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Equations

Yes, we now have to divide up our time like that, between politics and our equations. But to me our

equations are far more important, for politics are only a matter of present concern. A mathematical equation

stands forever. - Albert Einstein

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E = mc2 A Biography of the World’s Most Famous Equation

by David Bodanis Most of the main typographical symbols we use were in place by the end of the Middle Ages. Bibles of the fourteenth century often had text that looked much like telegrams: IN THE BEGINNING GOD CREATED THE HEAVEN

AND THE EARTH AND THE EARTH WAS WITHOUT

FORM AND VOID AND DARKNESS WAS UPON THE

FACE OF THE DEEP

One change that took place over time was to drop most of the letters to lowercase: In the beginning God created the heaven and the earth

and the earth was without form and void and

darkness was upon the face of the deep

Another shift was to insert tiny round circles to mark the major breathing pauses:

In the beginning God created the heaven and the

earth. And the earth was without form and void and

darkness was upon the face of the deep.

Smaller curves were used as well, for the minor breathing pauses:

In the beginning, God created the heaven and the

earth.

Major symbols were locked in rather quickly once printing began at the end of the 1400s. Texts began to be filled in with the old ? symbols and the newer ! marks. It was a bit like the Windows standard in personal computers driving out other operating systems. Minor symbols took longer. By now we take them so much for granted that, for example, we almost always blink when we see the period at the end of a sentence. (Watch someone when they're reading and you'll see it.) Yet this is an entirely learned response. For more than a thousand years, one of the world's major population centers used this symbol for addition since it showed someone walking toward you (and so was to be "added" to you), and for subtraction. These Egyptian symbols could easily have spread to become universally accepted, just as other Middle Eastern symbols had done. Phoenician symbols, for example, were the source of the Hebrew א and ב – aleph and beth – and also the Greek α and β – alpha and beta – as in our word alphabet.

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Through the mid-1500s there was still space for entrepreneurs to set their own mark by establishing the remaining minor symbols. In 1543, Robert Recorde, an eager textbook writer in England, tried to promote the new-style " + " sign, which had achieved some popularity on the Continent. The book he wrote didn't make his fortune, so in the next decade he tried again, this time with a symbol seemingly of his own creation that he was sure would take off. In the best style of advertising hype everywhere, he even tried to give it a unique selling point: “... And to avoide the tediouse repetition of these woordes: is equalle to: I will sette ... a pair of parallels, or ... lines of one lengthe, thus: ════ bicause noe .2. thynges, can be moare equalle. .. ." It doesn't seem that Recorde gained from his innovation, for it remained in bitter competition with the equally plausible / / and even with the bizarre [; symbol, which the powerful German printing houses were trying to promote. The full range of possibilities proffered at one place or another include, if we imagine them put in the equation:

e ││mc2 e → mc2

e .æqus. mc2 e ] [ mc2 There was even my favorite: e ════════ mc2 Not until Shakespeare's time, a generation later, was Recorde's victory finally certain. Pedants and schoolmasters since then have often used the equals sign just to summarize what's

already known, but a few thinkers had a better idea. If I say that 15 + 20 = 35, this is not very interesting. But imagine if I say: (go 15 degrees west) + (then go 20 degrees south) = (you'll find trade winds that can fling you across the Atlantic to a new continent in 35 days). Then I am telling you something new. A good equation is not simply a formula for computation. Nor is it a balance scale confirming that two items you suspected were nearly equal really are the same. Instead, scientists started using the = symbol as something of a telescope for new ideas – a device for directing attention to fresh, unsuspected realms. Equations simply happen to be written in symbols instead of words. This is how Einstein used the “ = ” in his 1905 equation as well. The Victorians had thought they'd found all possible sources of energy there were: chemical energy, heat energy, magnetic energy, and the rest. But by 1905 Einstein could say, No, there is another place you can look where you'll find more. His equation was like a telescope to lead there, but the hiding place wasn't far away in outer space. It was down here – it had been right in front of his professors all along. He found this vast energy source in the one place where no one had thought of looking. It was hidden away in solid matter itself.

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1. The scales in the three diagrams are balanced. How many triangles are needed to balance one rectangle?

2. A wrapping machine runs five days a week at a constant rate. On Monday it wrapped 60 parcels and 70 catalogues. On Tuesday it wrapped 80 parcels

and 40 catalogues. On Wednesday, Thursday and Friday it wrapped only catalogues. How many catalogues were wrapped during the week? (From Canadian Maths Competition Problems Vol. 1)

Two Problems

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Solve the following equations:

1. ( ) ( )5224 +−=+− aa 2. 426)32(4)5(3 +÷=+−−− bbb 3. 24

5 −=−c

c

4. 3

1

6

4

3

2

2−=+d

5. )2()1(54)2(3 −−−=−+ eeee 6. 212

4 =−f

7. 3)1(6)1()2(4 ++−=−−− gggg 8. 1805 2 =h 9. 1872 =− kk

10. 304

122031215)23(5

2

62)5(3

2

=−

−+÷−+−−+−−m

mmmmm

mm

Some Routine Equations

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1. If 102,0 =y , then y equals A) 0,02 B) 0,05 C) 0,5 D) 5 E) 50

2. If 6,03

2 =x , then x equals

A) 0,4 B) 0,9 C) 4 D) 0,09 E) 9 3. If axx =− 35 , then x equals

A) 3

5−a B) 5

3a− C)

3

5

+a D)

5

3

−a E)

a−5

3

4. A pen and pencil together cost R1,40. The pen costs R1 more than the pencil. The cost, in cents, of the pencil is

A) 120 B) 100 C) 40 D) 30 E) 20 5. If bdacdcba +=Ψ ),(),( and 3)5,2()3,( =−Ψx , then the value of x is

A) – 9 B) – 6 C) 5

9 D)

3

13 E) 6

6. A tank is 6

1 full of petrol. If 2 litres are added, then the tank is

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1 full. The total capacity of the tank, in litres, is

A) 6 B) 8 C) 12 D) 24 E) 30

7. If 11

4

1

3

1 =++n

, then the value of n is

A) 7

6 B)

12

5 C) 6− D)

5

12 E)

12

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Olympiad Problems on Equations

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8. If 4=x , xy 3= and yz 2= , then the value of z is

A) 12 B) 20 C) 40 D) 24 E) 36

9. If 2=x is a solution of the equation 1135 =−+ xq , the value of q is

A) 4 B) 7 C) 14 D) – 7 E) – 4

10. If 55 =−y and 82 =x , then yx + equals

A) 13 B) 28 C) 33 D) 35 E) 38

11. If aa5

216

5

2 −= , then 102

5

+a

a equals

A) 2 B) 4

1 C)

11

25 D)

9

10 E) 0

(From Canadian Mathematics Competitions Vol. 2 and 7)

I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful

mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.

- Paul Dirac