Handy Note 1 0 9 M

• Handy Note 1 0 9 M In Inches
• Handy Note 1 0 9 M 1 4
• Handy Note 1 0 9 M To Ft
• Handy Note 1 0 9 Multiplication

Problem: Mrs. Glosser asked Kyesha, Angie and Eduardo to list the set all of integers greater than ^-3.

Microsoft OneNote The digital note-taking app for your devices. 192 Pages Blank Pack Of 2 (2PK-502) - 10 customer reviews and a 4.9 star rating Staples® Arc Customizable Patent Quilted Leather Notebook System, Assorted, 9-1/2' x 11-1/2', Each (50061) - 14 customer reviews and a 4.9 star rating.

Analysis: Each student wrote this set using different notation.

Solution:

Each of the students in the problem above used correct notation! However, Mrs. Glosser told them that there was another way to write this set:

P = {x :x is an integer, x > ^-3 }, which is read as: 'P is the set of elements x such that x is an integer greater than ^-3.'

Mrs. Glosser used set-builder notation, a shorthand used to write sets, often sets with an infinite number of elements. Let's look at some more examples.

Each of these sets is read aloud exactly the same way when the colon : is replaced by a vertical line | as in {x | x > 0}. Both the colon and the vertical line represent the words 'such that'. Let's look at these examples again.

Note that the 'x' is just a place-holder, it could be anything, such as { q | q > 0 }.

The general form of set-builder notation is:

General Form: {formula for elements : restrictions} or {formula for elements | restrictions}

Types of Numbers

In the examples above, we examined values with set-builder notation. However, we did not specify what type of number these values can be. With set-builder notation, we normally show what type of number we are using. For example, look at x below:

{x | x > 3 }

Recall that means 'a member of', or simply 'in'. is the special symbol for Real Numbers. So x means 'all x in '.

Thus, {x | x > 3 } means 'the set of all x in such that x is any number greater than 3.' (In other words, x is all real numbers greater than 3.)

There are other types of numbers besides Real Numbers. Here are some common types used in mathematics.

Whole Numbers start at zero and go up by one forever (no fractions). The set of whole numbers is {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Counting Numbers are whole numbers greater than zero. (You cannot count with zero!) The set of counting numbers is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

Natural Numbers are whole, non-negative numbers,denoted by . This can mean either 'Counting Numbers', with = {1, 2, 3, ...}, or 'Whole Numbers', with = {0, 1, 2, 3, ...}.

Integers are the set of whole numbers and their opposites. These numbers can be negative, positive, or zero. Integers are denoted by , with = {..., ^-3, ^-2, ^-1, 0, ⁺1, ⁺2, ⁺3, ...}.

Real Numbers are denoted by the letter . Positive or negative, large or small, whole numbers or decimal numbers are all Real Numbers. A real number is any positive or negative number. This includes all integers and all rational and irrational numbers. Rational numbers, denoted by , may be expressed as a fraction (such as 7/8) and irrational numbers may be expressed by an infinite decimal representation (3.1415926535...). These numbers are called 'Real Numbers' because they are not Imaginary Numbers.

An Imaginary Number is a number which when squared, gives a negative result. There is such a number, called i, which when squared, equals negative 1. This is shown below:

When we take the square root of i, we get this algebraic result:

Thus, i is equal to the square root of negative 1. Imaginary numbers are defined as part of the Complex Numbers as shown below.

In short, a Complex Numberis a number of the form a+bi where a and b are real numbers and i is the square root of ^-1.

The definitions of these numbers may be somewhat elaborate. However, the important thing to realize is that each type of number listed above is an infinite set, and that set-builder notation is often used to describe such sets. Let's look at some examples of set-builder notation.

Note that we could also write this set as {6, 7, 8, ...}. Therefore, we can say that { K | k > 5 } = {6, 7, 8, ...}, and that these sets are equal. Set-Builder Notation is also useful when working with an interval of numbers, as shown in the examples below.

Why use set-builder notation?

You may be wondering about the need for such complex notation. If you have the set of all integers between 2 and 6, inclusive, you could simply use roster notation to write {2, 3, 4, 5, 6}, which is probably easier than using set-builder notation:

{ q: 2 ≤q≤ 6 }

But how would you list the Real Numbers in the same interval? Using roster notation doesn't make much sense in this case:

{2, 2.1, 2.01, 2.001, 2.0001, ... ??? }

To express the set of real numbers above, it is better to use set-builder notation. Start with all Real Numbers, then limit them to the interval between 2 and 6, inclusive.

{ x:x≥ 2and x≤ 6 }

You can also use set builder notation to express other sets, such as this algebraic one:

{ x:x = x² }

When you evaluate this equation algebraically, you get:

Thus { x:x = x² } = {0, 1}

Summary: Set-builder notation is a shorthand used to write sets, often for sets with an infinite number of elements. It is used with common types of numbers, such as integers, real numbers, and natural numbers. This notation can also be used to express sets with an interval or an equation.

Exercises

Directions: Read each question below. Select your answer by clicking on its button. Feedback to your answer is provided in the RESULTS BOX. If you make a mistake, rethink your answer, then choose a different button.

Handy Note 1 0 9 M In Inches

What is 'Standard Form'?

that depends on what you are dealing with!

I have gathered some common 'Standard Form's here for you..

Note: Standard Form is not the 'correct form', just a handy agreed-upon style. You may find some other form to be more useful.

Standard Form of a Decimal Number

In Britain this is another name for Scientific Notation, where you write down a number this way:

In this example, 5326.6 is written as 5.3266 × 10³,
because 5326.6 = 5.3266 × 1000 = 5.3266 × 10³

In other countries it means 'not in expanded form' (see Composing and Decomposing Numbers):

Standard Form of an Equation

The 'Standard Form' of an equation is:

(some expression)= 0

In other words, '= 0' is on the right, and everything else is on the left.

Example: Put x² = 7 into Standard Form

Answer:

x² − 7 = 0

Standard Form of a Polynomial

The 'Standard Form' for writing down a polynomial is to put the terms with the highest degree first (like the '2' in x² if there is one variable).

Example: Put this in Standard Form:

3x² − 7 + 4x³ + x⁶

The highest degree is 6, so that goes first, then 3, 2 and then the constant last:

x⁶ + 4x³ + 3x² − 7

Standard Form of a Linear Equation

The 'Standard Form' for writing down a Linear Equation is

Ax + By = C

A shouldn't be negative, A and B shouldn't both be zero, and A, B and C should be integers.

Example: Put this in Standard Form:

y = 3x + 2

Bring 3x to the left:

−3x + y = 2

Multiply all by −1:

3x − y = −2

Note: A = 3, B = −1, C = −2

This form:

Ax + By + C = 0

is sometimes called 'Standard Form', but is more properly called the 'General Form'.

Standard Form of a Quadratic Equation

The 'Standard Form' for writing down a Quadratic Equation is

(a not equal to zero)

Example: Put this in Standard Form:

Handy Note 1 0 9 M 1 4

x(x−1) = 3

Handy Note 1 0 9 M To Ft

Expand 'x(x-1)':

Handy Note 1 0 9 Multiplication

x² − x = 3

Bring 3 to left:

x² − x − 3 = 0

Note: a = 1, b = −1, c = −3