Big Numbers and Little Calculators

Some who’ve read What are the Odds may wonder how I calculated numbers like 7010,588 . Even Microsoft Excel and the scientific calculator bundled with Windows 7 balks at that number. Yes, I calculated it; the answer is 7.907 6 x 1019,535. I used nothing more than a standard scientific calculator. So can you.

To see how, let’s first look at scientific notation. Scientific notation is a way to work with large numbers by utilizing exponents. A number like 4.1 x 109 is nothing more than 4.1 x 1,000,000,000. 1,000,000,000 is what you get if you multiply 10 times itself 9 times, so 1,000,000,000 can also be written as 109. So, if we have a number like 4,100,000,000, we can write it as 4.1 x 109.

This not only saves space, but also time. If we can write two digits per second, it would take over 5 hours, 25 minutes to write out 7.908 6 x 1019,535. And, as I explained in the appendix of Learning the Metric System, using scientific notation makes it easier to work with large numbers.

Let’s say we’re going to divide 4.1 x 109 by 20. We can write 20 as 2 x 101 and the equation as 4.1 x 109 ÷ 2 x 101. That looks stupid because 20 is such as small number that we don’t really need to use scientific notation, and really, it is stupid. But writing the equation as 4.1 x 109 ÷ 2 x 101 let’s us better understand the next step. For when we divide, we subtract exponents. Our equation becomes 4.1 ÷ 2 x 109-1. This equals 2.05 x 108.

There are other rules. If we’re multiplying, we add exponents, so 4.1 x 109 x 20 = 4.1 x 109 x 2 x 101 = 4.1 x 2 x 109+1 = 8.2 x 1010. When we raise a number to a power, such as (3 x 108)2, we multiply the exponent by the power we’re raising the number: (3 x 108)2 = 32 x 108×2 = 9 x 1016. To take the root of a number we divide exponents: √(9 x 10)16 = √9 x 1016÷2 = 3 x 108.

Using this, we can work with large numbers through logarithms. That’s what we did in the days before pocket calculators. Logarithms convert numbers to exponents. Since we add and subtract exponents to multiply and divide them, and multiply and divide exponents to raise them to a power and to find their roots, this makes working with large numbers easier. That’s why science and technical books often had logarithm tables in the back.

In converting a number to a logarithm, we first pick a base number and ask what power we’d have to raise the base number to equal the number we’re converting? Base 10 logarithms were very popular in the day, so if we want to find the logarithm of 1,000, we ask what would we have to raise 10 to in order to equal 1,000? The answer is 1,000 = 103, so the base 10 logarithm (we’d write that as log10) of 1,000 is 3. If a number isn’t divisible by the base, things can get messy. The logarithm of 9 is about 0.954 2. But let’s say we want to multiply 9 by 1,000. With logarithms, 9 x 1,000 = log10(9) + log10(1,000) = 0.954 2 + 3 = 3.954 2.

You notice the number to the left of the decimal place, the characteristic, is a multiple of 10, since we’re dealing with base 10 logarithms. The number to the right of the decimal place, the mantissa, remains the same as log10(9). That makes sense, because 9 x 1,000 = 9,000.

Knowing this, we can extend the number range of a pocket scientific calculator. Back in the day, log10 was so common that it was simply called log, and that’s how it’s shown on most pocket scientific calculators. Log(70) x 10,588 = 1.845 1 x 10,588 = 19,535.898 1. The characteristic 19,535 tells us the power of 10, so we jot that down and subtract it: 19,535.898 1 – 19,535 = 0.898 1. Now we raise 10 to this power, usually shown by a 10x key: 10x(0.898 1) = 7.907 6. Remember that 19,535 we jotted down? That’s the same as 1019,535, so that means 7010,588 = 7.907 6 x 1019,535.

Be aware that this only works with base 10 logarithms. There’s another logarithm key on scientific calculators, ln, that stands for natural logarithm, which has a base of about 2.718 3, also known as e. Why in the world is there a logarithm based on e? Because it shows up in hyperbolic curves and all sorts of things. The value e, like π, is a transcendental number, which means a number that cannot be calculated by an algebraic equation. But thinking about this doesn’t count as Transcendental Meditation.

Cough. Anyway, the important thing is that since natural logarithms are based on e, we cannot use the characteristic as a shortcut to extend scientific calculator range. That would give us the wrong result.

It’s also why we have to be careful with computer languages, such as BASIC and C. In both, the log() function returns the natural, not base 10, logarithm. Fortunately, in Microsoft Excel, the log() function returns the base 10 logarithm, and LibreOffice Calc’s log() function will let you by with it. In Calc, the log() function wants the number and the base and defaults to base 10; the proper base 10 function in Calc is log10().

This method is a nod back to the days when science and technical textbooks always had a table of logarithms in the back. More exacting work called for books of logarithms. Once numbers were converted to logarithms, they could be processed by hand or by mechanical adding machines.

Slide rules operate by the same principle, and could give an answer as accurate as what we could get with the short tables in the back of the textbooks, so those were often used instead. With the advent of cheap pocket calculators, both slide rules and textbooks with logarithm tables have become a thing of the past.

Logarithms are still a handy thing to know. It helps when the numbers are so large that your calculator goes “TILT.”