10 pythagorean theorem, square roots and irrational numbers
TRANSCRIPT
Pythagorean Theorem and Irrational Numbers
Fractions measure quantities that are fragments of whole ones expressed as ratio of whole numbers.
Pythagorean Theorem and Irrational Numbers
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
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.
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.
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 beexpressed exactly using a fraction isthe length of the diagonal of a 1 x 1 square.
1
1 ?
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 beexpressed exactly using a fraction isthe 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.25
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 beexpressed exactly using a fraction isthe 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.25
We know this length is not a fraction because if it were it would violate a known fact–the Pythagorean Theorem.
A right triangle is a triangle with a right angle as shown.
A
BC
Pythagorean Theorem and Irrational Numbers
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
BC
Pythagorean Theorem and Irrational Numbers
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
BC
Pythagorean Theorem and Irrational Numbers
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 Theorema. Given a right triangle with A, B and C as the lengths of its sides, then A2 + B2 = C2.
hypotenuse
legs
A
BC
Pythagorean Theorem and Irrational Numbers
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 Theorema. Given a right triangle with A, B and C as the lengths of its sides, then A2 + B2 = C2.
hypotenuse
legs
A
BC
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
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 Theorema. Given a right triangle with A, B and C as the lengths of its sides, then A2 + B2 = C2.
hypotenuse
legs
A
BC
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
43
2
4
i. ii.
Pythagorean Theorem and Irrational Numbers
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 Theorema. Given a right triangle with A, B and C as the lengths of its sides, then A2 + B2 = C2.
hypotenuse
legs
A
BC
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
43
2
4
i. ii.
22 + 32 = 13
Pythagorean Theorem and Irrational Numbers
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 Theorema. Given a right triangle with A, B and C as the lengths of its sides, then A2 + B2 = C2.
hypotenuse
legs
A
BC
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
43
2
4
i. ii.
22 + 32 = 13 = 42 = 16so this is not a right triangle.
Pythagorean Theorem and Irrational Numbers
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 Theorema. Given a right triangle with A, B and C as the lengths of its sides, then A2 + B2 = C2.
hypotenuse
legs
A
BC
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
43
2
4
i. ii.
22 + 32 = 13 = 42 = 16so this is not a right triangle.
32 + 42 = 25 = 52
so this is a right triangle.
Pythagorean Theorem and Irrational Numbers
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
1c
c2 = 2
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
1c
1c2 = 2
This number c can’t be represented as a fraction, i.e. in the form of a ratio as p/q.
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
1c
1c2 = 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.
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
1c
1c2 = 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,
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
1c
1c2 = 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 100001 4 +…2+++= 1 +
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
1c
1c2 = 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.
410 100 1000 10000
1 4 +…2+++= 1 +
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.
Square Roots
Pythagorean Theorem and Irrational NumbersFrom here we assume the knowledge of signed (±) numbers.
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 NumbersFrom here we assume the knowledge of signed (±) numbers.
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.
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.
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.
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) = 4c. 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.
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) = 4c. 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.
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) = 4c. 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.
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) = 4c. 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.
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) = 4c. 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.
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
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
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
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 < 36hence 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
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 < 36hence 25 < 30 <36or 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
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 < 36hence 25 < 30 <36or 5 < 30 < 6Since 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
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 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 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
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 < 36hence 25 < 30 <36or 5 < 30 < 6Since 30 is about half way between 25 and 36, so we estimate that30 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
There are two types of problems that use the Pythagorean Theorem to find the sides of the right triangles
Pythagorean Theorem and Irrational Numbers
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
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
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
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
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
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
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
b. a = 5, c = 12, b = ?
Pythagorean Theorem and Irrational Numbers
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
Pythagorean Theorem and Irrational Numbers
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144
Pythagorean Theorem and Irrational Numbers
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119
Pythagorean Theorem and Irrational Numbers
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So b = 119 10.9
Pythagorean Theorem and Irrational Numbers
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So b = 119 10.9
Facts About Square Roots
Pythagorean Theorem and Irrational Numbers
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So 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
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So 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
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So 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
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So 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
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So 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
b. a = 5, c = 12, b = ?Since it is a right triangle,b2 + 52 = 122
b2 + 25 = 144b2 = 144 – 25 = 119So 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