104 mathematics magazine issue 2 april

8/9/2019 104 Mathematics Magazine Issue 2 April

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Created by Maths Rep of 104Created by Maths Rep of 104

Published in April 2010Published in April 2010Maths is Great PTE LTDMaths is Great PTE LTD

Issue no. 2

Cartesian Graphs?

GoldenRatio?

LinearGraphs?

Algebra?

Estima

tion?

Calculator Precision?

Approxim

ati

on?

Coordina


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Golden Ratio-----------------------pg3

Cartesian Graphs-----------------pg4

Coordinates & Linear Graphs--pg5

Estimation & Approximation----pg6Calculator Precision--------------pg7

Test your Calculator--------------pg8

Algebra------------------------------pg9Hall Of Fame: Al-Khwarizimi-pg10

Mathematics Quiz---------------pg11


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In mathematics and the arts, 2 quantities areIn mathematics and the arts, 2 quantities are

in the golden ratio if the ratio of (the sum ofin the golden ratio if the ratio of (the sum ofthe quantities: the larger quantity) is = to thethe quantities: the larger quantity) is = to the

ratio of (the larger quantity: the smaller one).ratio of (the larger quantity: the smaller one).

The golden ratio is an irrational mathematicalThe golden ratio is an irrational mathematical

constant, approximately 1.6180339887.constant, approximately 1.6180339887.

Other names frequently used for the goldenOther names frequently used for the golden

ratio are the golden section (Latin:ratio are the golden section (Latin: sectiosectioaureaaurea) and golden mean. Other terms) and golden mean. Other terms

encountered include extreme and mean ratio,encountered include extreme and mean ratio,

medial section, divine proportion, divinemedial section, divine proportion, divine

section (Latin:section (Latin: sectio divinasectio divina), golden), golden

proportion, golden cut, golden number, andproportion, golden cut, golden number, and

mean of Phidias. The golden ratio is oftenmean of Phidias. The golden ratio is oftendenoted by the Greek letter phi, usually lowerdenoted by the Greek letter phi, usually lower

case ().case ().

Larger quantity

Smaller quantity


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In the previous issue, we mention Rene Descartes. He wasIn the previous issue, we mention Rene Descartes. He was

the on who invented the Cartesian Coordinate Graphs.the on who invented the Cartesian Coordinate Graphs.

Once, there was a fly on the ceiling, he wanted to describeOnce, there was a fly on the ceiling, he wanted to describe

the position of the fly. So, that is why he had invented thethe position of the fly. So, that is why he had invented the

Cartesian Coordinate Graphs.Cartesian Coordinate Graphs.

Cartesian Coordinate Graphs is useful for locatingCartesian Coordinate Graphs is useful for locatingsomething .something .This graph is the combination of number lines,This graph is the combination of number lines,

algebra and geometry.algebra and geometry.

There are two types of axis: the x axis & the y axis.There are two types of axis: the x axis & the y axis.

The x axis goes horizontally but the y axis goes vertically.The x axis goes horizontally but the y axis goes vertically.

However, there is also a z axis. (not commonly used)However, there is also a z axis. (not commonly used)

The place where the x axis & the y axis intersect, its theThe place where the x axis & the y axis intersect, its the

origin, (O)origin, (O)

There are four quadrilaterals in a linear

graph.

The 1st quadrilateral has (+) x coordinate

&(+) y coordinates.

The 2nd quadrilateral has (-) x coordinate

&(+) y coordinates.

The 3rd quadrilateral has (-) x coordinate

&(-) y coordinates.

The 4th quadrilateral has (+) X

coordinates&(-) Y coordinates.


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Coordinate is a point.Coordinate is a point.

By joining two coordinates together, you will get a line.By joining two coordinates together, you will get a line. The line on the graph has a function (an algebra statement).The line on the graph has a function (an algebra statement).

A function is a relationship that expresses a dependantA function is a relationship that expresses a dependant

variable in terms of an independent variable and each valuevariable in terms of an independent variable and each value

of the independent variable will produce a unique value forof the independent variable will produce a unique value for

the dependant variable.E.g. y=2x+3the dependant variable.E.g. y=2x+3

The steepness of the line formed is a gradient.The steepness of the line formed is a gradient.

Gradient of a line can be derived by the formula: rise/runGradient of a line can be derived by the formula: rise/run

A negative gradient is a gradient sloping downwards fromA negative gradient is a gradient sloping downwards from

left to right.left to right.

A positive gradient is a gradient sloping upwards from leftA positive gradient is a gradient sloping upwards from leftto right.to right.

A horizontal line has a gradient 0A horizontal line has a gradient 0

A vertical line has a gradient, which is undefined.A vertical line has a gradient, which is undefined.


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We apply estimation in our daily lives and inWe apply estimation in our daily lives and in

measurements.measurements.We can round off very big or very small numbers byWe can round off very big or very small numbers byrounding off to significant numbers or decimal place orrounding off to significant numbers or decimal place orthe nearest integer. This makes things easier for us.the nearest integer. This makes things easier for us.

We approximate a measurement since there is no exacWe approximate a measurement since there is no exacor precise measurement.or precise measurement.

We estimate a value as is easier to calculate.We estimate a value as is easier to calculate.

In our daily life, we usually approximate a value to aIn our daily life, we usually approximate a value to acertain number of significant figures as it is easier for uscertain number of significant figures as it is easier for usto say it. For example, it is easier to read out theto say it. For example, it is easier to read out theapproximate value of the price of a housing estate thanapproximate value of the price of a housing estate thanto read out the exact price.to read out the exact price.

In measurements, like for example measuring theIn measurements, like for example measuring theamount of water in the beaker. From the reading belowamount of water in the beaker. From the reading belowthe meniscus, we might say the water level is aboutthe meniscus, we might say the water level is about6.5ml. If we wanted to read out the exact value, then we6.5ml. If we wanted to read out the exact value, then wewill expect to see lots of numbers.will expect to see lots of numbers.


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Calculator PrecisionCalculator Precision

Now, well look into the precision of calculators. Hmmmm oneNow, well look into the precision of calculators. Hmmmm onemight rely on calculator to do numeric computation, but calculatorsmight rely on calculator to do numeric computation, but calculatorsare also very limited, they dont show exact and precise value.are also very limited, they dont show exact and precise value.

The degree of precision of a calculator depends on the type ofThe degree of precision of a calculator depends on the type ofcalculator you are using. There are 2 different types: four-functioncalculator you are using. There are 2 different types: four-functioncalculator and the scientific calculator.calculator and the scientific calculator.

With the extensive use of calculators, we might notice that differentWith the extensive use of calculators, we might notice that differenttypes of calculators give different type of answers. Instead, theytypes of calculators give different type of answers. Instead, theyround off the value as calculator can also display at most ten digitsround off the value as calculator can also display at most ten digitson the screen.on the screen.

The number of digits stored and used also depends on the capacityThe number of digits stored and used also depends on the capacityof calculator Any number has exited the limit of the number ofof calculator Any number has exited the limit of the number ofdigits, the calculator will round off the value. These approximationsdigits, the calculator will round off the value. These approximationsresult in rounding and truncation errors especially in working veryresult in rounding and truncation errors especially in working verylarge or very small values.large or very small values.

2/3 can be written in 0.666666666(truncation error) or2/3 can be written in 0.666666666(truncation error) or

0.666666667(rounding error).0.666666667(rounding error).If you multiply 2/3 by 4, your mental sum gives you 8/3. However,If you multiply 2/3 by 4, your mental sum gives you 8/3. However,the calculator will either give you 2.666666667 or 2.666666664.the calculator will either give you 2.666666667 or 2.666666664.

In conclusion, dont rely too much on calculators. Instead, trust youIn conclusion, dont rely too much on calculators. Instead, trust youmental sum. Surprisingly, your mental sum may be more precise anmental sum. Surprisingly, your mental sum may be more precise anaccurate.accurate.


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Get a calculator. Lets test its precision.

Divide 100 by 3:

Then, multiply theanswer by 3:

Square root 2 three times:

Then square the answerthree times:

Cube root 5 six times:

Then cube the answer sixtimes:

Calculators screen value

Try this exercise with another type of calculator.

Different types of calculator gives you different

types of answer due to truncation errors and

rounding errors


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Algebra is a branch of Mathematics, which usedAlgebra is a branch of Mathematics, which usedletters and symbols to represent numbers orletters and symbols to represent numbers orvariables.variables.

Algebra can be applied in almost any branch ofAlgebra can be applied in almost any branch ofMathematics. Once you mastered it, no problemMathematics. Once you mastered it, no problem

is too hard for you!is too hard for you! Algebra is invented by a Muslim Mathematician,Algebra is invented by a Muslim Mathematician,

Al-Khwarizimi.Al-Khwarizimi.

When solving an algebraic expression, follow theWhen solving an algebraic expression, follow theBIDMAS rule.BIDMAS rule.

We add or subtract the like terms by adding orWe add or subtract the like terms by adding orsubtracting the coefficients. We do not add orsubtracting the coefficients. We do not add orsubtract the coefficients of unlike terms.subtract the coefficients of unlike terms.

If an expression within brackets is multiplied byIf an expression within brackets is multiplied bya number, each term within the brackets musta number, each term within the brackets mustbe multiplied by that number when the bracketsbe multiplied by that number when the brackets

are removed.are removed. Factorizing of algebraic expressions can be doneFactorizing of algebraic expressions can be done

by extracting common factors from all terms ofby extracting common factors from all terms ofthe expression or grouping terms such that thethe expression or grouping terms such that thenew terms obtained have a common factor.new terms obtained have a common factor.
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Hall Of Fame:Hall Of Fame: Al-KhwarizimiAl-Khwarizimi Al-Khwarizimi (c.780-c.850 CE) was a PersianAl-Khwarizimi (c.780-c.850 CE) was a Persian

mathematician, astronomer and geographer.mathematician, astronomer and geographer.

He is known as the founder of algebra.He thought ofHe is known as the founder of algebra.He thought ofrepresenting unknown values using alphabet.representing unknown values using alphabet.

Al-KhwarizmiAl-Khwarizmi was born at Khwarizm (Kheva), a townwas born at Khwarizm (Kheva), a townsouth of the river Oxus in present day Uzbekistan.south of the river Oxus in present day Uzbekistan.

He explained the use of ZERO. He developed the decimalHe explained the use of ZERO. He developed the decimalsystem, several arithmetical procedures includingsystem, several arithmetical procedures includingoperations on fractions, details of Trigonometric tablesoperations on fractions, details of Trigonometric tablescontaining Sine functions and tangent functions, calculuscontaining Sine functions and tangent functions, calculusof two errors, which led him to the concept ofof two errors, which led him to the concept ofdifferentiation.differentiation.

He also contributed to astronomy and geography.He also contributed to astronomy and geography.

Algebra and algorithms are enabling the building ofAlgebra and algorithms are enabling the building ofcomputers, and the creation of encryption.computers, and the creation of encryption.

The modern technology industry would not exist withoutThe modern technology industry would not exist withoutthe contributions of Muslim mathematicians like Al-the contributions of Muslim mathematicians like Al-

Khwarizmi.Khwarizmi.


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a t h e m a t ic s Q u iza t h e m a t ic s Q u izThere are 4 Quiz to solve.There are 4 Quiz to solve.

Q1. Simplify the following equations:Q1. Simplify the following equations:

(a) y=3x+5-4z-1x+5z(a) y=3x+5-4z-1x+5z

(b) z=3a2 x 5a2 x 4a3(b) z=3a2 x 5a2 x 4a3

Q2. Given that a line passes through theQ2. Given that a line passes through thecoordinates (3,5) and (-2,10). What is the gradientcoordinates (3,5) and (-2,10). What is the gradientof the line?of the line?

Q3. Given that a line passes through theQ3. Given that a line passes through the

coordinates (-8,-6) and (0,36). What is the gradientcoordinates (-8,-6) and (0,36). What is the gradientof the line?of the line?

Q4. Solve the problems and estimate the answersQ4. Solve the problems and estimate the answersto the acceptable form.to the acceptable form.

(a) 2.3456N+2345.6N(a) 2.3456N+2345.6N

(b) 6.5432m/s x 6.543.2m/s x 36kg(b) 6.5432m/s x 6.543.2m/s x 36kg

The top scorer of the quiz on the previous issueis() with the score of ().

Write your answers on a foolscap paper and submit it toTommy. The highest score for the Mathematics Quiz will bedisplayed on the next issue of the magazine. The deadline ofthe quiz is 30th of April 2010.(Prizes will also be awarded by

the welfare department.)


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104 mathematics magazine issue 2 april

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