DimensionsDate: 02/15/97 at 11:35:50 From: Suryadie Gemilang Subject: One Dimension I have a question: Does 1-D space exist? How about 5-D space? Thanks. Date: 02/16/97 at 02:28:50 From: Doctor Mike Subject: Re: One Dimension Hello Suryadie, Yes, such spaces exist. I will give examples having to do with temperature. You know that water freezes at 0 degrees Celsius and boils at 100 degrees Celsius. All other temperatures are somewhere between these two, above them, or below them. Temperatures are often shown on a thermometer that is a straight line, with colder at one end and hotter at the other end. This is a one-dimensional way to measure the world. Only one thing is measured. This value goes up or goes down. Nothing else is considered. A more complicated way of viewing the world is to consider physical position in space, temperature at that point and time. You could represent this as (X,Y,Z,C,T) where (X,Y,Z) are the usual space coordinates, C is Celsius temperature and T is time. Not all quintuples are possible. For instance, at a particular place and time, there is going to be only one true temperature. In three- dimensional space, it is common to describe a smooth surface by giving Z = F(X,Y) as a function of X and Y. Similarly, in five-dimensional space you could try to describe temperature by a function C = F(X,Y,Z,T) as a function of position and time. Weather forecasters try to do this all the time. People who listen to the weather forecasts want to know how cold or warm it will be at various times during the day tomorrow. Of course, the meteorologist can only give very approximate information about this. But such a function exists in thought. These are specific suggestions of why one might want to think of and use a 1-dimensional space or a 5-dimensional space. Actually, mathematicians have generalized this quite a bit. Because we know that there are these real-world situations where more than three quantities need to be measured, we have developed the idea of an N-dimensional space for any positive number N. The term Finite Dimensional Vector Space is used for this. This terminology comes from the fact that a triple such as (1,2,3) can not only be used to measure a position, but can also be used to indicate a direction (like 1 mile East, 2 miles North, and 3 miles Up for the direction, or vector, from an airport control tower to an airplane). You will encounter these ideas more and more if you continue your studies of mathematics. I hope this helps for now. -Doctor Mike, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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