NUMBERS IN SCIENTIFIC NOTATION. ERROR CONTROL 6

Test yourself

Express the following numbers in scientific notation: a)

The numbers 3.845 · 1015 and 9.8 · 10–11 are written in scientific notation because: advantageS of thiS notation

— They are made up of two factors: a decimal number and a power of 10.

— The decimal number is greater than or equal to 1 and less than 10.

— The power of 10 has an integer exponent. The first, 3.845 · 1015 = 3 845 000 000 000 000, is a ‘large’ number. The second, 9.8 · 10–11 = 0.000000000098, is a ‘small’ number.

As its name suggests, this form of notation is intended to be used in contexts that require precision, not in everyday language. Can you imagine saying something like this?

— How many children do you have? 8 5 · 100

— How many students are there in your school? 8 6.74 · 102

— Do you have any pins with a width of 2.5 · 10–4 m?

Obviously not. Clearly, this form of notation is not for use in everyday life. However, this form of expression is very useful for handling very large or very small approximate quantities, since:

• You can see the ‘size’ of the number at a glance. It can be seen in the second factor and it is given by the exponent of 10.

• We can see how accurate the quantity is. The more significant figures given in the first factor, the more accurate the number is.

Interesting fact

You might come across an explanation like this:

If an integer ends in one or more zeros, you can easily determine the number of significant figures it has by expressing it in scientific notation. However, another way to indicate that these zeros are significant figures is to put a decimal point at the end of the number. Like this:

3 200 would have 2 significant figures.

3 200. would have 4 significant figures. This rule is not approved by the scientific community (and we will not use it in this book), but if the teacher thinks it might help to clarify things, why not use it?

For example, we can see that 7.6 · 108 and 7.603 · 108 are approximately equal (‘very similar sizes’). However, the second one is more accurate, as it has four significant figures, whereas the first one has only two.

Operations with numbers given in scientific notation

We can easily do operations in scientific notation with a calculator. Remember that to write a number in scientific notation you use the key. For example, 7.6 · if we wanted to do this ‘by hand’, we would have to take great care.

Product And Quotient

We do the operations involving the decimal components separately from the powers of 10. Then, we adjust the result so that it is expressed correctly using scientific notation. For example:

Note that in the quotient 3.25 : 4.6 we have used four significant figures. If the context of the problem does not suggest the right number of significant figures to use, we have to make the decision subjectively.

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Operations with numbers in scientific notation.

Addition And Subtraction

We have to prepare the addends so that they have the same power of base 10. That way we can use it as a common factor. Then, once the addition has been performed, the result is readjusted. For example:

Attention

The information given by the following numbers is different: 2.5 · 102; 2.50 · 102; 2.500 · 102

The zeros added to the end of the decimal number are used to indicate the number of figures that are accurate.

Problem solved

We are told that 1 400 million people live in China.

a) Express this quantity in scientific notation.

b) Is it an exact or approximate amount?

c) Give a limit of absolute error based on how the figure was given.

d) Give a limit of relative error.

Controlling error in a number in scientific notation

If we are told that ‘there are 2 500 bags of flour in this warehouse’, this could be an approximate quantity. If so, the statement might only mean that there are approximately 25 hundreds of bags, with an error of less than 50 bags. Or it might mean there are 250 tens, with an error of less than 5 bags.

If we use scientific notation, the expression is unambiguous: 2.5 · 102 means that there are only two significant figures. If there are three, we write 2.50 · 102.

a) 1 400 million = 1.4 · 109 inhabitants b) Obviously, this is an approximate number, as it is impossible to calculalte such a large, dispersed and changeable amount accurately. c) and d) When we are told ‘1 400 million people’, we expect the first two figures to be accurate. However, it is possible that one of the zeros that comes after them is also accurate.

If only the two first figures in the measurement are accurate:

Measurement: 14 hundred millions of people.

Absolute error < 0.5 hundred millions = 50 000 000

Relative error < 0.5/14 < 0.036 = 3.6 %

If the first zero in the measurement is accurate:

Measurement: 140 ten millions of people. In this case, we can express it using scientific notification as follows: 1.40 · 109. Notice how the fact that the zero comes after the decimal point means that it is an accurate digit.

Absolute error < 0.5 ten millions = 5 000 000

Relative error < 0.5/140 < 0.0036 = 0.36 %

Think and practise

1 Calculate and check your answers with a calculator: a) (6.4 · 105) · (5.2 · 10– 6) b) (2.52 · 104) : (4 · 10– 6) c) 7.92 · 106 + 3.58 · 107 a) Express it using scientific notation. b) Express it in centimetres to two significant figures. c) Express it in centimetres to four significant figures. d) Give a limit of the absolute and relative errors for the three cases mentioned above. log2 8 is read ‘base-2 logarithm of 8’. Similarly, we can say: log5 125 = 3 because 53 = 125 log5 5 2 1 = because 51/2 = 5 log10 1 000 000 = 6 because 106 = 1 000 000 log10 0.0001 = – 4 because 10– 4 = 1/104 = 0.0001

2 The distance from the Earth to the Sun is 149 000 000 km.

The equality 23 = 8 can also be written: log2 8 = 3.

The exponent to which base a must be raised to get P (a > 0 y a ≠ 1)) is called base a logarithm of a number P > 0 and written as log a P.

Properties of logarithms

1. Two simple logarithms

The logarithm of the base is 1. The logarithm of 1 is 0 in any base.

2. Product and quotient

The logarithm of a product is the sum of the logarithms of the factors. The logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.

3.

The logarithm of a power (P k or PP / n n 1 = ) is equal to the exponent multiplied by the logarithm of the base of the power.

4. Base change

If we know how to calculate base-a logarithms, we can use the formula above to calculate logarithms with any base, b.

Interesting fact

At this level: log means decimal logarithm (base 10) and ln means Napierian logarithm (base e). However, in more advanced mathematics books log is used to refer to Napierian logarithms, since at these levels this type of logarithm is used almost exclusively.

Decimal logarithms

Base-10 logarithms are called decimal logarithms. For a long time, they were the most widely used logarithms. This is why we just write log, without the base. For example, log 10 = 1, log 100 = 2, log 1 000 = 3, log 0.001 = – 4.

And log 587 = 2. ... since 587 is greater than 100 but less than 1 000. There is a specific button for these logarithms on calculators. But in modern calculators, you have to press the SHIFT button to access this function: log 200 8 s 200 = 2.301029996

Napierian logarithms

Remember the number e? Its value is 2.71828… and at the beginning of this unit we explained that it is used to describe growth in plant or animal populations, radioactive decay, and catenaries.

Logarithms with base e are called Napierian logarithms and they are written ln (in other words, loge x = ln x). On calculators there is a button, , for accessing these logarithms.

Problems solved

1 Give the value of these logarithms, putting the numbers in the form of powers: a) log6 1 296 b) log2 0.125 a) 1 296 = 64. Therefore, log6 1 296 = 4. b) 0.125 = 1 000 125 8 1 2 1 3 == = 2–3 log 5 = 0.69897… log 50 = 1.69897… log 500 = 2.69897… log 5 000 = 3.69897…

2 With the calculator, find log 5, log 50, log 500 and log 5 000.

All of them have the same decimal part. Why?

Therefore, log2 0.125 = –3.

T hey have the same decimal part because all of them are the following type: log10 (5 · 10n) = log10 5 + n log10 10 = n + log10 5 log10 5 is the decimal part of all of them, with an integer, n, added in each case.

Think and practise anayaeducacion.es Practise calculating logarithms.

1 Use the definition above to find these logarithms: a) log5 125 b) log5 0.04 d) log2 0.0625 c) log2 128 e) log a 1 f ) log10 0.0001 g) log2 / 12` j h) log3 (1/3) log3 9 5

2 Work out the base of the following logarithms: a) log a 10 000 = 2 b) log b 216 = 3 c) log c 125 = 3 d) log d 3 = 2 1

3 Use the calculator to find log 7 and log 70. Explain why both have the same decimal part.

The importance of decimal logarithms

Look at these two numbers:

= 6 748 B = 67.48 = A/100 log B = log (A/100) = log A – log 100 = = log A – 2

This means that both logarithms have the same decimal part. In logarithm tables we would look for the decimal part of the logarithm of 6 748 and then add the integer part as appropriate, depending on whether it was 6 748; 67.48 or 67 480 000.

A bit of history

Several centuries before the emergence of calculators, logarithms were invented to cope with the enormous operations that had to be done by hand. This was done by converting products and quotients into additions and subtractions (which are operations that are much easier to work with than the others). To do this, people had to use enormous tables (in very thick books), which contained the exact or approximate values of the decimal logarithms of the factors. And what about Napierian logarithms? What role did they play? The number e arose naturally in the laborious process used to obtain the decimal logarithms for many numbers by hand, and therefore, so did Napierian logarithms. In other words, Napierian logarithms were the instrument used to obtain the values of decimal logarithms, which were the ones needed in practice. They were called natural logarithms, while decimal logarithms were known as common logarithms. Now that we have calculators, why do we need logarithms? Well, it is mostly a question of culture, but also because we find them in algebraic simplifications (for example, for solving some types of equations) and functional expressions in the world of science and technology.

Logarithms on a calculator

As you already know, the calculators we use today have three keys for calculating logarithms: , j and . The first two are easy to find. The third (decimal logarithm), however, is more complicated. It is the second function of the button, so to use it, we have to press s .

To find a logarithm with a base other than 10 and e, newer calculators also include the j button. However, sometimes it may be preferable to use property 4, base change (introduced in page 50), rather than this button. To do this, it is better to use the ln key, since that way you do not need to use the shift button. Let’s look at an example of the two ways of doing it:

Problem solved

Use a calculator to find the following logarithms both ways: using the j key and by changing the base.

Think and practise

4 Use the j and keys of your calculator to calculate these logarithms using both methods described above: