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Module 1 – Lesson 1

Rational & Irrational
Rational Numbers

Irrational Numbers

A rational number is a number that can be written as Irrational numbers are numbers that cannot be
a

a fraction /b (where b ≠ 0)

written as a fraction

This means that rational numbers could also be

When written as decimals, irrational numbers

represented as terminating or repeating decimals
...
25

29

3

/6

Π

0
...
1666666666…

?
?
??
0
...
1345698876623…

Terminating decimal: 3
...
84487

Converting fractions/decimals
Rational fraction to irrational decimal: Use long division to divide the denominator by the numerator
Irrational decimal to rational fraction: Count the number of repeating digits and put that as the numerator
over an equal number of 9’s (CAN BE PUT IN SIMPLEST FORM)

Module 1 – Lesson 2 & 3
Power: An expression that has a base and an exponent
...

Base: The number that is raised to an exponent; the factor that is being multiplied
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***Rule***

2

Square Root: If a = b, then a = ±√b

Non-Perfect Squares
When finding square roots, the radicand may not always be a perfect square
...

For example: √10

Approximating Square Roots

Approximating Non-Perfect Squares By Hand

Guess and check to approximate √10 to the tenths place

√9 < √10 < √16

Which perfect squares are closest to 10?

So let’s start with 3
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1 = 3
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1 = 9
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2

Use that square root as an estimated value for your
radical

3
...
2 • 3
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24 (A bit too high)

2

ANSWER: √10 ≈ 3

Decide the best estimate for 10

∑ The ≈ symbol means "approximately equal to
...
24 is the closer answer,
√10 ≈ 3
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01
100,000
1

Not a power of 10
350,000
0
...
00000004

Power of 10: A number that can be written in the
n

form 10 where n is an integer
...
(The same works when converting standard form to exponential form)
5

EX: 10 = 100,000

2

10 = 100

***Rule*** Standard Form to Exponential Form: If a decimal number has the digit 1 in the nth decimal
place preceded by zeros, it is written as 10
−5

EX: 10

-2

= 0
...
01

Scientific Notation
Scientific Notation: A number whose absolute value is between 1 and 10 multiplied by an integer power of 10
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Steps

Example

Create a number between 1 and 10

3
...
6 x 10

and a power of 10
The same can be done with numbers under 1, just
make the power of 10 negative

0
...
6 x 10

-5

Comparing Numbers in Scientific Notation
Comparing numbers in S
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: If the base number is the same, compare the exponent
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3

8

10 is 10 times as big as 10 since the exponent is one bigger
9

8

1

10 /10 = 10
9

Combine the two components

8

1

7×10 / 3×10 = 2
...
1 • 10 ) • (3 • 10 )

Steps

Example
3

18

Group the numerical factors together—multiply or
divide
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3 • (10 • 10 )

Use the properties of exponents to simplify powers
of 10—product or quotient rule

Using the product rule, add the exponents
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3 • 10

Convert to scientific notation if needed

This is already in scientific notation

Addition and Subtraction in Scientific Notation
5

8

EX: (2
...
0034 • 10 )

Steps

Example

Determine the smallest power of 10
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Add or subtract the new number factors and keep
the power of 10
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5

6
...
4 • 10

5

5

6005
...
005856 • 10

∑ Use good judgment when choosing appropriate units of measure for very large or very small numbers
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∑ Review calculator instruction manuals for specific directions, as many times E, EE, 10^, or 10x are used to
represent powers of 10
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