Chapter 1, Section 2 - UH



Chapter 1, Section 2

Numbers,

the basic building blocks. More vocabulary and some grammar, too.

Natural numbers – the counting numbers, correspond initially to body parts. They are usually the first numbers anyone learns about

{ , …}

Note: “…” is called an ellipsis and is read “and so on in this pattern”

Prime numbers – a subset of the natural numbers – are numbers greater than one that are divisible only by themselves and one (i.e. no remainder)

{ , …}

Composite numbers – a subset of the natural numbers greater than 1 that are not prime numbers.

{ , …}

Visually, consider that all the Natural numbers are rounded up into a big circle. Inside that circle the numbers are subdivided into groups by whether it is a composite or a prime. Fill in some numbers;

Where does 1 go?

Natural numbers

Even numbers – a subset of the natural numbers so that each even number is a natural number times 2.

Manglish: [pic]

Check this out with some even numbers

2(3)

2(11)

2(1)

28

18

Odd numbers – a subset of the natural numbers so that each odd number is an even number less one

Manglish: [pic]

2(1) – 1

2(4) – 1

7

13

Whole numbers – a set of numbers formed by appending 0 to the set of natural numbers

{ , …}

Whole numbers

What is an additive inverse?

So the additive inverse of 5 is _______________

Integers are the additive inverses of the natural numbers combined with the whole numbers.

{ …, , …}

note that there is no smallest integer!

How would we picture Integers with all the other numbers?

Integers

What do you know about the number 0?

Rational numbers – are numbers that can be written as a ratio of integers and, of course, the denominator is not zero.

Is 0 a rational number?

Rational numbers

What can you tell me about the number [pic]?

What about the number 1.25?

An alternate definition for Rational numbers notes that a rational number is a repeating or a terminating decimal.

So writing [pic] is true and means that the number is a rational number.

What about the number [pic][pic]?

Let’s revisit the number 2 – list all the adjectives that fit this number!

Irrational numbers are the numbers that are non-repeating, non-terminating decimals. They have many representations.

You will be expected to know that the square root of any prime number is irrational, that pi is irrational (() and that any multiple or fraction of an irrational number is irrational.

Another example of an irrational number is a decimal like the following – note the use of ellipsis!

3.01001000100001…

Does this fraction end? Is there anywhere to put a “repeat bar”?

We often use rational numbers to stand for irrational numbers – for example we’ll use 3.14 for pi ( π )…it’s not really π but it’s close. Some other common substitutions are 1.4 for[pic] and 1.7 for [pic].

Real numbers – are in a set composed of irrational numbers and rational numbers.

Fill in some irrational numbers below.

It turns out that there are more irrational numbers than there are rational numbers – a very famous and clever proof by Dr. Cantor showed this in the late nineteenth century.

The set of real numbers is represented by the horizontal number line. We can locate numbers on this line using a scale of our choosing.

Let’s locate the following set of numbers:

[pic]

Notice that I was being very nice with this example…I put all the numbers in order smallest to largest…this is not necessarily how sets are written!

Comparing Numbers

This brings us to comparing numbers – comparing is a very mathematical behavior. We have symbols that we use to compare numbers or math expressions. These symbols give us way more information that you might suppose.

The most common symbol is “ = “. This means that the expression on the left is the same as the expression on the right. You may use them interchangeably in computations. The next set of symbols gives approximate size information. Note that the symbol “points” toward the smaller expression. We use “ < “ when the LHS is definitely smaller than the RHS; we use “ ( ” when the LHS is smaller or might be equal to the RHS. A similar symbol notes that the LHS is definitely larger than the RHS ( “ > “). If the LHS is larger or might be the same as the RHS we use “ ( “.

Let’s look at some numbers and begin comparing them.

6 _________ 36

6 ________ [pic]

[pic] _________ 36

(0.06006000600006… __________ (1

3.14 _________ π

[pic] _________ [pic]

[pic] _________ 3

[pic] _________ [pic]

Now let’s look at what the inequalities are telling you.

Suppose I have an integer, x, and I know x < 0.

What do I know?

What’s the biggest number x can be? Why am I sure of this?

Let’s look at some arithmetic.

What’s 3 – 5?

What’s 7 – 9?

What’s 367 – 5000?

What do you know if I tell you x – y < 0?

What do you know if I tell you x – y > 0?

It’s these kind of facts that I’ll be expecting you to start picking up on as we progress through the course.

Absolute Value

Let’s look at another symbol and what it tells you. This one is a pair of straight bars around a number or a math expression. These are called absolute value bars and are written “[pic] “. Absolute value tells you distance without direction. If you want to be a distance 3 from 0 on the number line you’ll use the absolute value to say this.

Here’s the Manglish: [pic]. Here’s a picture:

x can be 3 or x can be (3 when absolute value bars are used.

You read the symbol [pic] by saying “the distance from x to 0 “. If you have an equation, [pic], you say “the distance from x to 0 is 3”. Thus there are two spots on the number that work for the value of x.

There is a formal definition of[pic] that we’ll use right away:

For any real number x, [pic]

We’ll apply it a few times to find

[pic]

There’s more practice in your homework!

Handout – 1.2

Given the following set, fill in the box with the appropriate word: Y for yes, N for no, and NA for not applicable. Make sure a grader can distinguish your “N” from your “NA”. Blank spots are wrong answers.

| [pic] | 1.25 | [pic] | 0 | [pic] | [pic] | (99 | [pic] |[pic] |

[pic]

| |Undefined | | | | | | | | | | | |Natural | | | | | | | | | | | |Whole | | | | | | | | | | | |Integer | | | | | | | | | | | |Rational | | | | | | | | | | | |Irrational | | | | | | | | | | | |Prime | | | | | | | | | | | |Composite | | | | | | | | | | | |Real | | | | | | | | | | | |

-----------------------

Prime

Composite

Composite

Prime

Natural

Rational numbers

Irrational

Integers

0

-x

x

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download