Rational Approximations

EXPLORING
IRRATIONAL

NUMBERS: FINDING

RATIONAL

APPROXIMATIONS

WHAT ARE RATIONAL
NUMBERS?

•Rational numbers are numbers that can be
expressed as a fraction a/b, where a and b
are integers and b ≠ 0

•Examples: 1/2, 3/4, -5/3, 2.5 (which can be
written as 5/2)

•Rational numbers have decimal expansions
that either terminate or repeat

Terminating decimal – decimals that end

Repeating decimal – decimals that repeat the same
digit or sequences of digits forever; written with a bar
over the repeating digits

WHAT ARE IRRATIONAL
NUMBERS?

•Irrational numbers cannot be expressed as
a simple fraction

•They have decimal expansions that never
end and never repeat

•Examples: π (pi), √2, e (Euler's number)

WHAT ARE REAL NUMBERS?

•Real numbers include both rational and
irrational numbers

•They can be represented on a number line

•Every point on a number line corresponds to
a real number

WHY DO WE NEED RATIONAL
APPROXIMATIONS?

IRRATIONAL NUMBERS HAVE

INFINITE DECIMAL

EXPANSIONS

WE OFTEN NEED TO USE

THESE NUMBERS IN

CALCULATIONS

RATIONAL APPROXIMATIONS
ALLOW US TO WORK WITH

IRRATIONAL NUMBERS

PRACTICALLY

Based on vocab we just learned….

Multiple Choice

√3  

• rational
• irrational

Multiple Choice

.141414141414...

• Rational 
• Irrational 

Multiple Choice

Which of the following is rational?

• .211232112421125. . . .
• π
• √25
• √12

Multiple Select

Select all sets that the number belongs to:

-6

• Irrational Number

• Rational Number

• Integer

• Whole Number

• Natural Number

FINDING
RATIONAL
APPROXIMATIONS:
SQUARE ROOTS

Let's start with √2 as an
example

We know that 1² = 1
and 2² = 4

So, 1 < √2 < 2

IMPROVING OUR
APPROXIMATION

• 1.4² = 1.96
• 1.5² = 2.25

Let's check values
between 1 and 2:

We can see that
1.4 < √2 < 1.5

GETTING EVEN CLOSER

So, 1.41 < √2 < 1.42

We can continue this process:

1.41² = 1.9881

1.42² = 2.0164

REVIEW: KEY POINTS

Irrational numbers
cannot be expressed
as simple fractions

We use rational
approximations to work
with irrational numbers

The process of finding
better approximations
can go on indefinitely

The level of precision
needed depends on
the specific situation

Multiple Choice

Between what two integers is the square root of 44?

• 3 and 4
• 4 and 5
• 5 and 6
• 6 and 7

Multiple Choice

Maddy has a square piece of craft paper that has an area of approximately $$120\ in^2$$ . What is a good approximation for the length of one side of her paper?

• 11 inches

• 16 inches

• 8 inches

• 6 inches

Multiple Choice

Which is the best estimate for: √15

• 5.6
• 2.6
• 3.9
• 3.6

Multiple Choice

Round to the Nearest Tenth: √119

• 10.9
• 10.5
• 11.1
• 10

Multiple Choice

put √7 in the right blank

• __, 2.5, 2.63, 2.65
• 2.5, __, 2.63, 2.65
• 2.5, 2.63, __, 2.65
• 2.5, 2.63, 2.65, __

Multiple Choice

What is the best ESTIMATE of $$\sqrt{40}$$ when rounded to the nearest hundredth?  

• 6.32

• 6.31

• 6.30