree diagram. There are two conditions for this step. This process applies only to sequences whose nature are either linear or quadratic.

Condition 1: If the first common difference is a constant, use the linear equation ax + b = 0 in finding the general term of the sequence.

a. Pick two pairs of numbers from the table and form two equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.

b. After forming the two equations, calculate a and b using the subtraction method.

c. Substitute a and b to the general term.

d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.

Condition 2: If the first difference is not constant and the second difference is constant, use the quadratic equation

a. Pick three pairs of numbers from the table and form three equations. The value of n from the table corresponds to the x in the linear equation, and the value of an corresponds to the 0 in the linear equation.


b. After forming the three equations, calculate a, b, and c using the subtraction method.

c. Substitute a, b, and c to the general term.

d. Check if the general term is correct by substituting the values in the general equation. If the general term does not meet the sequence, there is an error with your calculations.




This article was originally published on The owlcation. Read the original article.


Source:
https://owlcation.com/stem/How-to-Find-the-General-Term-of-Arithmetic-and-Geometric-Sequences

Definition of Set

A set is a collection of objects, things or symbols which are clearly defined.
The individual objects in a set are called the members or elements of the set.

The following table shows some Set Theory Symbols.

A set must be properly defined so that we can find out whether an object is a member of the set.

1. Listing the elements (Roster Method)
The set can be defined by listing all its elements, separated by commas and enclosed within braces. This is called the roster method.

Example:
B = {2, 4, 6, 8, 10}
X = {a, b, c, d, e}

However, in some instances, it may not be possible to list all the elements of a set. In such cases, we could define the set by method 2.

2. Describing the elements
The set can be defined, where possible, by describing the elements. This is called the set-builder notation.

Example:
C = {x : x is an integer, x > – 3 }
This is read as: “C is the set of elements x such that x is an integer greater than –3.”

D= {x: x is a river in a state}

We should describe a certain property which all the elements x, in a set, have in common so that we can know whether a particular thing belongs to the set.

We relate a member and a set using the symbol ∈. If an object x is an element of set A, we write x ∈ A. If an object z is not an element of set A, we write z ∉ A.

∈ denotes “is an element of’ or “is a member of” or “belongs to”

∉ denotes “is not an element of” or “is not a member of” or “does not belong to”

Example:
If A = {1, 3, 5} then 1 ∈ A and 2 ∉ A

Basic vocabulary used in set theory

A set is a collection of distinct objects. The objects can be called elements or members of the set.
A set does not list an element more than once since an element is either a member of the set or it is not.

There are three main ways to identify a set:
1. A written description
2. List or Roster method
3. Set builder Notation

The empty set or null set is the set that has no elements.
The cardinality or cardinal number of a set is the number of elements in a set.
Two sets are equivalent if they contain the same number of elements.
Two sets are equal if they contain the exact same elements although their order can be different.

Definition and notation used for subsets and proper subsets

If every member of set A is also a member of set B, then A is a subset of B, we write A ⊆ B. We can also say A is contained in B.

If A is a subset of B, but A is not equal B then A is a proper subset of B, we write A ⊂ B.

The empty set is a subset of any set.
If a set A has n elements that it has 2n subsets.

How to use Venn diagrams to show relationship between sets and set operations?

A Venn diagram is a visual diagram that shows the relationship of sets with one another.
The set of all elements being considered is called the universal set (U) and is represented by a rectangle.

The complement of A, A', is the set of elements in U that is not in A.
Sets are disjoint if they do not share any elements.
The intersection of A and B is the set of elements in both set A and set B.
The union of A and B is the set of elements in either set A or set B or both.



This article was originally published on The online math learning. Read the original article.

Source:
https://www.onlinemathlearning.com/describing-sets.html

Comparing Fractions

In this lesson, you will learn about comparing fractions. Before you start this lesson, I recommend that you study or review my lesson about fractions. There are many ways to compare fraction.

Comparing Fractions

Equivalent Fractions
Equivalent fractions are fractions that represent the same quantity.

example:
1/2 = 2/4 = 4/8

Fundamental Law of Fractions: the value of a fraction does not change when its numerator and denominator are both multiplied by the same number (not zero).
[Remembering that multiplication by ½ can also be considered division by 2, this law also extends to "division".]


Simplifying Fractions
Reducing Fractions

When working with fractions, it is a common practice to list fractional answers in their "simplest form". The simplest form usually gives the reader a better idea of what the fraction represents.

To simplify (or reduce) a fraction, look for a number that will divide exactly into BOTH the numerator and the denominator. Try to find the "largest" such value (the greatest common divisor).

Example:
3/6 = 1/2

If you don't choose the LARGEST common factor (the GCD), you can still reduce (or simplify) the fraction, but you will have to do the division process more than one time.

Less Than, Greater Than, Equal To
Comparing fractions as to their size.
A. Simple Cut-up Visual Comparisons
These diagrams show a pictorial representation of what is occurring in relation to the size of the fractions. Each of the whole bars are divided into 5, 3, or 6 equal pieces and compared. Unfortunately, it can be difficult to compare fractions in this simple cut-up visual manner. If we did not know that 2/3 = 4/6 in the last comparison, it would be difficult to tell from the diagram whether the fractions were equal or just really close to being equal.

B. Compare using a Common Denominator
1. Comparing fractions that already have the same denominator is easy. The fraction with the larger numerator will be the larger fraction. Since the denominators are the same, you are comparing "pieces" (parts) of the same size.

2. Comparing fractions that have different denominators is harder. The solution requires that we create equivalent fractions that have the same denominators and then compare them. When looking for this same denominator (called the common denominator), look for the smallest possible number (called the least common multiple or the least common denominator) to keep the calculations as simple as possible.

If we re-examine our visual models from above, we can see that when the whole bar is divided equally for BOTH fractions, using the common denominator, we have a more reliable and accurate portrayal of the fraction sizes.

3. Compare using the Cross Multipy Rule
We saw in Ratios and Proportions that the Cross Multiplication Algorithm (Rule) was the result of rewriting fractions to have the same denominator and then comparing the numerators.
Here's how this process works:
To compare two fractions,

• multiply the numerator from the fraction on the left by the denominator of the fraction on the right.
• Place this number above the fraction on the left.
• Now, multiply the denominator from the fraction on the left by the numerator of the fraction on the right. 
• Place this number above the fraction on the right.
• Place the appropriate inequalitiy symbol (< , >, or even =)

As with the Cross Multiplication Algorithm (Rule), this cross multiply application to comparing fractions is actually the result of rewriting fractions to have the same denominators (which we do not write down) and then comparing the numerators.

4. Ordering Fractions
Place in ascending order (from smallest on the left to largest on the right)




This article was originally published on The mathbits notebook. Read the original article.

Source:
https://mathbitsnotebook.com/JuniorMath/FractionsDecimals/FDcomparing.html

Operation of Algebraic Expression

Before we see how to add and subtract integers, we define terms and factors.

Terms and Factors

A term in an algebraic expression is an expression involving letters and/or numbers (called factors), multiplied together.

Example
The algebraic expression

5x

is an example of one single term. It has factors 5 and x.

The 5 is called the coefficient of the term and the x is a variable.

Like Terms

"Like terms" are terms that contain the same variables raised to the same power.

Example
3x and 7x are like terms.

Adding and Subtracting Terms

Important: We can only add or subtract like terms.

Why? Think of it like this. On a table we have 4 pencils and 2 books. We cannot add the 4 pencils to the 2 books - they are not the same kind of object.

We go get another 3 pencils and 6 books. Altogether we now have 7 pencils and 8 books. We can't combine these quantities, since they are different types of objects.

Next, our sister comes in and grabs 5 pencils. We are left with 2 pencils and we still have the 8 books.

Similarly with algebra, we can only add (or subtract) similar "objects", or those with the same letter raised to the same power.

Example
Simplify 13x + 7y − 2x + 6a

13x + 7y − 2x + 6a

The only like terms in this expression are 13x and −2x. We cannot do anything with the 7y or 6a.

So we group together the terms we can subtract, and just leave the rest:

(13x − 2x) + 6a + 7y

= 6a + 11x + 7y

We usually present our variables in alphabetical order, but it is not essential.

Multiplication of Algebraic Expressions

When we multiply algebraic expressions, we need to remember the Index Laws from the Numbers chapter.

Let's see how algebra multiplication works with a series of examples.

Example
Multiply (x + 5)(a − 6)

We multiply this out as follows. We take each term of the first bracket and multiply them by the second bracket. Then we expand out the result.

(x + 5)(a − 6)

= x(a − 6) + 5(a − 6)

= ax − 6x + 5a − 30

We cannot do any more with this answer. There are no like terms, so we cannot simplify it in any way.

Dividing by a Fraction

Recall the following when dividing algebraic expressions.

The reciprocal of a number x, is 1/x

For example, the reciprocal of 5 is 1/5 and the reciprocal of 5/3 is 3/5

To divide by a fraction, you multiply by the reciprocal of the fraction.




This article was originally published on The Interactive Mathematics. Read the original article.

Source:
https://www.intmath.com/basic-algebra/1-addition-subtraction-algebra.php

https://www.intmath.com/basic-algebra/2-multiplication-algebra.php

https://www.intmath.com/basic-algebra/3-division-algebra.php

Ratio

Proportions are built from ratios. A "ratio" is just a comparison between, or a relating of, two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Suppose there are thirty-five people, fifteen of whom are men. Then the rest are women, so:

35 – 15 = 20

...there are twenty women in the group. The language "the ratio of (this) to (that)" means that (this) comes before (that) in the comparison. So, if one were to express "the ratio of men to women", then the ratio, in English words, would be "15 men to 20 women" (or just "15 to 20").


The order of the items in a ratio is very important, and must be respected; whichever word came first in the ratio (when expressed in words), its number must come first in the ratio. If the expression had been "the ratio of women to men", then the in-words expression would have been "20 women to 15 men" (or just "20 to 15").

source: twinkl.co.uk

Expressing the ratio of men to women as "15 to 20" is expressing the ratio in words. There are two other notations for this "15 to 20" ratio:

odds notation: 15 : 20

fractional notation: 15/20

 You should be able to recognize all three notations; you will probably be expected to know them, and how to convert between them, on the next test. For example:

There are 16 ducks and 9 geese in a certain park. Express the ratio of ducks to geese as a ratio with a colon, as a fraction (do not reduce), and in words.
They want "the ratio of ducks to geese", so the number for the ducks comes first (or, for the fractional form, on top). So my answer is:

16:9, 9/16, 16 to 9

The ratio from a representative group can also be used to provide percentage information.

In the class above, what percentage of students passed the class? (Round your answer to one decimal place.)
I already know that the representative group contains 12 students, of which 7 passed the class. Converting this to a percentage (by dividing, and then moving the decimal point, as explained here), I get:

7/12 = 0.583333... = 58.3333...%

They want the answer rounded to one decimal place, so my answer is:

58.3% passed



This article was originally published on The purplemath. Read the original article.

Source:
https://www.purplemath.com/modules/ratio.htm

Aplication of the Pythagorean Theorem

The Pythagorean Theorem is a statement in geometry that shows the relationship between the lengths of the sides of a right triangle – a triangle with one 90-degree angle. Being able to find the length of a side, given the lengths of the two other sides makes the Pythagorean Theorem a useful technique for construction and navigation.

Source: passnownow.com

Architecture and Construction

Given two straight lines, the Pythagorean Theorem allows you to calculate the length of the diagonal connecting them. This application is frequently used in architecture, woodworking, or other physical construction projects. For instance, say you are building a sloped roof. If you know the height of the roof and the length for it to cover, you can use the Pythagorean Theorem to find the diagonal length of the roof's slope. You can use this information to cut properly sized beams to support the roof, or calculate the area of the roof that you would need to shingle.

Laying Out Square Angles

The Pythagorean Theorem is also used in construction to make sure buildings are square. A triangle whose side lengths correspond with the Pythagorean Theorem – such as a 3 foot by 4 foot by 5 foot triangle – will always be a right triangle. When laying out a foundation, or constructing a square corner between two walls, construction workers will set out a triangle from three strings that correspond with these lengths. If the string lengths were measured correctly, the corner opposite the triangle's hypotenuse will be a right angle, so the builders will know they are constructing their walls or foundations on the right lines.

Navigation

The Pythagorean Theorem is useful for two-dimensional navigation. You can use it and two lengths to find the shortest distance. For instance, if you are at sea and navigating to a point that is 300 miles north and 400 miles west, you can use the theorem to find the distance from your ship to that point and calculate how many degrees to the west of north you would need to follow to reach that point. The distances north and west will be the two legs of the triangle, and the shortest line connecting them will be the diagonal. The same principles can be used for air navigation. For instance, a plane can use its height above the ground and its distance from the destination airport to find the correct place to begin a descent to that airport.

Surveying

Surveying is the process by which cartographers calculate the numerical distances and heights between different points before creating a map. Because terrain is often uneven, surveyors must find ways to take measurements of distance in a systematic way. The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope's line of sight and the measuring stick form a right angle. Since the surveyor knows both the height of the measuring stick and the horizontal distance of the stick from the telescope, he can then use the theorem to find the length of the slope that covers that distance, and from that length, determine how steep it is.





This article was originally published on The sciencing. Read the original article.

Source:

https://sciencing.com/real-life-uses-pythagorean-theorem-8247514.html

Patterns

The Pattern is defined as the series or sequences that are replicates. The sequence of objects that are arranged based on particular rule is known as number pattern.

Algebraic pattern:

The sequence of number pattern based on the addition or subtraction is known as algebraic pattern.

algebraic pattern

Example:

Consider the sequence 1, 5, 9, 13, 17, 21, 25,...

Here, the numbers are arranged by following certain rule, which is explained below:

(i) First number in the given number pattern is 1.

(ii) Add 4 to the first number of the sequence. That is, 1 + 4 = 5, is the second number of the sequence.

(iii) Add 4 to the second number of the sequence. That is, 5+ 4 = 9, is the third number of the sequence.

(iv) Add 4 to the third number of the sequence. That is, 9 + 4 = 13, is the fourth number of the sequence.

(v) Add 4 to the fourth number of the sequence. That is, 13 + 4 = 17, is the fifth number of the sequence. Similarly, the remaining numbers will be obtained.

Thus, the given sequence is ordered by using the addition rule. Hence, the pattern followed by the sequence is an algebraic pattern.



Geometric pattern:

The sequence of number pattern based on the multiplication or division is known as geometric pattern.

Geometric pattern

Example:

Consider the sequence 90, 45, 22.5, 11.25,...

Here, the numbers are arranged by following certain rule, which is explained below:

(i) First number in the given number pattern is 90.

(ii) One-half of the first number gives the second number of the pattern. That is, , is the second number of the sequence.

(ii) One-half of the second number gives the third number. That is, , is the third number of the sequence. Similarly, the remaining numbers will be obtained.

Thus, the given sequence is ordered by using the division rule. Hence, the pattern followed by the sequence is a geometric pattern



This article was originally published on The cheeg. Read the original article.

Source:
https://www.chegg.com/homework-help/definitions/algebraic-and-geometric-patterns-67