Misplaced Pages

#48951

127-457: The line may be physical – as set by a road surveyor , pictorial as in a diagram of a road or roof, or abstract . An application of the mathematical concept is found in the grade or gradient in geography and civil engineering . The steepness , incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows: Special directions are: If two points of

254-423: A 0 {\displaystyle q={\tfrac {a}{0}}} is nonsensical, as the product q ⋅ 0 {\displaystyle q\cdot 0} is always 0 {\displaystyle 0} rather than some other number a . {\displaystyle a.} Following the ordinary rules of elementary algebra while allowing division by zero can create a mathematical fallacy ,

381-509: A lettuce wrap ). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant. The quotitive concept of division lends itself to calculation by repeated subtraction : dividing entails counting how many times the divisor can be subtracted before the dividend runs out. Because no finite number of subtractions of zero will ever exhaust a non-zero dividend, calculating division by zero in this way never terminates . Such an interminable division-by-zero algorithm

508-538: A one-point compactification , making the extended complex numbers topologically equivalent to a sphere . This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse stereographic projection , with the resulting spherical distance applied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and conversely complex arithmetic can be interpreted in terms of spherical geometry. As

635-495: A plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunter's chain was introduced in 1620 by English mathematician Edmund Gunter . It enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a theodolite that measured horizontal angles in his book A geometric practice named Pantometria (1571). Joshua Habermel ( Erasmus Habermehl ) created

762-464: A "value" of this distribution at x  = 0; a sophisticated answer refers to the singular support of the distribution. In matrix algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can be added and multiplied , and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by its inverse . Not all matrices have inverses. For example,

889-445: A GPS on large scale surveys makes them popular for major infrastructure or data gathering projects. One-person robotic-guided total stations allow surveyors to measure without extra workers to aim the telescope or record data. A fast but expensive way to measure large areas is with a helicopter, using a GPS to record the location of the helicopter and a laser scanner to measure the ground. To increase precision, surveyors place beacons on

1016-462: A consequence, the set of extended complex numbers is often called the Riemann sphere . The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example C ^ = C ∪ { ∞ } . {\displaystyle {\hat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.} In

1143-403: A curve, then the slope given by the above definition, is the slope of a secant line to the curve. For a line, the secant between any two points is the line itself, but this is not the case for any other type of curve. For example, the slope of the secant intersecting y = x at (0,0) and (3,9) is 3. (The slope of the tangent at x = 3 ⁄ 2 is also 3 −  a consequence of

1270-412: A division problem such as 6 3 = ? {\displaystyle {\tfrac {6}{3}}={?}} can be solved by rewriting it as an equivalent equation involving multiplication, ? × 3 = 6 , {\displaystyle {?}\times 3=6,} where ? {\displaystyle {?}} represents the same unknown quantity, and then finding

1397-477: A fixed base station and a second roving antenna. The position of the roving antenna can be tracked. The theodolite , total station and RTK GPS survey remain the primary methods in use. Remote sensing and satellite imagery continue to improve and become cheaper, allowing more commonplace use. Prominent new technologies include three-dimensional (3D) scanning and lidar -based topographical surveys. UAV technology along with photogrammetric image processing

SECTION 10

#1732773174049

1524-403: A fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth. Historically, one of

1651-428: A framework to support the extension of the realm of numbers to which they apply. For instance, to make it possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set of integers in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to the rational numbers . During this gradual expansion of

1778-734: A function has two distinct one-sided limits . A basic example of an infinite singularity is the reciprocal function , f ( x ) = 1 / x , {\displaystyle f(x)=1/x,} which tends to positive or negative infinity as x {\displaystyle x} tends to 0 {\displaystyle 0} : lim x → 0 + 1 x = + ∞ , lim x → 0 − 1 x = − ∞ . {\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x}}=+\infty ,\qquad \lim _{x\to 0^{-}}{\frac {1}{x}}=-\infty .} In most cases,

1905-583: A function is constructed by dividing two functions whose separate limits are both equal to 0 , {\displaystyle 0,} then the limit of the result cannot be determined from the separate limits, so is said to take an indeterminate form , informally written 0 0 . {\displaystyle {\tfrac {0}{0}}.} (Another indeterminate form, ∞ ∞ , {\displaystyle {\tfrac {\infty }{\infty }},} results from dividing two functions whose limits both tend to infinity.) Such

2032-423: A great step forward in the instrument's accuracy. William Gascoigne invented an instrument that used a telescope with an installed crosshair as a target device, in 1640. James Watt developed an optical meter for the measuring of distance in 1771; it measured the parallactic angle from which the distance to a point could be deduced. Dutch mathematician Willebrord Snellius (a.k.a. Snel van Royen) introduced

2159-502: A height above sea level. As the surveying profession grew it created Cartesian coordinate systems to simplify the mathematics for surveys over small parts of the Earth. The simplest coordinate systems assume that the Earth is flat and measure from an arbitrary point, known as a 'datum' (singular form of data). The coordinate system allows easy calculation of the distances and direction between objects over small areas. Large areas distort due to

2286-433: A known size. It was sometimes used before to the invention of EDM where rough ground made chain measurement impractical. Historically, horizontal angles were measured by using a compass to provide a magnetic bearing or azimuth. Later, more precise scribed discs improved angular resolution. Mounting telescopes with reticles atop the disc allowed more precise sighting (see theodolite ). Levels and calibrated circles allowed

2413-460: A limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions. For example, in lim x → 1 x 2 − 1 x − 1 , {\displaystyle \lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}},} the separate limits of the numerator and denominator are 0 {\displaystyle 0} , so we have

2540-624: A line in the plane containing the x and y axes is generally represented by the letter m , and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line. This is described by the following equation: (The Greek letter delta , Δ, is commonly used in mathematics to mean "difference" or "change".) Given two points ( x 1 , y 1 ) {\displaystyle (x_{1},y_{1})} and ( x 2 , y 2 ) {\displaystyle (x_{2},y_{2})} ,

2667-442: A line through the origin is the vertical coordinate of the intersection between the line and a vertical line at horizontal coordinate 1 , {\displaystyle 1,} dashed black in the figure. The vertical red and dashed black lines are parallel , so they have no intersection in the plane. Sometimes they are said to intersect at a point at infinity , and the ratio 1 : 0 {\displaystyle 1:0}

SECTION 20

#1732773174049

2794-434: A loop pattern or link between two prior reference marks so the surveyor can check their measurements. Many surveys do not calculate positions on the surface of the Earth, but instead, measure the relative positions of objects. However, often the surveyed items need to be compared to outside data, such as boundary lines or previous survey's objects. The oldest way of describing a position is via latitude and longitude, and often

2921-618: A multi frequency phase shift of light waves to find a distance. These instruments eliminated the need for days or weeks of chain measurement by measuring between points kilometers apart in one go. Advances in electronics allowed miniaturization of EDM. In the 1970s the first instruments combining angle and distance measurement appeared, becoming known as total stations . Manufacturers added more equipment by degrees, bringing improvements in accuracy and speed of measurement. Major advances include tilt compensators, data recorders and on-board calculation programs. The first satellite positioning system

3048-411: A plan or map, and the points at the ends of the offset lines could be joined to show the feature. Traversing is a common method of surveying smaller areas. The surveyor starts from an old reference mark or known position and places a network of reference marks covering the survey area. They then measure bearings and distances between the reference marks, and to the target features. Most traverses form

3175-406: A point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured more accurately than bearings of the vertices, which depended on a compass. His work established the idea of surveying a primary network of control points, and locating subsidiary points inside the primary network later. Between 1733 and 1740, Jacques Cassini and his son César undertook

3302-478: A profession. They established the basic measurements under which the Roman Empire was divided, such as a tax register of conquered lands (300 AD). Roman surveyors were known as Gromatici . In medieval Europe, beating the bounds maintained the boundaries of a village or parish. This was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of

3429-450: A ratio of 10 : 2 {\displaystyle 10:2} or, proportionally, 5 : 1. {\displaystyle 5:1.} To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to 5 : 1 {\displaystyle 5:1} could be maintained, for instance one cup of flour and one-fifth cup of sugar, or fifty cups of flour and ten cups of sugar. Now imagine

3556-449: A rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers. In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on set theory. First, the natural numbers (including zero) are established on an axiomatic basis such as Peano's axiom system and then this

3683-428: A reflector or prism to return the light pulses used for distance measurements. They are fully robotic, and can even e-mail point data to a remote computer and connect to satellite positioning systems , such as Global Positioning System . Real Time Kinematic GPS systems have significantly increased the speed of surveying, and they are now horizontally accurate to within 1 cm ± 1 ppm in real-time, while vertically it

3810-472: A road have altitudes y 1 and y 2 , the rise is the difference ( y 2 − y 1 ) = Δ y . Neglecting the Earth's curvature , if the two points have horizontal distance x 1 and x 2 from a fixed point, the run is ( x 2 − x 1 ) = Δ x . The slope between the two points is the difference ratio : Through trigonometry , the slope m of a line is related to its angle of inclination θ by

3937-562: A slope given as a percentage into an angle in degrees and vice versa are: and where angle is in degrees and the trigonometric functions operate in degrees. For example, a slope of 100 % or 1000 ‰ is an angle of 45°. A third way is to give one unit of rise in say 10, 20, 50 or 100 horizontal units, e.g. 1:10. 1:20, 1:50 or 1:100 (or "1 in 10", "1 in 20", etc.) 1:10 is steeper than 1:20. For example, steepness of 20% means 1:5 or an incline with angle 11.3°. Roads and railways have both longitudinal slopes and cross slopes. The concept of

Slope - Misplaced Pages Continue

4064-399: A slope is central to differential calculus . For non-linear functions, the rate of change varies along the curve. The derivative of the function at a point is the slope of the line tangent to the curve at the point and is thus equal to the rate of change of the function at that point. If we let Δ x and Δ y be the distances (along the x and y axes, respectively) between two points on

4191-487: A slope or gradient is also used as a basis for developing other applications in mathematics: Surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and the distances and angles between them. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership , locations, such as

4318-420: A special not-a-number value, or crash the program, among other possibilities. The division N / D = Q {\displaystyle N/D=Q} can be conceptually interpreted in several ways. In quotitive division , the dividend N {\displaystyle N} is imagined to be split up into parts of size D {\displaystyle D} (the divisor), and

4445-476: A star is determined, the bearing can be transferred to a reference point on Earth. The point can then be used as a base for further observations. Survey-accurate astronomic positions were difficult to observe and calculate and so tended to be a base off which many other measurements were made. Since the advent of the GPS system, astronomic observations are rare as GPS allows adequate positions to be determined over most of

4572-392: A subtle mistake leading to absurd results. To prevent this, the arithmetic of real numbers and more general numerical structures called fields leaves division by zero undefined , and situations where division by zero might occur must be treated with care. Since any number multiplied by zero is zero, the expression 0 0 {\displaystyle {\tfrac {0}{0}}}

4699-438: A sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio 10 : 0 , {\displaystyle 10:0,} or proportionally 1 : 0 , {\displaystyle 1:0,} is perfectly sensible: it just means that the cake has no sugar. However, the question "How many parts flour for each part sugar?" still has no meaningful numerical answer. A geometrical appearance of

4826-495: A theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725. In the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787. It was an instrument for measuring angles in the horizontal and vertical planes. He created his great theodolite using an accurate dividing engine of his own design. Ramsden's theodolite represented

4953-593: A time component. Before EDM (Electronic Distance Measurement) laser devices, distances were measured using a variety of means. In pre-colonial America Natives would use the "bow shot" as a distance reference ("as far as an arrow can slung out of a bow", or "flights of a Cherokee long bow"). Europeans used chains with links of a known length such as a Gunter's chain , or measuring tapes made of steel or invar . To measure horizontal distances, these chains or tapes were pulled taut to reduce sagging and slack. The distance had to be adjusted for heat expansion. Attempts to hold

5080-410: Is (−2,4). The derivative of this function is d y ⁄ d x = 2 x . So the slope of the line tangent to y at (−2,4) is 2 ⋅ (−2) = −4 . The equation of this tangent line is: y − 4 = (−4)( x − (−2)) or y = −4 x − 4 . An extension of the idea of angle follows from the difference of slopes. Consider the shear mapping Then ( 1 , 0 ) {\displaystyle (1,0)}

5207-542: Is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero. In 830, Mahāvīra unsuccessfully tried to correct the mistake Brahmagupta made in his book Ganita Sara Samgraha : "A number remains unchanged when divided by zero." Bhāskara II 's Līlāvatī (12th century) proposed that division by zero results in an infinite quantity, A quantity divided by zero becomes

Slope - Misplaced Pages Continue

5334-474: Is a staple of contemporary land surveying. Typically, much if not all of the drafting and some of the designing for plans and plats of the surveyed property is done by the surveyor, and nearly everyone working in the area of drafting today (2021) utilizes CAD software and hardware both on PC, and more and more in newer generation data collectors in the field as well. Other computer platforms and tools commonly used today by surveyors are offered online by

5461-399: Is a term used when referring to moving the level to take an elevation shot from a different location. To "turn" the level, one must first take a reading and record the elevation of the point the rod is located on. While the rod is being kept in exactly the same location, the level is moved to a new location where the rod is still visible. A reading is taken from the new location of the level and

5588-400: Is also appearing. The main surveying instruments in use around the world are the theodolite , measuring tape , total station , 3D scanners , GPS / GNSS , level and rod . Most instruments screw onto a tripod when in use. Tape measures are often used for measurement of smaller distances. 3D scanners and various forms of aerial imagery are also used. The theodolite is an instrument for

5715-621: Is also undefined. Calculus studies the behavior of functions in the limit as their input tends to some value. When a real function can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to " tend to infinity ", a type of mathematical singularity . For example, the reciprocal function , f ( x ) = 1 x , {\displaystyle f(x)={\tfrac {1}{x}},} tends to infinity as x {\displaystyle x} tends to 0. {\displaystyle 0.} When both

5842-412: Is an alternate method of determining the position of objects, and was often used to measure imprecise features such as riverbanks. The surveyor would mark and measure two known positions on the ground roughly parallel to the feature, and mark out a baseline between them. At regular intervals, a distance was measured at right angles from the first line to the feature. The measurements could then be plotted on

5969-505: Is because divergent conditions further away from the base reduce accuracy. Surveying instruments have characteristics that make them suitable for certain uses. Theodolites and levels are often used by constructors rather than surveyors in first world countries. The constructor can perform simple survey tasks using a relatively cheap instrument. Total stations are workhorses for many professional surveyors because they are versatile and reliable in all conditions. The productivity improvements from

6096-585: Is currently about half of that to within 2 cm ± 2 ppm. GPS surveying differs from other GPS uses in the equipment and methods used. Static GPS uses two receivers placed in position for a considerable length of time. The long span of time lets the receiver compare measurements as the satellites orbit. The changes as the satellites orbit also provide the measurement network with well conditioned geometry. This produces an accurate baseline that can be over 20 km long. RTK surveying uses one static antenna and one roving antenna. The static antenna tracks changes in

6223-405: Is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote 0 as x − 1 . The Brāhmasphuṭasiddhānta of Brahmagupta (c. 598–668) is the earliest text to treat zero as a number in its own right and to define operations involving zero. According to Brahmagupta, A positive or negative number when divided by zero

6350-444: Is expanded to the ring of integers . The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of ordered pairs of integers, {( a , b )} with b ≠ 0 , define a binary relation on this set by ( a , b ) ≃ ( c , d ) if and only if ad = bc . This relation

6477-654: Is mapped to ( 1 , v ) {\displaystyle (1,v)} . The slope of ( 1 , 0 ) {\displaystyle (1,0)} is zero and the slope of ( 1 , v ) {\displaystyle (1,v)} is v {\displaystyle v} . The shear mapping added a slope of v {\displaystyle v} . For two points on { ( 1 , y ) : y ∈ R } {\displaystyle \{(1,y):y\in \mathbb {R} \}} with slopes m {\displaystyle m} and n {\displaystyle n} ,

SECTION 50

#1732773174049

6604-404: Is necessary in this context. In this structure, a 0 = ∞ {\displaystyle {\frac {a}{0}}=\infty } can be defined for nonzero a , and a ∞ = 0 {\displaystyle {\frac {a}{\infty }}=0} when a is not ∞ {\displaystyle \infty } . It is the natural way to view the range of

6731-554: Is physically exhibited by some mechanical calculators . In partitive division , the dividend N {\displaystyle N} is imagined to be split into D {\displaystyle D} parts, and the quotient Q {\displaystyle Q} is the resulting size of each part. For example, imagine ten cookies are to be divided among two friends. Each friend will receive five cookies ( 10 2 = 5 {\displaystyle {\tfrac {10}{2}}=5} ). Now imagine instead that

6858-633: Is represented by a new number ∞ {\displaystyle \infty } ; see § Projectively extended real line below. Vertical lines are sometimes said to have an "infinitely steep" slope. Division is the inverse of multiplication , meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example ( 5 × 3 ) / 3 = {\displaystyle (5\times 3)/3={}} ( 5 / 3 ) × 3 = 5 {\displaystyle (5/3)\times 3=5} . Thus

6985-483: Is shown to be an equivalence relation and its equivalence classes are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying transitivity ). Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures. In

7112-416: Is that allowing it leads to fallacies . When working with numbers, it is easy to identify an illegal division by zero. For example: The fallacy here arises from the assumption that it is legitimate to cancel 0 like any other number, whereas, in fact, doing so is a form of division by 0 . Using algebra , it is possible to disguise a division by zero to obtain an invalid proof . For example: This

7239-406: Is the projectively extended real line , which is a one-point compactification of the real line. Here ∞ {\displaystyle \infty } means an unsigned infinity or point at infinity , an infinite quantity that is neither positive nor negative. This quantity satisfies − ∞ = ∞ {\displaystyle -\infty =\infty } , which

7366-471: Is the standard deviation of the y-values and s x {\displaystyle s_{x}} is the standard deviation of the x-values. This may also be written as a ratio of covariances : There are two common ways to describe the steepness of a road or railroad . One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also steep grade railway and rack railway . The formulae for converting

7493-425: Is the dividend (numerator). The usual definition of the quotient in elementary arithmetic is the number which yields the dividend when multiplied by the divisor. That is, c = a b {\displaystyle c={\tfrac {a}{b}}} is equivalent to c ⋅ b = a . {\displaystyle c\cdot b=a.} By this definition, the quotient q =

7620-460: Is the exact slope of the tangent. If y is dependent on x , then it is sufficient to take the limit where only Δ x approaches zero. Therefore, the slope of the tangent is the limit of Δ y /Δ x as Δ x approaches zero, or d y /d x . We call this limit the derivative . The value of the derivative at a specific point on the function provides us with the slope of the tangent at that precise location. For example, let y = x . A point on this function

7747-420: Is undefined in this extension of the real line. The subject of complex analysis applies the concepts of calculus in the complex numbers . Of major importance in this subject is the extended complex numbers C ∪ { ∞ } , {\displaystyle \mathbb {C} \cup \{\infty \},} the set of complex numbers with a single additional number appended, usually denoted by

SECTION 60

#1732773174049

7874-400: Is used in the planning and execution of most forms of construction . It is also used in transportation, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in many other scientific disciplines. The International Federation of Surveyors defines the function of surveying as follows: A surveyor is a professional person with

8001-432: Is with an altimeter  using air pressure to find the height. When more precise measurements are needed, means like precise levels (also known as differential leveling) are used. When precise leveling, a series of measurements between two points are taken using an instrument and a measuring rod. Differences in height between the measurements are added and subtracted in a series to get the net difference in elevation between

8128-471: Is zero rather than six, so there exists no number which can substitute for ? {\displaystyle {?}} to make a true statement. When the problem is changed to 0 0 = ? , {\displaystyle {\tfrac {0}{0}}={?},} the equivalent multiplicative statement is ? × 0 = 0 {\displaystyle {?}\times 0=0} ; in this case any value can be substituted for

8255-480: Is −1. So these two lines are perpendicular. In statistics , the gradient of the least-squares regression best-fitting line for a given sample of data may be written as: This quantity m is called as the regression slope for the line y = m x + c {\displaystyle y=mx+c} . The quantity r {\displaystyle r} is Pearson's correlation coefficient , s y {\displaystyle s_{y}}

8382-421: The x -axis is Consider the two lines: y = −3 x + 1 and y = −3 x − 2 . Both lines have slope m = −3 . They are not the same line. So they are parallel lines. Consider the two lines y = −3 x + 1 and y = ⁠ x / 3 ⁠ − 2 . The slope of the first line is m 1 = −3 . The slope of the second line is m 2 = ⁠ 1 / 3 ⁠ . The product of these two slopes

8509-509: The Principal Triangulation of Britain . The first Ramsden theodolite was built for this survey. The survey was finally completed in 1853. The Great Trigonometric Survey of India began in 1801. The Indian survey had an enormous scientific impact. It was responsible for one of the first accurate measurements of a section of an arc of longitude, and for measurements of the geodesic anomaly. It named and mapped Mount Everest and

8636-621: The Torrens system in South Australia in 1858. Torrens intended to simplify land transactions and provide reliable titles via a centralized register of land. The Torrens system was adopted in several other nations of the English-speaking world. Surveying became increasingly important with the arrival of railroads in the 1800s. Surveying was necessary so that railroads could plan technologically and financially viable routes. At

8763-799: The U.S. Federal Government and other governments' survey agencies, such as the National Geodetic Survey and the CORS network, to get automated corrections and conversions for collected GPS data, and the data coordinate systems themselves. Surveyors determine the position of objects by measuring angles and distances. The factors that can affect the accuracy of their observations are also measured. They then use this data to create vectors, bearings, coordinates, elevations, areas, volumes, plans and maps. Measurements are often split into horizontal and vertical components to simplify calculation. GPS and astronomic measurements also need measurement of

8890-434: The hyperreal numbers , division by zero is still impossible, but division by non-zero infinitesimals is possible. The same holds true in the surreal numbers . In distribution theory one can extend the function 1 x {\textstyle {\frac {1}{x}}} to a distribution on the whole space of real numbers (in effect by using Cauchy principal values ). It does not, however, make sense to ask for

9017-426: The infinity symbol ∞ {\displaystyle \infty } and representing a point at infinity , which is defined to be contained in every exterior domain , making those its topological neighborhoods . This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point ∞ , {\displaystyle \infty ,}

9144-406: The mean value theorem .) By moving the two points closer together so that Δ y and Δ x decrease, the secant line more closely approximates a tangent line to the curve, and as such the slope of the secant approaches that of the tangent. Using differential calculus , we can determine the limit , or the value that Δ y /Δ x approaches as Δ y and Δ x get closer to zero ; it follows that this limit

9271-456: The real numbers R {\displaystyle \mathbb {R} } by adding two new numbers + ∞ {\displaystyle +\infty } and − ∞ , {\displaystyle -\infty ,} read as "positive infinity" and "negative infinity" respectively, and representing points at infinity . With the addition of ± ∞ , {\displaystyle \pm \infty ,}

9398-417: The tangent function Thus, a 45° rising line has slope m = +1, and a 45° falling line has slope m = −1. Generalizing this, differential calculus defines the slope of a plane curve at a point as the slope of its tangent line at that point. When the curve is approximated by a series of points, the slope of the curve may be approximated by the slope of the secant line between two nearby points. When

9525-485: The tangent function and cotangent functions of trigonometry : tan( x ) approaches the single point at infinity as x approaches either + ⁠ π / 2 ⁠ or − ⁠ π / 2 ⁠ from either direction. This definition leads to many interesting results. However, the resulting algebraic structure is not a field , and should not be expected to behave like one. For example, ∞ + ∞ {\displaystyle \infty +\infty }

9652-425: The Earth's curvature. North is often defined as true north at the datum. Division by zero In mathematics , division by zero , division where the divisor (denominator) is zero , is a unique and problematic special case. Using fraction notation, the general example can be written as a 0 {\displaystyle {\tfrac {a}{0}}} , where a {\displaystyle a}

9779-585: The Egyptians' command of surveying. The groma instrument may have originated in Mesopotamia (early 1st millennium BC). The prehistoric monument at Stonehenge ( c.  2500 BC ) was set out by prehistoric surveyors using peg and rope geometry. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual , published in 263 AD. The Romans recognized land surveying as

9906-471: The above equation generates the formula: The formula fails for a vertical line, parallel to the y {\displaystyle y} axis (see Division by zero ), where the slope can be taken as infinite , so the slope of a vertical line is considered undefined. Suppose a line runs through two points: P  = (1, 2) and Q  = (13, 8). By dividing the difference in y {\displaystyle y} -coordinates by

10033-561: The academic qualifications and technical expertise to conduct one, or more, of the following activities; Surveying has occurred since humans built the first large structures. In ancient Egypt , a rope stretcher would use simple geometry to re-establish boundaries after the annual floods of the Nile River . The almost perfect squareness and north–south orientation of the Great Pyramid of Giza , built c.  2700 BC , affirm

10160-438: The basis for dividing the western territories into sections to allow the sale of land. The PLSS divided states into township grids which were further divided into sections and fractions of sections. Napoleon Bonaparte founded continental Europe 's first cadastre in 1808. This gathered data on the number of parcels of land, their value, land usage, and names. This system soon spread around Europe. Robert Torrens introduced

10287-457: The bearing from every vertex in a figure, a surveyor can measure around the figure. The final observation will be between the two points first observed, except with a 180° difference. This is called a close . If the first and last bearings are different, this shows the error in the survey, called the angular misclose . The surveyor can use this information to prove that the work meets the expected standards. The simplest method for measuring height

10414-468: The beginning of the century, surveyors had improved the older chains and ropes, but they still faced the problem of accurate measurement of long distances. Trevor Lloyd Wadley developed the Tellurometer during the 1950s. It measures long distances using two microwave transmitter/receivers. During the late 1950s Geodimeter introduced electronic distance measurement (EDM) equipment. EDM units use

10541-528: The boundaries. Young boys were included to ensure the memory lasted as long as possible. In England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the land owners, the area of land they owned, the quality of the land, and specific information of the area's content and inhabitants. It did not include maps showing exact locations. Abel Foullon described

10668-576: The case where the limit of the real function f {\displaystyle f} increases without bound as x {\displaystyle x} tends to c , {\displaystyle c,} the function is not defined at x , {\displaystyle x,} a type of mathematical singularity . Instead, the function is said to " tend to infinity ", denoted lim x → c f ( x ) = ∞ , {\textstyle \lim _{x\to c}f(x)=\infty ,} and its graph has

10795-406: The change in x {\displaystyle x} from one to the other is x 2 − x 1 {\displaystyle x_{2}-x_{1}} ( run ), while the change in y {\displaystyle y} is y 2 − y 1 {\displaystyle y_{2}-y_{1}} ( rise ). Substituting both quantities into

10922-560: The concept of a "limit at infinity" can be made to work like a finite limit. When dealing with both positive and negative extended real numbers, the expression 1 / 0 {\displaystyle 1/0} is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define 1 / 0 = + ∞ {\displaystyle 1/0=+\infty } . The set R ∪ { ∞ } {\displaystyle \mathbb {R} \cup \{\infty \}}

11049-597: The curve is given as the graph of an algebraic expression , calculus gives formulas for the slope at each point. Slope is thus one of the central ideas of calculus and its applications to design. There seems to be no clear answer as to why the letter m is used for slope, but it first appears in English in O'Brien (1844) who introduced the equation of a line as " y = mx + b " , and it can also be found in Todhunter (1888) who wrote " y = mx + c ". The slope of

11176-418: The designated positions of structural components for construction or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales. A professional in land surveying is called a land surveyor . Surveyors work with elements of geodesy , geometry , trigonometry , regression analysis , physics , engineering, metrology , programming languages , and

11303-482: The difference in x {\displaystyle x} -coordinates, one can obtain the slope of the line: As another example, consider a line which runs through the points (4, 15) and (3, 21). Then, the slope of the line is For example, consider a line running through points (2,8) and (3,20). This line has a slope, m , of One can then write the line's equation, in point-slope form: or: The angle θ between −90° and 90° that this line makes with

11430-543: The division-as-ratio interpretation is the slope of a straight line in the Cartesian plane . The slope is defined to be the "rise" (change in vertical coordinate) divided by the "run" (change in horizontal coordinate) along the line. When this is written using the symmetrical ratio notation, a horizontal line has slope 0 : 1 {\displaystyle 0:1} and a vertical line has slope 1 : 0. {\displaystyle 1:0.} However, if

11557-525: The earliest recorded references to the mathematical impossibility of assigning a value to a 0 {\textstyle {\tfrac {a}{0}}} is contained in Anglo-Irish philosopher George Berkeley 's criticism of infinitesimal calculus in 1734 in The Analyst ("ghosts of departed quantities"). Calculus studies the behavior of functions using the concept of a limit ,

11684-562: The early days of surveying, this was the primary method of determining accurate positions of objects for topographic maps of large areas. A surveyor first needs to know the horizontal distance between two of the objects, known as the baseline . Then the heights, distances and angular position of other objects can be derived, as long as they are visible from one of the original objects. High-accuracy transits or theodolites were used, and angle measurements were repeated for increased accuracy. See also Triangulation in three dimensions . Offsetting

11811-1214: The extended complex numbers, for any nonzero complex number z , {\displaystyle z,} ordinary complex arithmetic is extended by the additional rules z 0 = ∞ , {\displaystyle {\tfrac {z}{0}}=\infty ,} z ∞ = 0 , {\displaystyle {\tfrac {z}{\infty }}=0,} ∞ + 0 = ∞ , {\displaystyle \infty +0=\infty ,} ∞ + z = ∞ , {\displaystyle \infty +z=\infty ,} ∞ ⋅ z = ∞ . {\displaystyle \infty \cdot z=\infty .} However, 0 0 {\displaystyle {\tfrac {0}{0}}} , ∞ ∞ {\displaystyle {\tfrac {\infty }{\infty }}} , and 0 ⋅ ∞ {\displaystyle 0\cdot \infty } are left undefined. The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as

11938-544: The first prototype satellites of the Global Positioning System (GPS) in 1978. GPS used a larger constellation of satellites and improved signal transmission, thus improving accuracy. Early GPS observations required several hours of observations by a static receiver to reach survey accuracy requirements. Later improvements to both satellites and receivers allowed for Real Time Kinematic (RTK) surveying. RTK surveys provide high-accuracy measurements by using

12065-467: The first triangulation of France. They included a re-surveying of the meridian arc , leading to the publication in 1745 of the first map of France constructed on rigorous principles. By this time triangulation methods were well established for local map-making. It was only towards the end of the 18th century that detailed triangulation network surveys mapped whole countries. In 1784, a team from General William Roy 's Ordnance Survey of Great Britain began

12192-748: The ground (about 20 km (12 mi) apart). This method reaches precisions between 5–40 cm (depending on flight height). Surveyors use ancillary equipment such as tripods and instrument stands; staves and beacons used for sighting purposes; PPE ; vegetation clearing equipment; digging implements for finding survey markers buried over time; hammers for placements of markers in various surfaces and structures; and portable radios for communication over long lines of sight. Land surveyors, construction professionals, geomatics engineers and civil engineers using total station , GPS , 3D scanners, and other collector data use land surveying software to increase efficiency, accuracy, and productivity. Land Surveying Software

12319-509: The ground to large beacons that can be seen from long distances. The surveyors can set up their instruments in this position and measure to nearby objects. Sometimes a tall, distinctive feature such as a steeple or radio aerial has its position calculated as a reference point that angles can be measured against. Triangulation is a method of horizontal location favoured in the days before EDM and GPS measurement. It can determine distances, elevations and directions between distant objects. Since

12446-400: The height difference is used to find the new elevation of the level gun, which is why this method is referred to as differential levelling . This is repeated until the series of measurements is completed. The level must be horizontal to get a valid measurement. Because of this, if the horizontal crosshair of the instrument is lower than the base of the rod, the surveyor will not be able to sight

12573-437: The image has slope increased by v {\displaystyle v} , but the difference n − m {\displaystyle n-m} of slopes is the same before and after the shear. This invariance of slope differences makes slope an angular invariant measure , on a par with circular angle (invariant under rotation) and hyperbolic angle, with invariance group of squeeze mappings . The concept of

12700-695: The indeterminate form 0 0 {\displaystyle {\tfrac {0}{0}}} , but simplifying the quotient first shows that the limit exists: lim x → 1 x 2 − 1 x − 1 = lim x → 1 ( x − 1 ) ( x + 1 ) x − 1 = lim x → 1 ( x + 1 ) = 2. {\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\frac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}(x+1)=2.} The affinely extended real numbers are obtained from

12827-416: The law. They use equipment, such as total stations , robotic total stations, theodolites , GNSS receivers, retroreflectors , 3D scanners , lidar sensors, radios, inclinometer , handheld tablets, optical and digital levels , subsurface locators, drones, GIS , and surveying software. Surveying has been an element in the development of the human environment since the beginning of recorded history . It

12954-510: The limit of a quotient of functions is equal to the quotient of the limits of each function separately, lim x → c f ( x ) g ( x ) = lim x → c f ( x ) lim x → c g ( x ) . {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {\displaystyle \lim _{x\to c}f(x)}{\displaystyle \lim _{x\to c}g(x)}}.} However, when

13081-409: The line x = c {\displaystyle x=c} as a vertical asymptote . While such a function is not formally defined for x = c , {\displaystyle x=c,} and the infinity symbol ∞ {\displaystyle \infty } in this case does not represent any specific real number , such limits are informally said to "equal infinity". If

13208-464: The measurement of angles. It uses two separate circles , protractors or alidades to measure angles in the horizontal and the vertical plane. A telescope mounted on trunnions is aligned vertically with the target object. The whole upper section rotates for horizontal alignment. The vertical circle measures the angle that the telescope makes against the vertical, known as the zenith angle. The horizontal circle uses an upper and lower plate. When beginning

13335-409: The measurement of vertical angles. Verniers allowed measurement to a fraction of a degree, such as with a turn-of-the-century transit . The plane table provided a graphical method of recording and measuring angles, which reduced the amount of mathematics required. In 1829 Francis Ronalds invented a reflecting instrument for recording angles graphically by modifying the octant . By observing

13462-424: The measuring instrument level would also be made. When measuring up a slope, the surveyor might have to "break" (break chain) the measurement- use an increment less than the total length of the chain. Perambulators , or measuring wheels, were used to measure longer distances but not to a high level of accuracy. Tacheometry is the science of measuring distances by measuring the angle between two ends of an object with

13589-411: The modern systematic use of triangulation . In 1615 he surveyed the distance from Alkmaar to Breda , approximately 72 miles (116 km). He underestimated this distance by 3.5%. The survey was a chain of quadrangles containing 33 triangles in all. Snell showed how planar formulae could be corrected to allow for the curvature of the Earth . He also showed how to resect , or calculate, the position of

13716-433: The number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (is undefined ) in the whole number setting, this remains true as the setting expands to the real or even complex numbers . As the realm of numbers to which these operations can be applied expands there are also changes in how

13843-410: The numerator and the denominator tend to zero at the same input, the expression is said to take an indeterminate form , as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits. As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define

13970-430: The operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers. Similarly, when the realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't

14097-672: The other Himalayan peaks. Surveying became a professional occupation in high demand at the turn of the 19th century with the onset of the Industrial Revolution . The profession developed more accurate instruments to aid its work. Industrial infrastructure projects used surveyors to lay out canals , roads and rail. In the US, the Land Ordinance of 1785 created the Public Land Survey System . It formed

14224-417: The quotient Q {\displaystyle Q} is the number of resulting parts. For example, imagine ten slices of bread are to be made into sandwiches, each requiring two slices of bread. A total of five sandwiches can be made ( 10 2 = 5 {\displaystyle {\tfrac {10}{2}}=5} ). Now imagine instead that zero slices of bread are required per sandwich (perhaps

14351-480: The result of division by zero in other ways, resulting in different number systems. For example, the quotient a 0 {\displaystyle {\tfrac {a}{0}}} can be defined to equal zero; it can be defined to equal a new explicit point at infinity , sometimes denoted by the infinity symbol ∞ {\displaystyle \infty } ; or it can be defined to result in signed infinity, with positive or negative sign depending on

14478-477: The rod and get a reading. The rod can usually be raised up to 25 feet (7.6 m) high, allowing the level to be set much higher than the base of the rod. The primary way of determining one's position on the Earth's surface when no known positions are nearby is by astronomic observations. Observations to the Sun, Moon and stars could all be made using navigational techniques. Once the instrument's position and bearing to

14605-422: The satellite positions and atmospheric conditions. The surveyor uses the roving antenna to measure the points needed for the survey. The two antennas use a radio link that allows the static antenna to send corrections to the roving antenna. The roving antenna then applies those corrections to the GPS signals it is receiving to calculate its own position. RTK surveying covers smaller distances than static methods. This

14732-400: The sign of the dividend. In these number systems division by zero is no longer a special exception per se, but the point or points at infinity involve their own new types of exceptional behavior. In computing , an error may result from an attempt to divide by zero. Depending on the context and the type of number involved, dividing by zero may evaluate to positive or negative infinity , return

14859-454: The slope is taken to be a single real number then a horizontal line has slope 0 1 = 0 {\displaystyle {\tfrac {0}{1}}=0} while a vertical line has an undefined slope, since in real-number arithmetic the quotient 1 0 {\displaystyle {\tfrac {1}{0}}} is undefined. The real-valued slope y x {\displaystyle {\tfrac {y}{x}}} of

14986-423: The surface of the Earth. Few survey positions are derived from the first principles. Instead, most surveys points are measured relative to previously measured points. This forms a reference or control network where each point can be used by a surveyor to determine their own position when beginning a new survey. Survey points are usually marked on the earth's surface by objects ranging from small nails driven into

15113-406: The survey, the surveyor points the instrument in a known direction (bearing), and clamps the lower plate in place. The instrument can then rotate to measure the bearing to other objects. If no bearing is known or direct angle measurement is wanted, the instrument can be set to zero during the initial sight. It will then read the angle between the initial object, the theodolite itself, and the item that

15240-536: The telescope aligns with. The gyrotheodolite is a form of theodolite that uses a gyroscope to orient itself in the absence of reference marks. It is used in underground applications. The total station is a development of the theodolite with an electronic distance measurement device (EDM). A total station can be used for leveling when set to the horizontal plane. Since their introduction, total stations have shifted from optical-mechanical to fully electronic devices. Modern top-of-the-line total stations no longer need

15367-406: The ten cookies are to be divided among zero friends. How many cookies will each friend receive? Since there are no friends, this is an absurdity. In another interpretation, the quotient Q {\displaystyle Q} represents the ratio N : D . {\displaystyle N:D.} For example, a cake recipe might call for ten cups of flour and two cups of sugar,

15494-459: The two endpoints. With the Global Positioning System (GPS), elevation can be measured with satellite receivers. Usually, GPS is somewhat less accurate than traditional precise leveling, but may be similar over long distances. When using an optical level, the endpoint may be out of the effective range of the instrument. There may be obstructions or large changes of elevation between the endpoints. In these situations, extra setups are needed. Turning

15621-404: The unknown quantity to yield a true statement, so there is no single number which can be assigned as the quotient 0 0 . {\displaystyle {\tfrac {0}{0}}.} Because of these difficulties, quotients where the divisor is zero are traditionally taken to be undefined , and division by zero is not allowed. A compelling reason for not allowing division by zero

15748-669: The value for which the statement is true; in this case the unknown quantity is 2 , {\displaystyle 2,} because 2 × 3 = 6 , {\displaystyle 2\times 3=6,} so therefore 6 3 = 2. {\displaystyle {\tfrac {6}{3}}=2.} An analogous problem involving division by zero, 6 0 = ? , {\displaystyle {\tfrac {6}{0}}={?},} requires determining an unknown quantity satisfying ? × 0 = 6. {\displaystyle {?}\times 0=6.} However, any number multiplied by zero

15875-559: The value of the function decreases without bound, the function is said to "tend to negative infinity", − ∞ . {\displaystyle -\infty .} In some cases a function tends to two different values when x {\displaystyle x} tends to c {\displaystyle c} from above ( x → c + {\displaystyle x\to c^{+}} ) and below ( x → c − {\displaystyle x\to c^{-}} ) ; such

16002-524: The value to which a function's output tends as its input tends to some specific value. The notation lim x → c f ( x ) = L {\textstyle \lim _{x\to c}f(x)=L} means that the value of the function f {\displaystyle f} can be made arbitrarily close to L {\displaystyle L} by choosing x {\displaystyle x} sufficiently close to c . {\displaystyle c.} In

16129-504: Was the US Navy TRANSIT system . The first successful launch took place in 1960. The system's main purpose was to provide position information to Polaris missile submarines. Surveyors found they could use field receivers to determine the location of a point. Sparse satellite cover and large equipment made observations laborious and inaccurate. The main use was establishing benchmarks in remote locations. The US Air Force launched

#48951