# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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292 möbius strip

m

F

F’

0

References: Axis of Symmetry, Bisect, Mediator.

MIXED NUMBER

Reference: Fraction.

MKS SYSTEM

This system of units was based upon the meter, the kilogram, and the second. It became the SI system of units.

References: CGS System of Units, SI Units.

MÖBIUS STRIP

The Möbius strip, or band, is a continuous surface that has only one side and only one edge. It is named after A. F. Möbius (1790-1868).

A Möbius strip is extremely simple to make. Take a strip of paper in the shape of a long, thin rectangle, say 30 cm long (see figure a). Give one end a twist through half a turn. Now bring the two ends together and glue them onto each other forming a join at A. The resulting shape is a Möbius strip.

Using a crayon, color one side of the paper that makes up the Möbius strip, and you will discovery that you have colored all the paper. If you place your finger at the point A and traverse the whole surface of the strip, your finger will arrive back at the same point A after traveling a distance of twice 30 cm = 60 cm. This shows that the Möbius strip has only one surface, unlike the original rectangle, which had two surfaces. The Möbius strip also has only one edge.

The property of the Möbius strip of having only one surface can be made use of in machines that are belt-driven. With a normal belt only the inside of the belt is in contact with the drive shaft. If the belt is in the shape of a Möbius strip, the surface in contact with the drive shaft is twice as long and therefore the belt will last longer,

IX

A

(a)

MULTIPLE 293

because there is less wear. Because of the twist in the belt, the two shafts it connects turn in opposite directions.

If you cut a Möbius strip into two pieces by cutting with a pair of scissors along the dashed centerline as shown in figure b, you obtain two Möbius strips, which are linked together like a chain.

MODE

References: Bimodal, Central Tendency.

MODELING

Reference: Königsberg Bridge Problem.

MODULUS

Reference: Absolute Value.

MONOMINO

Reference: Polyominoes.

MOSAICS

Reference: Tessellations.

MOVING AVERAGES

Reference: Time Series.

(b)

MULTIPLE

Reference: Factor.

294 MULTIPLYING FRACTIONS

MULTIPLICAND

Suppose two numbers, say 6 and 8, are multiplied together: 6 x 8 = 48. The first number (6) is the multiplicand, the second number (8) is the multiplier, and the answer (48) is the product.

MULTIPLIER

Reference: Multiplicand.

MULTIPLYING FRACTIONS

Multiplying fractions is easier than adding and subtracting them. When we multiply two fractions together we multiply the numerators together, multiply the denominators together, and write the result as a single fraction.

Example 1. Multiply the two fractions | x ^.

Solution. Write 5 4 5x4

- x — = -—— Multiplying the numerators and the denominators

8 15 8 x 15

If the two fractions are mixed numbers, it is best to change them into improper fractions before multiplying.

Example 2. Multiply the two mixed numbers 2| x l|.

Solution. Under the entry Fractions you can see how to change mixed numbers into fractions and back again. These skills are needed to solve this problem. Write

3 1 11 3

2-xl- = — x- Changing mixed numbers into improper fractions

_ 33 “ Y

= 4- Writing the improper fraction back as a mixed number

8

= —— This fraction needs cancelling down

120 6

1 20

= - x — 20 is a common factor of numerator and denominator

6 20

1

6

MUTUALLY EXCLUSIVE EVENTS 295

Example 3. Uncle Harry leaves $52,000 in his will to be shared between his two nephews, John and Bill. John is to inherit | of his uncle’s money and Bill is to get the rest. John decides to give ^ of his money to his favorite charity. How much money does John give away?

Solution. John gives away ^ of | of his uncle’s money. Write

12^ 1 2 52,000

— x - x 52,000 = —- x - x —-—

10 3 10 3 1

1 x 2 x 52,000 = 10 x 3 x 1

104,000 “ 30

= 3467 to the nearest dollar

John gives away $3467.

Division of two fractions is done by multiplying the first fraction by the reciprocal of the second fraction.

Example 4. Work out 1 \ -f §.

Solution. Write

l|^f = §x§ Changing 11 into | and multiplying by the reciprocal of |

- U

— 6

= 21 Writing the fraction ^ as a mixed number

The methods of adding and subtracting fractions are explained in the entry Fractions. Fractions can be added, subtracted, multiplied, and divided using a scientific calculator, and the processes are explained in the calculator handbook.

References: Canceling, Fractions, Reciprocal.

MUTUALLY EXCLUSIVE EVENTS

Reference: Complementary Events.

N

NATURAL NUMBERS

Reference: Integers.

NEGATIVE

This word describes a quantity which is less than zero, whereas positive describes a quantity which is greater than zero. Negative numbers are numbers that are to the left of zero on the number line, and positive numbers are to the right of zero. Negative numbers should not be confused with “minus,” which is an instruction to subtract one number from another.

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