Pythagorean theorem, square roots and irrational numbers

1. Pythagorean Theorem and Irrational Numbers

2. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers.
Pythagorean Theorem and Irrational Numbers

3. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers. Decimals and
percentage are fractions (in different formats) hence all these
numbers are referred to as rational numbers.
Pythagorean Theorem and Irrational Numbers

4. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers. Decimals and
percentage are fractions (in different formats) hence all these
numbers are referred to as rational numbers.
Pythagorean Theorem and Irrational Numbers
However, rational numbers are not all the numbers there are.

5. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers. Decimals and
percentage are fractions (in different formats) hence all these
numbers are referred to as rational numbers.
Pythagorean Theorem and Irrational Numbers
However, rational numbers are not all the numbers there are.
There are quantities that can’t be recorded precisely using
rational numbers.

6. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers. Decimals and
percentage are fractions (in different formats) hence all these
numbers are referred to as rational numbers.
Pythagorean Theorem and Irrational Numbers
However, rational numbers are not all the numbers there are.
There are quantities that can’t be recorded precisely using
rational numbers. The simplest example
of a measurement that can not be
expressed exactly using a fraction is
the length of the diagonal of
a 1 x 1 square.
1
1
?

7. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers. Decimals and
percentage are fractions (in different formats) hence all these
numbers are referred to as rational numbers.
Pythagorean Theorem and Irrational Numbers
However, rational numbers are not all the numbers there are.
There are quantities that can’t be recorded precisely using
rational numbers. The simplest example
of a measurement that can not be
expressed exactly using a fraction is
the length of the diagonal of
a 1 x 1 square.
1
1
?
The length of this diagonal is
approximately 1 or 1.4 but its precise value is not a fraction.2
5

8. Fractions measure quantities that are fragments of whole ones
expressed as ratio of whole numbers. Decimals and
percentage are fractions (in different formats) hence all these
numbers are referred to as rational numbers.
Pythagorean Theorem and Irrational Numbers
However, rational numbers are not all the numbers there are.
There are quantities that can’t be recorded precisely using
rational numbers. The simplest example
of a measurement that can not be
expressed exactly using a fraction is
the length of the diagonal of
a 1 x 1 square.
1
1
?
The length of this diagonal is
approximately 1 or 1.4 but its precise value is not a fraction.2
5
We know this length is not a fraction because if it were
it would violate a known fact–the Pythagorean Theorem.

9. A right triangle is a triangle with a right angle as shown.
A
B
C
Pythagorean Theorem and Irrational Numbers

10. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
hypotenuse
A
B
C
Pythagorean Theorem and Irrational Numbers

11. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
hypotenuse
legs
A
B
C
Pythagorean Theorem and Irrational Numbers

12. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
Pythagorean Theorem
a. Given a right triangle with A, B and C
as the lengths of its sides,
then A2 + B2 = C2.
hypotenuse
legs
A
B
C
Pythagorean Theorem and Irrational Numbers

13. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
Pythagorean Theorem
a. Given a right triangle with A, B and C
as the lengths of its sides,
then A2 + B2 = C2.
hypotenuse
legs
A
B
C
b. If the lengths of the sides of a triangle satisfies
A2 + B2 = C2 then the triangle is a right triangle.
Pythagorean Theorem and Irrational Numbers

14. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
Pythagorean Theorem
a. Given a right triangle with A, B and C
as the lengths of its sides,
then A2 + B2 = C2.
hypotenuse
legs
A
B
C
b. If the lengths of the sides of a triangle satisfies
A2 + B2 = C2 then the triangle is a right triangle.
Example A.
Which of the
following is a
right triangle
and which is not?
3
5
4
3
2
4
i. ii.
Pythagorean Theorem and Irrational Numbers

15. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
Pythagorean Theorem
a. Given a right triangle with A, B and C
as the lengths of its sides,
then A2 + B2 = C2.
hypotenuse
legs
A
B
C
b. If the lengths of the sides of a triangle satisfies
A2 + B2 = C2 then the triangle is a right triangle.
Example A.
Which of the
following is a
right triangle
and which is not?
3
5
4
3
2
4
i. ii.
22 + 32 = 13
Pythagorean Theorem and Irrational Numbers

16. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
Pythagorean Theorem
a. Given a right triangle with A, B and C
as the lengths of its sides,
then A2 + B2 = C2.
hypotenuse
legs
A
B
C
b. If the lengths of the sides of a triangle satisfies
A2 + B2 = C2 then the triangle is a right triangle.
Example A.
Which of the
following is a
right triangle
and which is not?
3
5
4
3
2
4
i. ii.
22 + 32 = 13 = 42 = 16
so this is
not a right triangle.
Pythagorean Theorem and Irrational Numbers

17. A right triangle is a triangle with a right angle as shown.
The longest side C of a right triangle is called the hypotenuse,
the two shorter sides A and B forming the right angle are
called the legs.
Pythagorean Theorem
a. Given a right triangle with A, B and C
as the lengths of its sides,
then A2 + B2 = C2.
hypotenuse
legs
A
B
C
b. If the lengths of the sides of a triangle satisfies
A2 + B2 = C2 then the triangle is a right triangle.
Example A.
Which of the
following is a
right triangle
and which is not?
3
5
4
3
2
4
i. ii.
22 + 32 = 13 = 42 = 16
so this is
not a right triangle.
32 + 42 = 25 = 52
so this is
a right triangle.
Pythagorean Theorem and Irrational Numbers

18. From the Pythagorean Theorem,
the length c of the diagonal of the 1 x 1 square
must satisfy the relation c2 = 12 + 12 or that
Pythagorean Theorem and Irrational Numbers
1
c
c2 = 2

19. From the Pythagorean Theorem,
the length c of the diagonal of the 1 x 1 square
must satisfy the relation c2 = 12 + 12 or that
Pythagorean Theorem and Irrational Numbers
1
c
1
c2 = 2
This number c can’t be represented as a fraction,
i.e. in the form of a ratio as p/q.

20. From the Pythagorean Theorem,
the length c of the diagonal of the 1 x 1 square
must satisfy the relation c2 = 12 + 12 or that
Pythagorean Theorem and Irrational Numbers
1
c
1
c2 = 2
This number c can’t be represented as a fraction,
i.e. in the form of a ratio as p/q.
We call this number the square root of 2 and denoted it as √2,
and we say that √2 is irrational, i.e. it's not a ratio–number.

21. From the Pythagorean Theorem,
the length c of the diagonal of the 1 x 1 square
must satisfy the relation c2 = 12 + 12 or that
Pythagorean Theorem and Irrational Numbers
1
c
1
c2 = 2
This number c can’t be represented as a fraction,
i.e. in the form of a ratio as p/q.
We call this number the square root of 2 and denoted it as √2,
and we say that √2 is irrational, i.e. it's not a ratio–number.
Using a calculator, we see that
√2 ≈ 1.4142135… as an infinite decimal expansion,

22. From the Pythagorean Theorem,
the length c of the diagonal of the 1 x 1 square
must satisfy the relation c2 = 12 + 12 or that
Pythagorean Theorem and Irrational Numbers
1
c
1
c2 = 2
This number c can’t be represented as a fraction,
i.e. in the form of a ratio as p/q.
We call this number the square root of 2 and denoted it as √2,
and we say that √2 is irrational, i.e. it's not a ratio–number.
Using a calculator, we see that
as a fractional expansion.
√2 ≈ 1.4142135… as an infinite decimal expansion,
4
10 100 1000 10000
1 4
+…
2
+++= 1 +

23. From the Pythagorean Theorem,
the length c of the diagonal of the 1 x 1 square
must satisfy the relation c2 = 12 + 12 or that
Pythagorean Theorem and Irrational Numbers
1
c
1
c2 = 2
This number c can’t be represented as a fraction,
i.e. in the form of a ratio as p/q.
We call this number the square root of 2 and denoted it as √2,
and we say that √2 is irrational, i.e. it's not a ratio–number.
Using a calculator, we see that
as a fractional expansion.
√2 ≈ 1.4142135… as an infinite decimal expansion,
It turns out that most numbers are irrational numbers
requiring infinitely long expansions to be presented precisely.
4
10 100 1000 10000
1 4
+…
2
+++= 1 +

24. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
From here we assume the knowledge of signed (±) numbers.

25. Square Roots
Pythagorean Theorem and Irrational Numbers
From here we assume the knowledge of signed (±) numbers.

26. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Square Roots
Pythagorean Theorem and Irrational Numbers
From here we assume the knowledge of signed (±) numbers.

27. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Definition:
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
From here we assume the knowledge of signed (±) numbers.

28. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
From here we assume the knowledge of signed (±) numbers.

29. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Example B.
a. Sqrt(16) =
c. 3 =
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
b. 1/9 =
d.  =
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
3
From here we assume the knowledge of signed (±) numbers.

30. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Example B.
a. Sqrt(16) = 4
c. 3 =
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
b. 1/9 =
d.  =
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
3
From here we assume the knowledge of signed (±) numbers.

31. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Example B.
a. Sqrt(16) = 4
c. 3 =
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
b. 1/9 = 1/3
d.  =
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
3
From here we assume the knowledge of signed (±) numbers.

32. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Example B.
a. Sqrt(16) = 4
c. 3 = 1.732.. (calculator)
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
b. 1/9 = 1/3
d.  = 1.732..
= 1.316074 (calculator)
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
3
From here we assume the knowledge of signed (±) numbers.

33. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Example B.
a. Sqrt(16) = 4
c. 3 = 1.732.. (calculator)
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
b. 1/9 = 1/3
d.  = 1.732..
= 1.316074 (calculator)
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
3
L
L = ?
Example C. The area of a L x L square is L2.
If a square rug covers an area of 20 ft2,
what’s the length of its side? 20 ft2
From here we assume the knowledge of signed (±) numbers.

34. In the identity 32 = 9, we say that “9 is the square of 3”,
and that “3 is the square root of 9” and we write 3 = 9.
Example B.
a. Sqrt(16) = 4
c. 3 = 1.732.. (calculator)
Definition: If a is > 0 and a2 = x, then we say that
“a is the square root of x” and we write a = sqrt(x), or a =x.
b. 1/9 = 1/3
d.  = 1.732..
= 1.316074 (calculator)
Square Roots
Pythagorean Theorem and Irrational Numbers
We define the square root of non–negative numbers here.
3
L
L = ?
Example C. The area of a L x L square is L2.
If a square rug covers an area of 20 ft2,
what’s the length of its side?
We have that L2 = 20
so L = 20 ≈ 4.4721.. or a little less than 4½ ft.
20 ft2
From here we assume the knowledge of signed (±) numbers.

35. 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
Following 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 Irrational Numbers

36. 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.
Following 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 Irrational Numbers

37. 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
Following 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 Irrational Numbers

38. 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
Following 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 Irrational Numbers

39. 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
Following 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 Irrational Numbers

40. 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,
Following 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 Irrational Numbers

41. 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 that30  5.5.
Following 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 Irrational Numbers

42. 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 that30  5.5.
In fact 30  5.47722….
Following 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 Irrational Numbers

43. There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles
Pythagorean Theorem and Irrational Numbers

44. There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

45. Example D. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

46. Example D. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

47. Example D. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

48. Example D. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

49. Example D. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

50. Example D. Find the missing side of the following right
triangles. Find the exact answer and the approximate answer.
Draw.
a. a = 5, b = 12, c = ?
Since it is a right triangle,
122 + 52 = c2
144 + 25 = c2
169 = c2
So c = 169 = 13.
There are two types of problems that use the Pythagorean
Theorem to find the sides of the right triangles-finding the
hypotenuse versus finding the legs.
Pythagorean Theorem and Irrational Numbers

51. b. a = 5, c = 12, b = ?
Pythagorean Theorem and Irrational Numbers

52. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
Pythagorean Theorem and Irrational Numbers

53. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
Pythagorean Theorem and Irrational Numbers

54. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
Pythagorean Theorem and Irrational Numbers

55. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Pythagorean Theorem and Irrational Numbers

56. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
Pythagorean Theorem and Irrational Numbers

57. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
I. Let x be a non–negative number, then x*x = x,
e.g. 2*2 = 2.
Pythagorean Theorem and Irrational Numbers

58. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
I. Let x be a non–negative number, then x*x = x,
e.g. 2*2 = 2.
II. Let x and y be two non–negative number, then x*y = xy.
e.g. 2*3 = 6.
Pythagorean Theorem and Irrational Numbers

59. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
I. Let x be a non–negative number, then x*x = x,
e.g. 2*2 = 2.
Example E.
II. Let x and y be two non–negative number, then x*y = xy.
e.g. 2*3 = 6.
3*3 * 2*8
Pythagorean Theorem and Irrational Numbers

60. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
I. Let x be a non–negative number, then x*x = x,
e.g. 2*2 = 2.
Example E.
II. Let x and y be two non–negative number, then x*y = xy.
e.g. 2*3 = 6.
3*3 * 2*8
= 3 * 16
Pythagorean Theorem and Irrational Numbers

61. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
I. Let x be a non–negative number, then x*x = x,
e.g. 2*2 = 2.
Example E.
II. Let x and y be two non–negative number, then x*y = xy.
e.g. 2*3 = 6.
3*3 * 2*8
= 3 * 16 = 3*4 = 12.
Pythagorean Theorem and Irrational Numbers

62. b. a = 5, c = 12, b = ?
Since it is a right triangle,
b2 + 52 = 122
b2 + 25 = 144
b2 = 144 – 25 = 119
So b = 119  10.9
Facts About Square Roots
I. Let x be a non–negative number, then x*x = x,
e.g. 2*2 = 2.
Example E.
II. Let x and y be two non–negative number, then x*y = xy.
e.g. 2*3 = 6.
3*3 * 2*8
= 3 * 16 = 3*4 = 12.
Note: x + y = x + y. e.g. 2 + 2 = 4 = 2.
Pythagorean Theorem and Irrational Numbers