Chicago's streets are arranged in a typical metropolitan grid fashion, but they also feature a consistency in numbering that makes it nearly impossible to get lost in the city once you commit it to memory. Observe these rules:
• For east-west streets, zero occurs at Madison Street.
• For north-south streets, zero occurs at State Street.
• Thus, any north-south street will always have a constant east-west coordinate.
• Likewise, any east-west street will always have a constant north-south coordinate.
• 800 units on the grid system equals one mile.
• On the The South Side of the city, the numbering of the east-west streets corresponds to the grid system number divided by 100; there are 8 major streets per mile. Thus, 42nd Street is at 4200 South.
Knowing these rules, one can easily find their way around the city, and know the distance from their destination by noting the coordinate numbers below the street name on the street signs at major intersections. Signage at stops on the El also includes the complete set of coordinates for your location.

Applying the Pythagorean Theorem will give one the distance between any two points. Note the following story problem:

You are at the corner of State and Lake. How far away is Sheffield and Belmont?

Solution:
State is 0 E/W and Lake is 200 N. Sheffield is 1000W and Belmont is 3200N.
Total distance north = 200 + 3200 = 3400
Total distance west: 0 + 1000 = 1000
Apply Pythagoras, and we see that the two points are roughly 5.5 miles apart.

Of course, you probably won't be walking through buildings and other solid obstructions. You can determine the rough walking distance by a quick caculation in your head of the north-south distance plus the east-west distance. This is an excellent way of learning how to divide by eight. Living on the South Side, I generally use the north-south distances to calculate roughly how far I need to go: my favorite bar is at 2900 North, and my own address is 5100 South, which conveniently sums to 8000, thus I am about 10 miles south of my favorite location to become intoxicated.

Let the reader make careful note of an important exception, however: diagonal streets. There are a few of these in the city, most notably Clark St., Milwaukee Ave., and Elston St. You're on your own if you care to memorize the grid coordinates of major intersections involving those streets.

If you move to the city, you'll find that with a little effort you'll rapidly commit to memory the grid numbers of major streets: for example, going north, Madison St. is 0, Chicago is 800, North Ave. is 1600, Fullerton Pkwy. 2400, Belmont Ave. 3200, Irving Park 4000, and so on.

The real value of the grid system lies in knowing that a N/S or E/W address is at a fixed point. For example, living at 5100 South, thus I know that I am more or less due east from the housing projects that lie at 5130 South State. (There are more useful applications, but you get the idea.)

Chicago's unusually organized grid was the brainchild of Daniel Burnham (1846-1912), an eminent architect and urban planner. The grid arose out of the 1909 Plan for Chicago, which was developed in response to the city's rebuilding efforts after the Great Chicago Fire of 1871. To understand the place of the grid system in a larger urban planning context, research about the 1909 Plan would be an excellent resource.

References:
http://www.thursdayarchitects.com/Texts/bigplans.html
http://www.chipublib.org/004chicago/timeline/plan.html

Though the Chicago grid system is regular for the most part, there are a few quirks to it. The first has to do with street numberings on the South Side of the City. Due to (from what I've heard) a mistake during the original re-numbering of the city, the "800 to a mile" rule does not apply between Madison Ave and 31st St. Instead, the streets have been "condensed" so that Roosevelt Rd (1200 S) is one mile south of Madison, Cermak Rd or 22nd St (2200 S) is two miles south of Madison, and 31st St. (3100 S) is three miles south of Madison. Within each mile, streets are spaced relatively evenly, so that Harrison St (600 S) is a half-mile south of Madison, and 17th St (1700 S) is a half-mile south of Roosevelt.

The other confusion about Chicago's grid system has to do with the full names of numbered streets. In certain western suburbs, north-south streets are numbered by their hundred, like streets on Chicago's south side. This can be confusing, until you notice that the suburbs refer to their north-south roadways as "Avenue" (as in the terminus of the Douglas Branch of the Blue Line train at 54th Avenue and Cermak, or 5400 W/2200 S), while the east-west roadways in the city are named "Street" (like the 31st Street, which is 3100 S). If a numbered street has "Place" in its name, it refers to an east-west roadway that runs parallel to the namesake Street, usually about a half-block north or south. So, 50th St is 5000 S while 50th Pl is 5032 S. A numbered "Drive" is the continuation of that same "Street" inside a park (though at times, the Drives may be a half-block off from their namesake "Street"), but other named "Drives" ("King" and "Lake Shore", to give two well-known examples), can also run north-south.

What's the moral of this story? Know the hundreds of the streets you're going to and traveling on. Trying to get from 50th Pl. and Halsted to Broadway and Carmen sounds like a pain, but when you think of it as going from 5032 S/800 W to 1200 W/5100 N, things get a lot easier.

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