Lesson 1: Subsets of Real Numbers Lesson 2: Arranging Real

Lesson 1: Subsets of Real Numbers Lesson 2: Arranging Real

Multiple Select

1. All whole numbers are rational numbers?

• True

• False

Multiple Select

2. All rational numbers are integers.

• True

• False

Multiple Select

3. All rational numbers are whole numbers.

• True

• False

Multiple Choice

4.Zero (0) is a whole number.

• True

• False

Multiple Choice

5. 𝜋 is a rational number.

• True

• False

 Based on the above illustration,

What are the two classifications of real numbers? Yes, rational and irrational

numbers. From module 3, you have learned when is a number considered rational?

Do you still remember? Would you like to go back to your notes?  

 Rational numbers are those which can be written in the form a/b where b ≠ 0.

Irrational therefore, are those that are not rational, or numbers that cannot be written

in fraction form.

Based on the diagram, All set of numbers under rational number are rational number and are considered subsets of real number.  

 What are then the 3 main subsets of rational number? Look at the diagram. The 3

boxes immediately below the set of rational numbers, are the 3 main subsets of rational numbers.

Those are: Fractions, Integers, and take note, some Decimals. As

you can see, one type of decimals is under the set of irrational number. Have you

seen it? In other words, “only some decimals are rational number.” 

Those decimals

are: Terminating and Repeating, as you can see in the diagram. Why do you think

so? Let us take a look at these rational numbers.  

FRACTIONS

Fractions, of course, is a rational number, because all fractions are written in the

form a/b , where a is the numerator, b is the denominator and b must not be zero.

You have learned so much about fractions in the previous module including its operations, right?  

DECIMALS

As we have learned, only terminating and repeating decimals are rational number.

Recall, when is a decimal considered terminating? Observe the illustrations below:  

 Terminating decimal results when you obtain a zero remainder upon dividing the numerator by the denominator of a given fraction. And how will you prove that a terminating decimal is a rational number? See below.  

Can you prove now that 0.75 is a rational number? What is the fraction form of

0.75, if there is any? Yes, you are right. 0.75 = 3/4 , hence, 0.75 is a rational number.

How about Repeating Decimals? Why are they considered rational? Look at the

next illustration. 

And how can we justify that

a repeating decimal is a rational number. Again, we have to show that the decimal

can be written in fraction form.  

From the Real Number diagram above, notice that under the set of Integers,

we have counting or natural numbers, its opposites and zero. Right? How does the set of whole numbers differ from the set of counting numbers?

Set of whole numbers = { 0, 1, 2 ,3, 4, 5, …}

Set of counting/natural numbers = { 1, 2, 3,4,5,6,… }

(Note: Counting numbers and Natural numbers are the same)

 Have you seen the difference? Yes, we do not start counting from zero, right?

Hence, the set of counting / natural numbers start from 1 and zero is not a part of

the set of counting/natural numbers. Hope that is clear. 

Let us these further examples:

Positive whole numbers less than 5 = { 1, 2, 3, 4 }

Whole numbers less than 5 = {0, 1, 2, 3, 4}

Counting/Natural numbers less than 5 = { 1, 2, 3, 4 }  

 From here we can see that, “every counting number is a whole number”, also,

“every whole number is an integer” and “zero (0) is not a counting number, but it is

an integer and a whole number”.  

–4 is an integer. How can this be written in fraction form?

Yes, – 4 = − 4/1

Every integer / whole number can be written in fraction form where the denominator

is 1, hence, every integer / whole number is a rational number. Got it?  

Arranging Real Numbers 

A Real Number Line allows us to visually display real numbers and makes it

easy to tell which numbers are greater or lesser. It clearly shows the order of real

numbers. A number on the left is always less than a number on the right. Similarly, a

number on the right is greater than a number on the left.

Remember:

 Any real number can be associated with a point on a line.

 Create a number line by first identifying the origin and marking off a scale appropriate for the given problem.

 Negative numbers lie to the left of the origin and positive numbers lie to the right.

 Smaller number always lie to the left of a larger number on the number line.

 The opposite of a positive number is negative and the opposite of a negative number is positive. 

Module 9

Writing numbers in scientific notation & vice versa.

scientific notation

is a system of notation used to express very large or very small numbers conveniently. Its uses exponents so as to require the use of so many zeros which can be confusing and lead to errors.

Multiple Choice

1. How would you write 0.0005 in scientific notation?

• 50 x 10⁵

• 5 x 10³

• 5 x 10^-4

• .5 x 10³

Multiple Select

2. How would you write -5.6 x 10^-3 in standard form?

• 0.0056

• 0.00056

• -5,600

• -0.0056

Multiple Select

3. The rule of scientific notation is to write all exponents with a base of ____.

• 50

• 100

• 5

• 10

Multiple Choice

4. How do you write

1001

in scientific notation?

• 1.0001 x 10⁴

• 1.001 x 10³

• 1.01 x 10⁵

• 10.1 x 10³

Multiple Select

5. How do you write

8.317 x 10⁶

in standard form?

• 8, 371, 000

• 837, 100

• 83, 170, 000

• 8, 317, 000

Multiple Choice

1. – 1.25 ___ − 3/4

• <

• >

• =

Multiple Select

2. – 23 ___ – 32

• >

• <

• =

Multiple Choice

3. −√9 ___ −𝜋

• <

• >

• =

Multiple Choice

4. ) √5 ,

• Real, irrational

• Real, Rational

• Real, Integers

Multiple Choice

5. 37

• Real,Rational, Integers, Whole numbers

• Real, Irrational, Integers, Whole number