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Math Investigation: Conclusion

Although my ruler has a lot of accuracy, this is not enough. The ruler measures in

centimeters and millimeters. However, the ruler is quite hard to measure with and should

result in the measurement being at least one millimeter off. Also, in diagonal lines and

midpoints, it is usually unlikely that the line will be a whole number therefore meaning that

the ruler needs to be very accurate. However, in this task the midpoints and lengths do not

have to be exactly correct. This is because the point of the investigation was to check

whether the opposite sides were equal in length and therefore it is only needed to check

whether the lines were same or similar in length or if they were completely different in

length.

When drawing lines with a ruler, if using an old or broken ruler sometimes parts of the ruler

are chipped off making the line not straight. Also when using a ruler the line sometimes

does not lie in the exact place we mean for it to.

The thickness of pencil affects the accuracy of the pencil slightly. This is because the

thickness of the pencil may cause the lines to be longer or shorter than they actually are.

The reason this happens is when the pencil is thick causing the incorrect place to be

measured. However, this inaccuracy is very slight as most pencils are usually less than 1mm

thick. The investigation does not require an absolutely precise degree of measurement

therefore it is not needed to make the measurements exactly precise.

A tool I could use to be more accurate is a protractor. This is because I am able to check the

angles of my shapes. In a parallelogram opposite angles are equal, also when all the angles

are added together the result is 360 degrees. By using a protractor also it is possible to

double the accuracy since by both measuring the angles and lengths this helps ensure that

the measurements are correct. Many protractors are also able to measure at an accurate

degree being around less than one degree inaccurate. This allows for extremely precise

measurements.

When the coordinate grid is added, it makes the measurements much more accurate. This

is because a coordinate grid allows us to find midpoints and line lengths using formulas and

logic rather than measurement. These formulas have been proven by mathematicians to be

accurate and therefore are able to give a much more accurate result than from measuring

by hand. Both the formula used to find the midpoint and the line lengths will give a

completely precise answer. Although a problem is sometimes the answer may be in

decimals and not completely accurate.

There are many real world examples of coordinate geometry. A very simple way is by the

game Battleship. To locate ships coordinate geometry is used. However this is modified

into a simpler method as most of the time the two coordinates are a number and alphabet

since many people do not know how to use geometry and will be confused about the order

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of the x and y axis if two numbers are used instead. Also, there are no negative numbers in

battleship as only one quarter is used. GPS also uses coordinate geometry to navigate and

function. Although in a GPS the system is more 3-D and very complicated. By using

coordinate geometry this allows the GPS to show a location and find locations. Architects

and designers also use coordinate geometry to design things. For example, when designinga house architects would be able to make lines equal and in correct proportions it also

allows them to make measurements on lengths and midpoints that are accurate and do not

need measuring tools. This is very useful since they may have to measure long lengths and

need to do it accurately. Another machine that uses coordinate geometry is scanners.

Scanners use coordinates to help measure and determine location when they transfer the

image of something into a computer. A PDF file also uses coordinate geometry. PDF files

combine images and text into a file that is easy to read and cannot be modified. This is done

by using a coordinate graph to place all the text and images. Mathematical proof is used in

all math formulas. This is because by using math proof we can be sure that the formula is

correct. Math proof involves deductive reasoning which is much more useful than guessing.

There have also been other mathematical proofs. There are many of these and some are

very complex. Two ofthese are the proof that 0.999 is equal to 1 and that is an

irrational number.

Algebraic proof is a lot more desirable than proof by exhaustion. This is because proof by

exhaustion looks a lot less elegant since it requires many tests and trials. Proof by

exhaustion also does not have any basis or foundation as it is more of a guess. An algebraic

proof has used deductive reasoning and thought rather from inductive or empiricalarguments. Also an algebraic is capable of showing that a statement is always true. An

algebraic proof also makes sure that the statement is proven and true. However, a proof by

exhaustion could merely be showing a coincidence as it has no reason or thought behind it.

Using algebraic proof will also help in solving other problems since algebra allows us to

changing the letters for whatever we desire helping us solve other problems quickly.

However, there are many conjectures such as the Kepler Conjecture which has no algebraic

proof and the only available proof is through proof by exhaustion. In conclusion, algebraic

proof is a lot greater than proof by exhaustion. It does not require large amount of tests but

instead needs thinking and thought. Algebraic proof has reasoning and concrete proof thatshows that the statement is always true unlike proof by exhaustion which could also be a

result of coincidence. There it is best to choose algebraic proof over proof by exhaustion.