The binary number system

Maria and Lina usually play soccer together after school. They live on the opposite sides of the street. They both have a lamp in their kitchen windows. Maria can see Lina's lamp and Lina can see Maria's. They agreed that if the lamp is lit it means, see you at the soccer field.

Sometimes they would rather go swimming than play soccer. Lina suggested they should each put another lamp in their kitchen windows which will mean, see you at the beach. This is important, have you thought about this? If we have a third lamp in our kitchen windows it could mean would you like to come to my house to watch a movie? We don't need another lamp, two is enough.

If both lamps are lit it means come to watch a movie. Clever. We already have three lamps, or rather, we have three messages by using just two lamps. If we get another lamp, how many messages could we send? Let's try. I'll write it like this.

Message one - first lamp - soccer, that's obvious. Message two - second lamp - the beach. Message three - both first and second lamp - watch a movie as we agreed. Message four - only the third lamp - what should that mean? Study?

A bit boring, but we are the best in our class. Yes. Check this out. Three lamps can mean a lot. Five - first and last lamps.

Six - the last two lamps. Seven - all of them. Seven different combinations with three lamps. All of them can be turned off as well. If all the lamps are off, that means nothing today.

Good. That is eight combinations, three lamps - eight messages. Maria and Lena have just discovered the binary number system. The binary number system, or base two, has two numbers instead of ten like we are used to. Even though Maria and Lina are using on and off instead of numbers, it works just the same.

The number of messages that Lina and Maria can send with their binary system is two to the power of the number of lamps they have. Two to the power of three is eight. If they get a fourth lamp, they can send 16 messages. With five lamps, they have 32 options. For every new lamp, the number of possible combinations is doubled.

Look at it this way. When they added the third lamp, they could first let it be off and still do the same combinations with the other two lamps. They then light the new lamp and do all the old combinations again. Compare the upper and lower half of the table, they are the same aside from the lamp on the left. This way of coding information with the number based two - on or off - is more common than you might think.

Electronic parts in your computer and telephone handle all the information this way. But we usually write it like this, with ones and zeroes. Just like with regular decimal number systems, the binary number system is a positional system. Where every position or place, has a value. The position on the right is worth one, like always.

But where we usually have tens, we have twos and to then where we used to have hundreds, we have fours instead. Look at the row that means five, it has one four, zero twos, and one one. Four plus one, is five. Here's a trickier example, a five digit binary number, the left most positional is 16, next goes eight then four, two, and one. The binary number 10100 means one 16, zero eights, one four, zero twos, and zero ones.

16 plus zero, plus four, plus zero, plus zero, equals 20. Number bases are connected to both positional value and exponents. If you understand those concepts, this table might help you to understand number bases. Look here, every positional value can be written as an exponent, where the base of the exponent is the number base, and the number base is two since we are using the binary number system. For the power, we use the position and start with the right most zero.

Number base to the power of position equals positional value. You can now get the binary number's value by multiplying the positional value by the number in that position, and then summing everything up, like this. One times 16. We write 16 as two to the power of four plus zero eights, five, one, four, and we write fours as two to the power of two, plus zero twos, plus zero ones. This equals 20.

Look at how units are written as two to the power of zero. Any number to the power of zero equals one. This looks a bit complicated, but if you write the number like this, it's easier to understand the relation to the decimal number system. For comparison, we can write decimal numbers in the same way. Every positional value is written as an exponent.

In this exponential expression, the base is ten, same as the base number, and the power is the position. There are ten thousands, zero thousands, and zero hundreds. Then come the tens, there are two of them, and then zero ones. The sum is 20, so one zero, one zero zero in the binary system is the same as 20 in the decimal system. But this looks very confusing, so in order to distinguish the number systems, we write the base number after the number, otherwise misunderstanding might happen.

However, misunderstanding can be fun, as well. Lena's new tee-shirt, for example. Pay attention to the image. Take your time to read and understand.