She is referring to my article about the square root algorithm. However, I've never meant that kids would need to learn that long algorithm in school work. In the article, I'm actually advocating the method of finding the approximate square root by "guess and check".I read your suggestion for calculating square root without a calculator. I teach Math for Elementary Teachers and developmental math courses (algebra) to adults. I feel that the focus should be on understanding the number rather than an exercise in following a memorized algorithm. I suggest you have the student determine the pair of perfect squares the number falls between. For example, if finding the sqrt of 645, it falls between the sqrt of 625 which equals 25 and the sqrt of 676 which equals 26. So the sqrt of 645 has to be between 25 and 26. Where does it fall between? There are 50 numbers between 676 and 625. 645 is 20 numbers beyond 625, so 20/50 = 0.4
So the sqrt of 645 is very close to 25.4
This method provides the student with a process that improves their understanding of numbers without expecting them to memorize an algorithm, and it provides an answer to the nearest tenth.
Andrea S. Levy, Ed.D.
Showing posts with label square root. Show all posts
Showing posts with label square root. Show all posts
Thursday, January 8, 2009
A little trick for square roots (mental math)
Someone sent me this little mental math trick for square roots. I liked it, didn't know it before, so here goes:
A little trick for square roots (mental math)
Someone sent me this little mental math trick for square roots. I liked it, didn't know it before, so here goes:
She is referring to my article about the square root algorithm. However, I've never meant that kids would need to learn that long algorithm in school work. In the article, I'm actually advocating the method of finding the approximate square root by "guess and check".I read your suggestion for calculating square root without a calculator. I teach Math for Elementary Teachers and developmental math courses (algebra) to adults. I feel that the focus should be on understanding the number rather than an exercise in following a memorized algorithm. I suggest you have the student determine the pair of perfect squares the number falls between. For example, if finding the sqrt of 645, it falls between the sqrt of 625 which equals 25 and the sqrt of 676 which equals 26. So the sqrt of 645 has to be between 25 and 26. Where does it fall between? There are 50 numbers between 676 and 625. 645 is 20 numbers beyond 625, so 20/50 = 0.4
So the sqrt of 645 is very close to 25.4
This method provides the student with a process that improves their understanding of numbers without expecting them to memorize an algorithm, and it provides an answer to the nearest tenth.
Andrea S. Levy, Ed.D.
Monday, December 18, 2006
Square root of 11
Is square root of 11 an irrational number? How do you know from using a calculator? Thank you.
Well, I happen to know that if the square root of a natural number is NOT a whole number, then it is an irrational number. There are no other possibilities.
Of course you can't tell by the calculator. The calculator will show you 8 or 10 decimals, but you won't know from that if it's going to continue or not, or if it is periodical or not.
But pure mathematics and established, proven theorems will tell you that! : )
A proof that the square root of 2 is irrational
Lots of proofs of the same... plus one proving that any root is irrational if it's not a whole number
Square root of 11
Is square root of 11 an irrational number? How do you know from using a calculator? Thank you.
Well, I happen to know that if the square root of a natural number is NOT a whole number, then it is an irrational number. There are no other possibilities.
Of course you can't tell by the calculator. The calculator will show you 8 or 10 decimals, but you won't know from that if it's going to continue or not, or if it is periodical or not.
But pure mathematics and established, proven theorems will tell you that! : )
A proof that the square root of 2 is irrational
Lots of proofs of the same... plus one proving that any root is irrational if it's not a whole number
Wednesday, November 29, 2006
Square root problem
prove that
5√20 × √45 × √5 = 150√5
This is a quite easy problem. You use two basic "ideas" or properties relating to square roots:
* that √b × b is b - or that you can "pull out" a number times itself from under the root (this is just the definition of a square root of course).
* that √a √b = √ab - or you can combine the radicands under the same root when they are multiplied.
So √20 * √45 is equal to √20*45 = √4 × 5 × 5 × 9
= √2 × 2 × 5 × 5 × 3 × 3 = 2 × 5 × 3 = 30.
So that's the crux of that problem.
Square root problem
prove that
5√20 × √45 × √5 = 150√5
This is a quite easy problem. You use two basic "ideas" or properties relating to square roots:
* that √b × b is b - or that you can "pull out" a number times itself from under the root (this is just the definition of a square root of course).
* that √a √b = √ab - or you can combine the radicands under the same root when they are multiplied.
So √20 * √45 is equal to √20*45 = √4 × 5 × 5 × 9
= √2 × 2 × 5 × 5 × 3 × 3 = 2 × 5 × 3 = 30.
So that's the crux of that problem.
Wednesday, January 11, 2006
Square root of 2 and Pythagoreans' shock
Last week I asked you two questions pertaining to the image below:
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
Square root of 2 and Pythagoreans' shock
Last week I asked you two questions pertaining to the image below:
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
1) How does this connect with irrational numbers?
Well, the two sides and the diagonal form a right triangle. You can use Pythagorean theorem to find the length of the diagonal. My picture doesn't have any lengths but I was thinking about having the side to be 1 (that's the simplest way).
If both sides are 1 and diagonal is d, then Pythagorean theorem says:
d2 = 12 + 12
Solving that, you get d = √2
And, √2 is an irrational number.
So if the sides of the square are 1, then the diagonal is an irrational number (square root of 2).
2) How does this connect with history of mathematics?
Pythagoras was a philosopher in the 6th century B.C. who founded a philosophical school or 'cult' in southern Italy. They had some mystical beliefs, such as reality is mathematical in nature, certain symbols have mystical significance, that the whole cosmos is a scale and a number. Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. They believed the whole universe was ruled by whole numbers.
These Pythagoreans apparently proved the Pythagorean theorem at some point (though they weren't the first ones to find it), plus various other mathematical results.
Then, they also found that some numbers are irrational... And that was a "biggie".
See, before that time, everyone had believed that ALL numbers are rational (all numbers are whole numbers or fractions). But Pythagoreans were able to prove that the diagonal of square and its side are "incommensurable" - that you can't find a fraction so that the diagonal would be this fraction times the side. This discovery is usually attributed to Hippas.
The LEGEND says that Pythagoreans were at sea when Hippatus discovered this, and when he told his comrades about his finding, they became so upset that they threw him in the sea!
You see, to Pythagoreans, the whole universe was ruled by whole numbers - so finding a number that was not a ratio of two whole numbers was a disastrous blow to their philosophical beliefs. Went against the dogma, so to speak.
See more:
P.S. Remember what I said: history can help get kids interested in math. It provides further insight into the topic and can help us remember better, and even understand better.
Categories: geometry, history
Saturday, January 7, 2006
Square and its diagonal
Two questions for you to think about:
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
Square and its diagonal
Two questions for you to think about:
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
1) How does the above image of a square with one diagonal relate to irrational numbers?
2) How does this image connect with history of mathematics?
Answers will be here next week... : )
Wednesday, December 7, 2005
The Golden Section
Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?
The discussion below covers the basics of golden section. I'll try to keep it short.
Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.
Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).
This is how we get this golden section or golden cut:
Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.
From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).
And the golden ratio is the ratio Phi:1.
Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.
And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.
My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.
The discussion below covers the basics of golden section. I'll try to keep it short.
Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.
Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).
This is how we get this golden section or golden cut:
Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.
S:L = L:W
From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).
And the golden ratio is the ratio Phi:1.
Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2 |
Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
here is one golden rectangle |
Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.
And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.
My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
- The Golden section in architecture
- Dividing the line in golden section using compass and ruler
- Phi in pentagons and pentagrams
- Where Fibonacci numbers are found in nature
- Where golden section is found in nature
P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.
The Golden Section
Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?
The discussion below covers the basics of golden section. I'll try to keep it short.
Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.
Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).
This is how we get this golden section or golden cut:
Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.
From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).
And the golden ratio is the ratio Phi:1.
Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.
And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.
My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.
The discussion below covers the basics of golden section. I'll try to keep it short.
Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.
Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).
This is how we get this golden section or golden cut:
Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.
S:L = L:W
From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).
And the golden ratio is the ratio Phi:1.
Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2 |
Golden rectangle
Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.
here is one golden rectangle |
Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.
And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.
My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.
- The Golden section in architecture
- Dividing the line in golden section using compass and ruler
- Phi in pentagons and pentagrams
- Where Fibonacci numbers are found in nature
- Where golden section is found in nature
P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.
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