Harnessing direct solar energy – a progress report
Box 1 | Eliminating the zeroes
Very large numbers are cumbersome to write because of all the zeroes. They are hard to compare – a number with 20 zeroes at the end looks about the same as one with 19, yet it is ten times larger. Large numbers are also hard to name. Once you start talking about a thousand million million million, not only is it long-winded, it’s impossible to keep track and to imagine how the number would look when written down.
The quickest way around all these problems is to use the simple mathematical method of writing very large or very small numbers, a method used by all scientists too. This form of writing numbers essentially relies on counting zeroes and is called using powers of 10 or exponents. Everyone knows what squaring a number means – it is multiplying a number by itself. Thus, 3 squared is 9. The mathematical way of writing this is 32 = 9. The little number 2 above the 3 is called an exponent. 32 can also be spoken of as three to the power of two. The square of 10 is 100; so 102 = 100. You can say this as ‘ten to the power of two’, or just ‘ten to the two’. If the exponent is 3, then the number is multiplied by itself once again. So, 33 = 3 × 3 × 3 = 27, and 103 = 10 × 10 × 10 = 1000. You may have noticed that when dealing in powers of 10, the exponent is the same as the number of zeroes in the number, as this table shows:
101 = 10
102 = 100 (one hundred)
103 = 1000 (one thousand)
104 = 10,000 (ten thousand)
105 = 100,000 (one hundred thousand)
106 = 1,000,000 (one million)
107 = 10,000,000 (ten million)
108 = 100,000,000 (one hundred million)
109 = 1,000,000,000 (one thousand million or one billion in some countries).
Of course, not all large numbers are always full of zeroes. Suppose we have the number 2,720,000 (two million, seven hundred and twenty thousand). You could call this 2.72 million, or just multiply 2.72 by the appropriate power of 10 for one million. As the million has six zeroes, the exponent would be 6, and the number would be expressed as 2.72 × 106
Rounding numbers up or down
If a number is more precise, such as 2,738,458, we can either express it as it is, express it using an exponential but keeping all the figures (2.738458 × 106), or round it up or down. Whether we round it or not depends on whether the number is describing a precisely quantifiable thing. For example, if the figure is a calculated average (eg, the number of bacteria found on each square metre of rainforest topsoil), then rounding it is valid. We cannot be sure of the number down to the last bacterium, and even as we are estimating it, the bacterial population is changing. In this case, we would approximate the number as 2.74 × 106 or 2.7 × 106, or even just 3 × 106.
Very small numbers
Very small numbers are also conveniently expressed using powers of 10. In this case, a minus sign is written before the exponent. For example, one-thousandth is 10-3, one-millionth is 10-6, which we say as ‘ten to the minus six’. The exponents represent the zeroes that would come after the decimal point, and the one that comes before it. A concept like 0.0000345 millimetre is hard to grasp and not easily compared with 0.00000345 millimetre (who bothers to count the zeroes?), but the difference is more obvious when they are written as 3.45 × 10-5 millimetre and 3.45 × 10-6 millimetre respectively as there is a clear difference between the exponents 5 and 6. The first number is 10 times less small (which means 10 times larger) that the second.
You can practise using power-of-ten notation either with pure numbers or in combination with units. For example, using exponents, express:
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12 kilometres in metres
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1,234,567,890
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one hundred thousandth of a gram
- 0.00023
Boxes
Box 2. Driving on a sunbeam
Box 3. Light to electricity
Box 4. The Big Dish
Box 5. Chemical fuels from the sun
Posted February 1997.