Mathematics for Land Surveying

• This lecture covers mathematics in surveying, including significant figures, round-off, algebra, geometry, and trigonometry principles.

Importance of Mathematics in Surveying

• Math is necessary to measure property accurately using basic math courses like algebra, trigonometry, and geometry.
• Land surveyors use math to measure precise angles, distances, and directions.
• Data research and reconciliation of conflicting documents necessitates complex mathematical skills.
• Drafting requires plotting lines and coordinates using calculus, differential equations, geometry, statistics, probability, and trigonometry.
• Computer-based data analysis still requires mathematical skills to troubleshoot software and computer bugs.

Algebra vs Geometry and Trigonometry

• Algebra is the mathematics of abstraction, which uses equations and variables to solve problems.
• Algebra trains reasoning abilities and is the basis for all other mathematics.
• Geometry relates to shapes and graphical representations, using axioms and logic to derive theorems.
• The geometric method teaches logical reasoning and proves if something is true or false.
• Trigonometry is the starting point of advanced mathematics useful in real-world applications, focusing on the "trigonometric functions".
• Trigonometric functions describe the relationship between angles and the length of sides in a triangle.

Significant Figures

• Significant figures/digits are the reliable digits in a measurement.
• Measurements are recorded with all known reliable digits plus one uncertain digit.
• Significant figures indicate measurement precision, dependent on the measuring device's least count.
• Significant figures are identified from left to right, starting with non-zero digits and ending with the last digit.
• All non-zero digits are considered significant (e.g., 91 has two, 123.45 has five significant figures).
• Zeros between non-zero digits are significant (e.g., 101.1203 has seven significant figures).
• Leading zeros are not significant (e.g., 0.00052 has two significant figures).
• Trailing zeros in a number with a decimal point are significant (e.g., 12.2300 and 0.000122300 both have six significant figures).
• The significance of trailing zeros in a number lacking a decimal point can be ambiguous (e.g., 1400 could have two, while 1400. has four).
• Scientific notation helps express the number of significant digits; the pre-exponential factor indicates the number of significant digits.
• For example, 2.5 x 10^5 has 2 sig figs, 2.50 x 10^5 has 3 sig figs and 2.500 x 10^5 has 4 sig figs.

Addition and Subtraction with Significant Figures

• The answer should have the same number of decimal places as the term with the fewest decimal places.
• Example: 12.793 + 4.58 + 3.25794 = 20.63094, rounds to 20.63 since 4.58 has two decimal places.
• Example: 215.47 g + 918.251 g - 0.000458 g = 1133.72 g

Multiplication and Division with Significant Figures

• The answer should have the same number of significant figures as the term with the fewest number of significant figures.
• Example: 56.937 / 0.46 = 130.29782609, reported as 1.3 x 10^2 since 0.46 has only 2 significant figures.
• Without scientific notation, the reading would be ambiguous
• Example: (214.21 g) x (11.2 cm) / (17.413 g) = 138 cm

Significance of Significant Figures

• Significant figures carry meaning and contribute to a number's precision.
• The number of significant digits depends on the least count of the measuring instrument.
• Significant figures help in rounding.
• Significant figures ensure computational results reflect the accuracy of initial measurements.
• Accuracy is the relationship between measure and true value.
• Precision is the degree of refinement with which the measurement is made.
• A more precise method yields more accurate results.

Round-Off Rules

• If the digit is smaller than 5, drop it (e.g., 1.68497 rounded to three significant figures becomes 1.68).
• If the digit is 5 or larger, drop it and add 1 to the preceding digit (e.g., 1.24712 rounded to three significant figures becomes 1.25).
• If the digit to be dropped is 5: increase the last remaining digit by one if it is odd, but left as it is if even (e.g., 11.5 rounds to 12 and 12.5 rounds to 12).

Examples of Rounding

• 6746.583 rounded to 5 significant figures is 6746.6.
• 6746.583 rounded to 2 significant figures is 6.7x10².
• Examples with a 5 as the first non-significant figure show 4.7475 becomes 4.748 to make the last digit even, and 4.7465 becomes 4.746 because the last digit is already even.
• When adding or subtracting, round to the last common digit furthest to the right in all components.
• 100 + 23.643 = 123.643 becomes 124.
• When multiplying or dividing, round to the least number of significant figures in any component.
• 3.0 x 12.60 = 37.8000 becomes 38.

Algebra in Surveying

• Algebra analyzes and solves problems within the physical world and society.
• Algebra focuses on problem-solving techniques up to theoretical mathematics.
• Three main types of equations in algebra are Linear, Quadratic, and Exponential Models.
• A linear equation has no exponent.
• A quadratic equation has a highest exponent of two.
• An exponential equation has a variable in the exponent.

Geometry in Surveying

• Geometry is fundamental for surveyors to understand and calculate angles, distances, and elevations.
• Dimensions are the different facets of perceived reality.
• The three dimensions are length, width, and depth in the x, y, and z axes.
• A point has no dimensions and is defined by a pair of numbers like (x, y).
• Straight lines are the shortest distance between two points.
• Straight lines are one dimensional.
• A two-dimensional coordinate system contains a horizontal (x) and a vertical (y) measurement scale.
• The 2D origin point occurs at both x and y equal zero in the coordinate system.
• Cartesian Coordinates mark a point on a graph by how far along and how far up it is (x, y).
• Polar Coordinates mark a point by how far away and what angle it is (r, Θ).
• To convert from Cartesian points to Polar points, Pythagoras' Thoerem with the equation r = √(x² + y²) must be computed to find the long side.
• Then the Tangent function must be used to calculate the angle 0 = tan¯¹ (y/x).
• In summary, Cartesian points (x,y) can be converted Polar points (r,0) with the equations r = √(x² + y²) and 0 = tan¯¹ (y/x).
• Convert to Cartesian points from Polar using x = r × cos( θ ) and y = r × sin( θ ).

Trigonometry in Surveying

• Right angle trigonometry and oblique triangle trigonometry is used.
• The equation c² = a² + b² is used for right angle based problems.
• Sine = Opposite / Hypotenuse for a right angle.
• Cosine = Adjacent / Hypotenuse for a right angle.
• Tangent = Opposite / Adjacent for a right angle.
• A quiz example in the lecture provided the following data to calculate distance.
• A man in a boat sees the top of the Barnegat Lighthouse at an angle of 20 degrees above the horizon. He knows that the lighthouse is 165 feet tall. How far away from the shore is the boat?
• Tangent = Opposite / Adjacent so Adj = Opposite/Tangent, therefore boat is 453 feet from the shore.