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how to teach your child numbers arithmetic mathematics

orders of magnitude, indices (powers) and logarithms

orders of magnitude

When you write a number such as 10¹, you are writing “once times ten”, or one lot of ten, while 10² means “ten times ten” or 10 x 10, while 10³ means 10 x10 x10. The little 1 or two in the air is called the exponent, or index, and shows how many times a number is multiplied by itself.

As we go from 10¹, 10², 10³, and so on, each jump is bigger than the one that went before. Thus the line on a graph rises at an ever steeper angle. This is called a power series or a geometric series. Ordinary counting: 1, 2, 3, 4 etc, is called a linear series. As usual, there will be many more delights to come. The jump between one power of ten and the next is often called an order of magnitude.

So if two numbers differ by one order of magnitude, then one number is about ten times bigger than the other. If the numbers differ by two orders of magnitude, one number is about a hundred times bigger than the other. This is, of course, a way of communicating approximate differences in scale. It does not pretend to precision.

Communication is a matter of purpose.You choose the degree of accuracy you want according to your purpose. When trying to fit a piston into a cylinder, you will need much more precision than when getting an idea of the relative size of planets.

Of course if you work in a different base from base 10, the number of orders of magnitude will also change.

watch your units!

Orders of magnitude will become confused if you do not work to a standard unit of measurement. An order of magnitude expressed in yards (distance) will not give you the same results as an order of magnitude expressed in miles, or in time or temperature.

10⁷ could be a million miles or a million inches. Keep the units you are using consistent if you intend to compare different orders of magnitude.

Here is a rather droll table with selected list of different orders of magnitude, in terms of length. As you can see, the units are not consistent. This has resulted in strange juxtapositions such as the moon’s diameter being an order of magnitude 10³, while Mount Everest is listed as of order 10⁴!

A normal way of expressing an order, using as the example the height of Mount Everest in feet, would be 2.9029 x 10⁴. That means moving the decimal point four places to the right to become 29,029 feet.

The Scale of the Universe is an interesting online depiction of the size of the universe and what is within it, from neutrons to the known universe.

indices

Indices are written above and top the right of the number (or expression) to which they are applied. For example, 2² (said as ‘two to the power of two’) means two multiplied by (times) two. That is 2 x 2 which, of course, is four (4). 2³ means two times two times two and equals eight (2 x 2 x 2 = 8).

As discussed a little in multiplication and still more counting, our widely used system works in base ten. So using indices with 10, 10² is ten times ten - 10 x 10 is a hundred (100), and 10⁶ is a million (1,000,000). Now notice with tens, the number of the index matches the number of zeroes after the one.

Moving on to our normally misbehaved numbers one and zero, 10¹ is in fact ten (10) and 10⁰ is one (1). As you can see, the rule for indices is continuing to work. 10⁰ has no zeroes after the one. But this goes further still, 10^-1 is ·1, 10^-2 is ·01 and 10^-3 is ·001.

Here is a graph plotting y = 2n, y = 3n and y = 4n	The points plotted are:
	n	2n	3n	4n
	-4	1/16
	-3	1/8	1/27
	-2	1/4	1/9	1/16
	-1	1/2	1/3	1/4
	0	1	1	1
	1	2	3	4
	2	4	9	16
	3	8	27
	4	16
Note: y =2-n is the same as saying 1/2n.

As you can see with this graph, it is not just 10⁰ that equals 1.
2⁰, 3⁰ and 4⁰ also equal 1. In fact, any number to the 0th power equals 1.

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using indices with multiplication and division

Multiplication with indices can be done easily only with numbers of the same base, for instance base 10, or base 2. Although the numbers as a whole are multiplied together, it is done by adding together the indices. In general, such a multiplication is written as y^a × y^b = y^a+b.

An example multiplication with indices:

• 2²× 2⁴ = 2 x 2    x    2 x 2 x 2 x 2
  which, obviously equals 2 x 2 x 2 x 2 x 2 x 2
  and which is, therefore, 2⁶.
  Thus, 2²× 2⁴ = 2⁶ [2^{2 + 4} = 2⁶].
  Of course, 2² = 4, and 2⁴ = 16,
  and 4 x 16 = 64. 2⁶ = 64.

So, you can see that adding indices is equivalent to multiplication.

No, this won’t work for multiplying 3⁶ by 2⁴ - more on this later on.

But will work with divisions. So 2⁶ ÷ 2² is 2⁴ [2^{6 - 2} = 2⁴].

And here’s a fairly complicated multiplication sum involving indices:
2³ x 2^-2 = 8 x ¼
             = 2
             = 2¹
So 2³ x 2^-2 = 2¹.
If you look at just the indices in this sum, they make the subtraction 3 - 2 = 1.

Here is a summary for base 2,

of the relationships between indices (or exponents) {1},
indexed numbers (or numbers to different powers) {2}
and the numbers equivalent to the indexed numbers (resolved numbers) {3, 4}.

And a similar summary table for numbers to base 10:

Of course, additions and subtractions are simpler to carry out than multiplications and divisions.

On this basis is generated the idea of logarithms, although we will have to go a bit further first by looking at fractional indices.

the marvels of logarithms

In these days of calculators and computers, logarithms are sort of out of date and becoming part of history. However, they remain useful for understanding mathematics and how various parts of maths fit together. They may even be useful for exams!

Logarithms are a development of indices.

Logarithms are numbers that are known in algebra as exponents (exponents means the same as indices). Exponents can be used as a shortcut for multiplication that are used to express repeated multiplications of a single number. For instance, 10⁵ is another way of writing 10 x 10 x 10 x 10 x 10 x 10.

With some graphs, because geometric series leap upward in an inconvenient manner, we must modify the Y scale on the graph to accommodate the rapidly expanding power numbers.

Click for complete, unmarked, four-figure log tables to print out. [Opens in new tab/window.]

the inventor of logarithms and log tables

Previously to John Napier, complicated trigonometric tables and tiresome calculations were used in astronomical studies: calculating planetary orbits and distances, predicting astronomical events.

John Napier, 1550-1617, spent over twenty years to simplify some of these tedious calculations and computations. Finally, he decided to use the indices of numbers, adding or subtracting the indices, rather than multiplying or dividing the actual large numbers.

Napier used an obscure base for his logarithms, and Henry Briggs,1556 – 1630, Professor of Geometry at Oxford, suggested that base 10 would more useful. He constructed such log tables after visiting Napier.

what is a logarithm?

You remember indices, and how 10² is another way of writing 100, while 10⁶ is another way of writing 1,000,000? Well, the index, the little 2 in the air, or the little 6, is the logarithm of 100, or of 1,000,000.

A logarithm is the power to which a number can be raised to equal the number. So for 100, its logarithm = 2; for 1000, its log = 3. Logs to base 10 are numbers expressed as a power of 10. So that 100 = 10 squared, so log 100 is 2.

Logarithms can be calculated for different bases, the ‘common logarithm’ being logs to base ten: log₁₀. For example, 10³ = 1000, so log₁₀1000 = 3. [Note that ‘common logarithms’ are usually written as, say, log 1000.]

Here are some examples of logs to other bases, the log being highlighted in pink, while the base is coloured in blue:

• 2⁵ = 32, so log₂32 = 5
• 3³ = 27, so log₃27 = 3
• 5⁸ = 390,625, so log₅390,625 = 8

fractional indices, or fractional powers

Right, we’ve looked at indices that are whole numbers, like 2², or two squared, which can also be written as 2 x 2. And the graph, a way above in the Indices section, illustrates how 2^-1 (or 2ⁿ, where n = -1) is ½. But what about 2^½?

When 2^½ is squared, that is multiplied by itself, 2^½ x 2^½,
you can write this as (2^½)² .
(2^½)² is the same as 2^{½ x 2} or 2¹, and that’s 2.
So 2^½ means the square root of 2.

Repeating for clarity, 2^½ x 2^½ = 2¹.
That is, adding indices (powers) is still working out (is consistent).

Notice that, at each stage, the numbers must have consistent meanings. This is a great deal of how the structure of mathematics has been built. The objective is expressed in the concept of consistency. This objective amounts to a prime directive to avoid contradictions.

Mathematics is an idealised world, it is not the real world. It rests on various make-believes, such as that two different things can be entirely the same. But the real world is not like that. So mathematics is a sort of formalised game, sometimes with arbitrary rules that can often be very useful as a model of the world.

why are logarithms useful?

Logs are very useful for making complicated sums more simple. Instead of doing ginormous long multiplications or divisions, for instance, all you have to do is add (or subtraction for division) the indices of the numbers involved, and then convert the result back into a number.

Simple you say, but I’ve never done this before.
No problem, we’ll walk you through a very simple example to set you on your way.

First look at these indices for numbers to base 2. We have included indices intermediate to the whole numbers shown in the earlier table, and we show how they are resolved to decimal numbers:

And here below is a similar table for indices of numbers to base 10. Again, we have included indices intermediate to the whole numbers shown in the equivalent earlier table, and we show how they are resolved to decimal numbers

And now follows a more extended table of indices of numbers to base 10, again resolved to decimal numbers. We will be referring shortly to the columns highlighted in pink and crimson.

Now for that simple example of using logs.

Suppose we want to multiply 4 and 5, and it’s too difficult to do easily (imagine that 4 and 5 are horribly complicated numbers).

Using the table above (which is rather like a shrunk version of log tables), we see that log 4 is ·6021 and log 5 is ·6990. Adding these indices is another way of multiplying the numbers together.
Thus, ·6021 + ·6990 =1·3011.

Now, looking at the table above, the index, or log, ·3010 resolves to 2 ( ·3011 is a rounding error).
But the resulting index is 1·3011. As 10¹ equals 10 and 10² equals 100, then 10^1·3011 must be between 10 and 100.

Thus the answer is 20, not 2. So 4 x 5 = 20.

Notice carefully that we are dealing with base 10 indices, therefore each whole number in the index represents an order of magnitude. That is, 1 is 10 upwards, 2 is 100 upwards, 3 is 1,000 upwards and so on.

And here’s a graph of y = log x. This provides a picture of numbers and their logarithms (logs). The arrow shows that the logarithm of 2 is just over ·3 (·3010). This is another way to find the logs of numbers, or to resolve a log into a number. That latter process is also known as finding the ‘anti-log’.

An example of a logarithmic scale encountered in everyday life is the decibel [dB] scale, which measures the noise level of different sounds.

To continue, also see writing down logarithms.

end notes

1. In the United States of America, the spelling of metre, and all words that have metre at the end (the suffix), is altered to meter. Thus you can encounter the Americanisms of attometer, picometer, nanometer etc.
   
   
2. Indices is the plural of the word index. So indices means “more than one index”.
   
   
3. Rounding errors
   In the text above, we are dealing with logarithms to four figures. Logarithms can be calculated to much greater accuracy than this. When we do a calculation to an approximation (as we always must), then differences will creep in. There is no such thing as complete accuracy in the real world. Remember regularly, to draw the learner’s attention to the idealised/fictional (but useful) nature of mathematics relative to the real world.
   
   The real world must always dominate, not the ‘theory’. Did you run 100 metres, or 100 metres plus 1 millimetre, or even minus 2 millimetres? Is the time ‘exactly’ 2 o’clock, or has the clock ticked again?
   
   
4. Note that logs (or logarithms) to base 10 are normally written as log. Logs to other bases are written as log₂ for logs to base 2, or log₅ for logs to base 5, and so on. It is important that the number indicating the base of the logarithm is clearly written as smaller and below the word ‘log’, otherwise log₂ might be confused for log 2, a very different animal.

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