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Numbers

Rules of Commutation

Commutation is just a big-person word for the rules which determine which order to do mathematical operations.

Operators and Operands

Operators are the symbols for: + (addition), - (subtraction), x (multiplication, also '*' or '⋅'), ÷ (division, also '/').

Operands are the quantities on which the operators act:

e.g. in the equation 6 + 3 = 9, 6 and 3 are the operands, and + is the operator.

For example, it does not matter which order we do addition: a + b = b + a, but it does with subtraction: a - b ≠ b - a.

Similarly, a × b = b × a, but a/b ≠ b/a.

When we combine operands we use parentheses () to indicate in which order the operators should be executed:

$$(3 + 4) × 5 = 7 × 5 = 35$$ $$3 + (4 × 5) = 3 + 20 = 23$$

If not otherwise specified, multiplication and division are given higher priority than addition and subtraction:

$$12 - 3 × 4 = 12 - 12 = 0$$ $$6 ÷ 3 × 2 = 2 × 2 = 4$$ $$6 ÷ (3 × 2) = 6 ÷ 6 = 1$$

Inequalities

x = 2. This simple statement tells us that x is exactly equal to the value 2.

x < 2. This tells us not what x is equal to, but what it is not equal to: its 'inequality'.

The possibilities for inequalities are:

Domains

The domain of an equation is the permitted range of values x may have. This is often defined after an equation.

e.g. $y = 1/x$, where x ≠ 0

If x were allowed to be zero, the equation has no value, and calculators go bananas on you.

The domain can also be defined using inequalities: e.g. {-∞ < x < 2} ∪ {2 < x < +∞} means x < 2 and x > 2, but not 2.

While we are on the subject, the set of all the values y can take is called the 'range'.

  • Any value except. e.g. x ≠ 2 means x < 2 and x > 2, but not 2: We can write this formally as: {-∞ < x < 2} ∪ {2 < x < +∞}, but simply x ≠ 2 is enough for those of us with a train to catch.

  • <
  • Any value less than. e.g. x < 2 means {-∞ < x < 2}, or all values from negative infinity right up to but not including 2.

  • >
  • Any value greater than. e.g. x > 2 means {2 < x < +∞}, or all values from 2, but not including 2, right up to positive infinity.

  • Any value less than or equal to. e.g. x ≤ 2 means {-∞ < x ≤ 2}, or all values from negative infinity right up to and including 2.

  • Any value greater than or equal to. e.g. x ≥ 2 means {2 ≤ x < +∞}, or all values from 2, including 2, right up to positive infinity.

Decimal places and Accuracy

If you ask a calculator what is $1/2$, he tells you '0.5'. Ask him what $1/3$ is and he gives you '0.333333333333'. He would be happy to go on forever, if his screen were wide enough. So, what is going on? Why is there such a big difference in the number of decimal places?

Calculators do their thinking in decimals. 'Decimal' means 'on the basis of ten' (dec is a prefix for ten). Three does not go into ten easily: ten divided by three is 3 plus one. This one divided by three is 0.3 plus 0.1. This 0.1 divided by 3 is 0.03 plus 0.01. See where I am going with this?

To be able to make decimal calculations we have to understand 'accuracy' and his loyal friend 'uncertainty'.

Uncertainty

Good mathematicians and scientists are proud of their uncertainty.

When we make a measurement we have to decide how accurate we can say it is. For example, if you measure the length of your table with a small ruler, you may get an answer of, say, 120cm. But, using a small ruler means introducing a lot of imprecision in the measurement. You could easily get an answer of 121 cm a second time, or 119cm a third time. To report your certainty about the result, you could write your measurement as: 120 cm ± 1 cm. This tells the reader that your best guess is 120 cm, but it could just as easily be one centimetre more or less.

How well do you know something?

Precision

How well you did the measurement (also the quality of the instrument you used) is the precision. To check your precision you could make a number of measurements and see how much they vary.

Accuracy

How close your measurement is to the real value is the accuracy. To check your accuracy would require knowing the true value. In real science you probably don't know what result you should have got, so understanding and reporting precision and uncertainty becomes very important.

Uncertainty

How sure you are of your result is your uncertainty. This is usually expressed as a number or percentage. e.g. 100cm ± 1 cm = 100 ± 1%

Now, if someone asks you how long four tables put side to side would be, what can you say? You could say they are about 480cm ± 4cm, meaning that you have a fair degree of certainty that the real answer lies somewhere between 476 cm and 484 cm.

On the other hand, if you measured the table with a long tapemeasure, you could probably get a more precise measurement, of which you feel more certain: 120 cm ± 0.1 cm, or plus or minus a millimetre. This would allow you to say four tables together have a length of 480 cm ± 0.4 cm.

The version of assuming the greatest error on the low side is called the 'lower' bound'. In the ruler and table example, the lower bound is assuming each measurement was 119cm, and the combined calculation is 476cm.

The upper bound is therefore 121cm and 484cm, respectively.

Significant Figures

If your measurement is 100 cm, how many decimal places should you use?

Well, if you write 100.00 cm, this would mean you are certain about the next two zeroes. This means your uncertainty is very low: in fact it means you think the measurement is 100 cm ± 0.005 cm. You wouldn't be able to get this precision with a tape measure!

With a tape measure, the precision is about a millimetre. So you would write 100.0 ± 0.1cm.

You could also write this as 1.00 ± 0.001 m, or 1.00 ± 0.1%.

The number of significant figures therefore informs the reader of how close you think the value is to the real value.

Golden Ratio

Golden Ratio
The golden ratio is an ancient concept, identifying one of the natural proportions.

Symbol φ. Approximately 1.61. This ratio is a commonly found pattern in nature, and is used in design and architecture for proportions, such as rectangles, since it is instinctively pleasing to the eye.

The golden ratio is the ratio, where C/A = A/B, and A is a portion of a line length C, and B the remaining portion. A + B = C.

$φ={√5+1}/2$

$1/{φ}= φ-1$

φ=1.61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 28621 35448 62270 52604 62818 90244 97072 07204 18939 11375 ...

Content © Andrew Bone. All rights reserved. Created : December 22, 2013 Last updated :February 14, 2016

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