Podcast
Questions and Answers
What determines whether a small quantity can be omitted from consideration in calculations?
What determines whether a small quantity can be omitted from consideration in calculations?
- Its relative minuteness compared to other quantities. (correct)
- Its absolute size.
- The units it is measured in.
- Whether it involves time or money.
A quantity that is considered small in comparison to one value will always be considered small, regardless of the reference value.
A quantity that is considered small in comparison to one value will always be considered small, regardless of the reference value.
False (B)
What term did people in Queen Elizabeth's days use for the subdivisions of a minute, and what does it literally mean?
What term did people in Queen Elizabeth's days use for the subdivisions of a minute, and what does it literally mean?
Second minutes, small quantities of the second order of minuteness.
If 1/100 is considered a small fraction, then 1/10,000 would be considered a small fraction of the ______ order of smallness.
If 1/100 is considered a small fraction, then 1/10,000 would be considered a small fraction of the ______ order of smallness.
Match the following fractions with their order of smallness, assuming 1/100 is a 'small fraction'.
Match the following fractions with their order of smallness, assuming 1/100 is a 'small fraction'.
What concept is often confused with the second order of smallness by mathematicians?
What concept is often confused with the second order of smallness by mathematicians?
According to the examples given, a minute is a relatively small quantity of time when compared to a week.
According to the examples given, a minute is a relatively small quantity of time when compared to a week.
Explain the relationship between a 'minute,' a 'second minute,' and a 'second' in the context of historical time measurement.
Explain the relationship between a 'minute,' a 'second minute,' and a 'second' in the context of historical time measurement.
In the context of calculus, what is the primary reason for neglecting small quantities of the second order or higher?
In the context of calculus, what is the primary reason for neglecting small quantities of the second order or higher?
If dx represents a small quantity, then x * dx is always negligible.
If dx represents a small quantity, then x * dx is always negligible.
Explain, in your own words, the difference between a small quantity of the first order and a small quantity of the second order.
Explain, in your own words, the difference between a small quantity of the first order and a small quantity of the second order.
In calculus, a small bit of x is represented by the term ______.
In calculus, a small bit of x is represented by the term ______.
Match the mathematical expressions with their corresponding order of smallness, given that dx is a small quantity:
Match the mathematical expressions with their corresponding order of smallness, given that dx is a small quantity:
Why might a very small quantity still be important in a calculation?
Why might a very small quantity still be important in a calculation?
If we consider 1/1,000,000 as 'small', then 1/1,000,000,000,000 is a small quantity of the second order and can typically be disregarded in comparison.
If we consider 1/1,000,000 as 'small', then 1/1,000,000,000,000 is a small quantity of the second order and can typically be disregarded in comparison.
Explain the geometric illustration provided using a square (Figure 1) to demonstrate the concept of small quantities and their orders.
Explain the geometric illustration provided using a square (Figure 1) to demonstrate the concept of small quantities and their orders.
The text uses the analogy of a millionaire, his secretary, and the secretary's boy to illustrate the concept of ______ of smallness.
The text uses the analogy of a millionaire, his secretary, and the secretary's boy to illustrate the concept of ______ of smallness.
What analogy does the text use from Dean Swift to illustrate the concept of small quantities?
What analogy does the text use from Dean Swift to illustrate the concept of small quantities?
Flashcards
Small quantities
Small quantities
Quantities considered negligible when compared to larger ones.
First order smallness
First order smallness
The smallest level of quantity that can be significant in calculations.
Second order smallness
Second order smallness
A small quantity that is negligible compared to first order smallness.
Differentials
Differentials
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Negligible quantities
Negligible quantities
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x⋅dx
x⋅dx
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(dx)²
(dx)²
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First-order vs Second-order terms
First-order vs Second-order terms
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Growth of x
Growth of x
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Proportional fractions
Proportional fractions
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Minute
Minute
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Second
Second
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Relative minuteness
Relative minuteness
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Small fraction
Small fraction
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Second order of smallness
Second order of smallness
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Farthing
Farthing
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Sovereign
Sovereign
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Third order of smallness
Third order of smallness
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Study Notes
Small Quantities in Calculations
- Small quantities, varying in degree, are common in calculations.
- Determining when to disregard a small quantity depends on its relative size (minuteness).
- Time units illustrate relative minuteness:
- A minute is small compared to a week.
- A second (second order of smallness) is even smaller compared to a day or a week.
- The concept of "minute" and "second" originates from their fractions of larger units.
Orders of Smallness
- Relative size is key: a farthing is small relative to a sovereign, but negligible relative to a large sum like £1000.
- Defining a "small" fraction allows establishing smaller orders:
- If 1/60 is small, then (1/60) * (1/60) (1/3600) is a second-order smallness.
Importance in Precise Measurements
- Extremely precise measurements (e.g., a chronometer error of 0.5 minutes per year) require considering extremely small quantities.
- One millionth (1/1,000,000) is small for these purposes; and further orders (one billionths) are extremely negligible in comparison.
- The smaller the first-order small quantity, the more negligible the second-order quantity becomes.
- Small order quantities can still become important if multiplied by a large factor.
Differentials and Higher-Order Smallness
- In calculus,
dx,du,dyrepresent small changes. - Small quantities, such as x ⋅ dx or x 2 ⋅ dx, might not be considered negligible if not multiplied by another small fraction.
- (dx)2 (or a quantity raised to a higher power) is always considered a smaller/negligible quantity in the context of a small change.
Visual Representation (Geometric Example)
- Imagine enlarging a square by
dxon each side. - The added areas (
2 * x * dx) are first-order small quantities. - The tiny corner square (
(dx)2) is a second-order small quantity and becomes negligible in comparison with the first order ifdxis itself a sufficiently small value.
Analogy and Further Explanation
- Analogy of a millionaire, his secretary, and a boy receiving portions.
- Smaller portions of decreasing significance/order demonstrate that second order (or smaller) quantities are very negligible in proportion to first order quantities.
- Emphasizes that even something small (
dx) could be further divided/made smaller and smaller and become insignificant.
Example from Nature
- Illustrates how successive orders of smallness exist in nature.
- An ox cares little about a flea, but even less about a flea's flea (the second-order smallness).
- This principle of successive smallness holds true across various fields.
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Description
This text explains how to deal with small quantities in calculations, emphasizing the importance of relative size. It also explains different orders of smallness, using time units as examples. It concludes by highlighting the significance of considering extremely small quantities in precise measurements.
