Pythagorean Theorem and Square Roots

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A right triangle is a triangle with a right angle as one of its

angle.

Pythagorean Theorem and Square Roots

A right triangle is a triangle with a right angle as one of its

angles. The longest side C of a right triangle is called the

hypotenuse,

Pythagorean Theorem and Square Roots

hypotenuse

C

A right triangle is a triangle with a right angle as one of its

angles. The longest side C of a right triangle is called the

hypotenuse, the two sides A and B forming the right angle

are called the legs.

Pythagorean Theorem and Square Roots

hypotenuse

legs

A

B

C

A right triangle is a triangle with a right angle as one of its

angles. The longest side C of a right triangle is called the

hypotenuse, the two sides A and B forming the right angle

are called the legs.

Pythagorean Theorem

Given a right triangle as shown and A, B, and C

be the length of the sides, then A2 + B2 = C2.

Pythagorean Theorem and Square Roots

hypotenuse

legs

A

B

C

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

?

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

? = h

Let h be the height of the wall.

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

? = h

Let h be the height of the wall.

The wall and the ground form a right triangle,

hence by the Pythagorean Theorem

we have that h2 + 32 = 52

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

? = h

Let h be the height of the wall.

The wall and the ground form a right triangle,

hence by the Pythagorean Theorem

we have that h2 + 32 = 52

h2 + 9 = 25

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

? = h

Let h be the height of the wall.

The wall and the ground form a right triangle,

hence by the Pythagorean Theorem

we have that h2 + 32 = 52

h2 + 9 = 25

–9 –9

subtract 9

from both sides

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

? = h

Let h be the height of the wall.

The wall and the ground form a right triangle,

hence by the Pythagorean Theorem

we have that h2 + 32 = 52

h2 + 9 = 25

–9 –9h2 = 16

subtract 9

from both sides

Pythagorean Theorem

Given a right triangle

with labeling as shown,

then A2 + B2 = C2

Pythagorean Theorem and Square RootsPythagorean Theorem allows us

to compute a length, i.e. a distance,

without measuring it directly.

Example A. A 5–meter ladder leans

against a wall as shown. Its base is

3 meters from the wall. How high is

the wall?

5 m

3 m

? = h

Let h be the height of the wall.

The wall and the ground form a right triangle,

hence by the Pythagorean Theorem

we have that h2 + 32 = 52

h2 + 9 = 25

–9 –9h2 = 16

By trying different numbers for h, we find that 42 = 16

so h = 4 or that the wall is 4–meter high.

subtract 9

from both sides

Pythagorean Theorem and Square RootsSquare Root

Pythagorean Theorem and Square RootsSquare Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”,

Pythagorean Theorem and Square RootsSquare Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square RootsSquare Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square RootsSquare Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Note that both +4 and –4, when squared, give 16.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x.

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x. This is written as sqrt(x) = a, or x = a.

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) =

c.3 =

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 =

d. –3 =

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4

c.3 =

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 =

d. –3 =

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4

c.3 =

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 = 1/3

d. –3 =

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4

c.3 = 1.732.. by calculator

or that 3 ≈ 1.7 (approx.)

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 = 1/3

d. –3 =

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

We also state this relation as “the square–root of 16 is 4”,

i.e. 4 is the source for output “16”, and it’s written as 16 = 4:

Example A.

a. Sqrt(16) = 4

c.3 = 1.732.. by calculator

or that 3 ≈ 1.7 (approx.)

Pythagorean Theorem and Square Roots

Definition: If a2 → x and a is not negative, then a is called the

square root of x. This is written as sqrt(x) = a, or x = a.

b. 1/9 = 1/3

d. –3 = doesn’t exist (why?),

and the calculator returns “Error”.

Note that both +4 and –4, when squared, give 16. But we

designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.

We refer “–4” as the “negative of the square root of 16”.

Square Root

From example A, we encountered that “the square of 4 is16”:

4 16(#)2

16 = 4 16 #

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table.

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table. For example,

25 < 30 < 36

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table. For example,

25 < 30 < 36

hence

25 < 30 <36

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table. For example,

25 < 30 < 36

hence

25 < 30 <36

or 5 < 30 < 6

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table. For example,

25 < 30 < 36

hence

25 < 30 <36

or 5 < 30 < 6

Since 30 is about half way

between 25 and 36,

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table. For example,

25 < 30 < 36

hence

25 < 30 <36

or 5 < 30 < 6

Since 30 is about half way

between 25 and 36,

so we estimate that30 5.5.

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

0 02 = 0 0 = 0

1 12 = 1 1 = 1

2 22 = 4 4 = 2

3 32 = 9 9 = 3

4 42 = 16 16 = 4

5 52 = 25 25 = 5

6 62 = 36 36 = 6

7 72 = 49 49 = 7

8 82 = 64 64 = 8

9 92 = 81 81 = 9

10 102 = 100 100 = 10

11 112 = 121 121 = 11

We may estimate the sqrt

of other small numbers using

this table. For example,

25 < 30 < 36

hence

25 < 30 <36

or 5 < 30 < 6

Since 30 is about half way

between 25 and 36,

so we estimate that30 5.5.

In fact 30 5.47722….

Pythagorean Theorem and Square RootsFollowing are the square numbers and square-roots that one

needs to memorize. These numbers are special because

many mathematics exercises utilize square numbers.

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

we are to find the hypotenuse,

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

we are to find the hypotenuse,

so 122 + 52 = c2

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

Example B.

Find the missing side of the following right triangles.

b. a = 5, c = 12,

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

Example B.

Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

25 + b2 = 144

Example B.

Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

25 + b2 = 144

b2 = 144 – 25 = 119

Example B.

Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

a. We have the legs a = 5, b = 12,

Pythagorean Theorem and Square RootsDepending on which is the missing side, there are two versions

of calculation based on the Pythagorean Theorem –

finding the hypotenuse versus finding a leg.

so 52 + b2 = 122

25 + b2 = 144

b2 = 144 – 25 = 119

Hence b = 119 10.9.

Example B.

Find the missing side of the following right triangles.

b. a = 5, c = 12, we are to find a leg,

we are to find the hypotenuse,

so 122 + 52 = c2

144 + 25 = c2

169 = c2

Hence c = 169 = 13.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be

represented as a ratio of whole numbers i.e.

P/Q, where P and Q are integers.

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be

represented as a ratio of whole numbers i.e.

P/Q, where P and Q are integers.

Hence these numbers are called irrational (non–ratio)

numbers.

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be

represented as a ratio of whole numbers i.e.

P/Q, where P and Q are integers.

Hence these numbers are called irrational (non–ratio)

numbers. Most real numbers are irrational, not fractions, i.e.

they can’t be represented as ratios of two integers.

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be

represented as a ratio of whole numbers i.e.

P/Q, where P and Q are integers.

Hence these numbers are called irrational (non–ratio)

numbers. Most real numbers are irrational, not fractions, i.e.

they can’t be represented as ratios of two integers. The real

line is populated sparsely by fractional locations.

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be

represented as a ratio of whole numbers i.e.

P/Q, where P and Q are integers.

Hence these numbers are called irrational (non–ratio)

numbers. Most real numbers are irrational, not fractions, i.e.

they can’t be represented as ratios of two integers. The real

line is populated sparsely by fractional locations. The

Pythagorean school of the ancient Greeks had believed that

all the measurable quantities in the universe are fractional

quantities. The “discovery” of these extra irrational numbers

caused a profound intellectual crisis.

The number 2 is the length of the

hypotenuse of the right triangle as shown.

Pythagorean Theorem and Square RootsRational and Irrational Numbers

21

1

It can be shown that 2 can not be

represented as a ratio of whole numbers i.e.

P/Q, where P and Q are integers.

Hence these numbers are called irrational (non–ratio)

numbers. Most real numbers are irrational, not fractions, i.e.

they can’t be represented as ratios of two integers. The real

line is populated sparsely by fractional locations. The

Pythagorean school of the ancient Greeks had believed that

all the measurable quantities in the universe are fractional

quantities. The “discovery” of these extra irrational numbers

caused a profound intellectual crisis. It wasn’t until the last two

centuries that mathematicians clarified the strange questions

“How many and what kind of numbers are there?”

Pythagorean Theorem and Square Roots

x

3

4

Exercise C. Solve for x. Give the square–root answer and

approximate answers to the tenth place using a calculator.

1.4

3

x2. x

12

53.

x

1

14.2

1

x5. 6

x

6.10

1. sqrt(0) = 2. 1 =

Exercise A. find the following square–root (no calculator).

3. 25 3. 100

5. sqrt(1/9) = 6. sqrt(1/16) = 7. sqrt(4/49)

Exercise A. Give the approximate answers to the tenth place

using a calculator.

1. sqrt(2) = 2. 3 = 3. 10 3. 0.6