Where Marketing and Maths Meet

binary

 

Think of a number. Any number. I’ll think of 76 and i’ll talk to you about that number for a short while.

Base 10

I don’t want to dive into the detail of binary numbers without first talking briefly about Base 10, the number system you’ll be most familiar with. The number 76 can be calculated in the following way: We multiply 10 7 times and add 6.  If you cast your mind back to primary school, that’s 7 10s and 6 units. Whenever we right numbers down in English, we are implicitly talking about base 10 or the decimal system.

Any number we write down in base 10 is related to a power of 10. For example, if we have the number 13263, we have 3 units, 6 tens, 2 hundreds, 3 thousands and 1 ten thousands. We’d usually say that the number was thirteen thousand, two hundred and sixty three. If you said one ten thousand, three thousand, two hundreds, six tens and three units you’d get a strange look, but you’ll be technically correct.

As we work from right to left, each digit refers to a different power of 10.The last digit of any base 10 number represents the value up to the base; in this case, the value up to 10. This is because 10 to the power 0 is 1. The second last digit, represents how many 10′s we have: i.e. 10 x 1. The third last digit, represents how many 100′s we have: i.e. 10 x 2. In the decimal system, each digit used can be between 0 and 9. We have 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 available to us when we count.

You need two digits to represent the number 10. This might seem obvious, but it’s really important when it comes to understanding other base systems. For any base, we can use any number up to one less than the base number for any digit of that number. For base 8, we could use the numbers 0, 1, 2, 3, 4, 5, 6 and 7 to count. We can’t use 8; we have to stick with numbers one less than the base.

Binary: Base 2

If we have a binary number, each place value represents a power of 2, rather than a power of 10 in base 10 decimal notation. As we are working in base 2, we can only use a 0 or a 1 for a digit of the number.The value of zero in base 2 is 0. It is represented as 0.If we the decimal number 1, that number in base 2 is 1. It’s when we get to representing the decimal number 2 in base 2 where things get interesting. Remember that in base 2 we can only use a 0 or a 1 when we count. We would write the decimal number 2 as 10 in base 2.

That is, we have: The number 76 in binary is 1001100. There are 7 digits in this number. Computers use base 2, because an electronic current can be either off or on.The power of a computer is built on its ability to count in binary. As computers have evolved, so has their ability to count. The Sega Master System I played on with my friends in the early 1990s had an 8-bit CPU. The CPU could count using 8 bit precision: i.e. it could use 8 digits to store something whilst processing.

If that something was a number, the biggest number it could handle was 255.The decimal number 255 in binary is 11111111. If we can only use 8 digits to store the number, we can’t store the decimal number 256 as, in binary, this number is 100000000 and it would require 9 digits.

The Sega Mega Drive I finally bought myself in 1992 had a 16-bit processor. This number may be familiar to you if you used versions of Excel prior to Excel 2007. Prior to Excel 2007, each workbook could only hold 65,535 rows, a limit imposed because of the bit depth of that version of Excel. My Sega Mega Drive could handle numbers up to 65,535 but only if sign wasn’t important. If it needed to represent negative numbers, it would only be capable of dealing of numbers up to 32,767, as one bit would be required to hold the sign of the number.In 32-bit computing, the largest signed integer which can be stored is 2147483647. This number crops up in computing when you least expect it.