How to generalize upper and lower bounds rules
I want to create a rule on finding the upper and lower bound of a number.
Examples:
For $29.8$ (3 s.f.) we write $1$ in the place of the $8$ and zero in every other position, giving us $0.01$. After that we divide $frac{0.01}{2} = 0.005$ and use that number to add and subtract $29.8$ by, giving us $29.75$ and $29.85$ as the lower and upper bounds, respectively.
For $0.661$ (3 s.f.) we leave the $1$ in its place and zero in every other position, giving us $0.001$. After that we divide $frac{0.001}{2} = 0.0005$ and use that number to add and subtract $0.661$ by, giving us $0.6615$ and $0.6605$ as the lower and upper bounds, respectively.
My question is why does this method fail on the following question: Find the upper and lower bounds of $1000$ (one s.f.)?
The method would give us $1500$ and $500$ whereas the correct answer is $1500$ and $950$ respectively.
algebra-precalculus rounding-error
edited Nov 29 '18 at 12:18
Henry
98.3k475162
asked Nov 29 '18 at 11:24
bru1987
1,0671022
add a comment |
I want to create a rule on finding the upper and lower bound of a number.
Examples:
For $29.8$ (3 s.f.) we write $1$ in the place of the $8$ and zero in every other position, giving us $0.01$. After that we divide $frac{0.01}{2} = 0.005$ and use that number to add and subtract $29.8$ by, giving us $29.75$ and $29.85$ as the lower and upper bounds, respectively.
For $0.661$ (3 s.f.) we leave the $1$ in its place and zero in every other position, giving us $0.001$. After that we divide $frac{0.001}{2} = 0.0005$ and use that number to add and subtract $0.661$ by, giving us $0.6615$ and $0.6605$ as the lower and upper bounds, respectively.
My question is why does this method fail on the following question: Find the upper and lower bounds of $1000$ (one s.f.)?
The method would give us $1500$ and $500$ whereas the correct answer is $1500$ and $950$ respectively.
algebra-precalculus rounding-error
edited Nov 29 '18 at 12:18
Henry
98.3k475162
asked Nov 29 '18 at 11:24
bru1987
1,0671022
add a comment |
I want to create a rule on finding the upper and lower bound of a number.
Examples:
For $29.8$ (3 s.f.) we write $1$ in the place of the $8$ and zero in every other position, giving us $0.01$. After that we divide $frac{0.01}{2} = 0.005$ and use that number to add and subtract $29.8$ by, giving us $29.75$ and $29.85$ as the lower and upper bounds, respectively.
For $0.661$ (3 s.f.) we leave the $1$ in its place and zero in every other position, giving us $0.001$. After that we divide $frac{0.001}{2} = 0.0005$ and use that number to add and subtract $0.661$ by, giving us $0.6615$ and $0.6605$ as the lower and upper bounds, respectively.
My question is why does this method fail on the following question: Find the upper and lower bounds of $1000$ (one s.f.)?
The method would give us $1500$ and $500$ whereas the correct answer is $1500$ and $950$ respectively.
algebra-precalculus rounding-error
edited Nov 29 '18 at 12:18
Henry
98.3k475162
asked Nov 29 '18 at 11:24
bru1987
1,0671022
I want to create a rule on finding the upper and lower bound of a number.
Examples:
For $29.8$ (3 s.f.) we write $1$ in the place of the $8$ and zero in every other position, giving us $0.01$. After that we divide $frac{0.01}{2} = 0.005$ and use that number to add and subtract $29.8$ by, giving us $29.75$ and $29.85$ as the lower and upper bounds, respectively.
For $0.661$ (3 s.f.) we leave the $1$ in its place and zero in every other position, giving us $0.001$. After that we divide $frac{0.001}{2} = 0.0005$ and use that number to add and subtract $0.661$ by, giving us $0.6615$ and $0.6605$ as the lower and upper bounds, respectively.
My question is why does this method fail on the following question: Find the upper and lower bounds of $1000$ (one s.f.)?
The method would give us $1500$ and $500$ whereas the correct answer is $1500$ and $950$ respectively.
algebra-precalculus rounding-error
algebra-precalculus rounding-error
edited Nov 29 '18 at 12:18
Henry
98.3k475162
asked Nov 29 '18 at 11:24
bru1987
1,0671022
edited Nov 29 '18 at 12:18
Henry
98.3k475162
asked Nov 29 '18 at 11:24
bru1987
1,0671022
edited Nov 29 '18 at 12:18
Henry
98.3k475162
edited Nov 29 '18 at 12:18
Henry
98.3k475162
edited Nov 29 '18 at 12:18
Henry
98.3k475162
98.3k475162
asked Nov 29 '18 at 11:24
bru1987
1,0671022
asked Nov 29 '18 at 11:24
bru1987
1,0671022
asked Nov 29 '18 at 11:24
bru1987
1,0671022
1,0671022
add a comment |
add a comment |
1 Answer
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It fails because your rounded number is a power of $10$
You would for example have a similar issue with $0.0100$ as three significant figures: the bounds would be $0.01005$ and $0.009995$ rather than $0.00995$ (spot the extra $9$)
So your rule needs to check you are starting from a power of $10$. If so, the amount to subtract for the lower bound is a tenth of the amount to add for the upper bound
answered Nov 29 '18 at 12:18
Henry
98.3k475162
add a comment |
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It fails because your rounded number is a power of $10$
You would for example have a similar issue with $0.0100$ as three significant figures: the bounds would be $0.01005$ and $0.009995$ rather than $0.00995$ (spot the extra $9$)
So your rule needs to check you are starting from a power of $10$. If so, the amount to subtract for the lower bound is a tenth of the amount to add for the upper bound
answered Nov 29 '18 at 12:18
Henry
98.3k475162
add a comment |
It fails because your rounded number is a power of $10$
You would for example have a similar issue with $0.0100$ as three significant figures: the bounds would be $0.01005$ and $0.009995$ rather than $0.00995$ (spot the extra $9$)
So your rule needs to check you are starting from a power of $10$. If so, the amount to subtract for the lower bound is a tenth of the amount to add for the upper bound
answered Nov 29 '18 at 12:18
Henry
98.3k475162
add a comment |
It fails because your rounded number is a power of $10$
You would for example have a similar issue with $0.0100$ as three significant figures: the bounds would be $0.01005$ and $0.009995$ rather than $0.00995$ (spot the extra $9$)
So your rule needs to check you are starting from a power of $10$. If so, the amount to subtract for the lower bound is a tenth of the amount to add for the upper bound
answered Nov 29 '18 at 12:18
Henry
98.3k475162
It fails because your rounded number is a power of $10$
You would for example have a similar issue with $0.0100$ as three significant figures: the bounds would be $0.01005$ and $0.009995$ rather than $0.00995$ (spot the extra $9$)
So your rule needs to check you are starting from a power of $10$. If so, the amount to subtract for the lower bound is a tenth of the amount to add for the upper bound
answered Nov 29 '18 at 12:18
Henry
98.3k475162
answered Nov 29 '18 at 12:18
Henry
98.3k475162
answered Nov 29 '18 at 12:18
Henry
98.3k475162
answered Nov 29 '18 at 12:18
Henry
98.3k475162
98.3k475162
add a comment |
add a comment |
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