Are we born with the ability to tell which things are close and which are far away? Or, do we learn it? Two psychologists at Cornell University, Eleanor Gibson and Richard Walk, did an interesting experiment to answer this question. The kitten in the photograph above is sitting on a strip in the middle of a sheet of glass. Behind the kitten is a floor directly underneath the glass; in front of it is another floor several feet below the glass. Even when the kitten is just old enough to move about, it is afraid to move off the "cliff" to the "deep" side. Chicks and baby goats just 1 day old and human infants just old enough to crawl behave exactly the same way. What explains this phenomenon? Surprisingly, it all about inequalities! The figure above shows geometrically the difference in appearance between the two sides of the visual cliff. If point E represents the viewer's eye, the angles of the equal squares are unequal and there is a sudden change in their size at the edge of the cliff.
This difference is a visual cue that helps give a sense of depth. Looking at the numbered angles, we see that angle 3 is larger than angle 4, which we can write as angle 3 > angle 4. Comparing the sizes of the angles on either side of the cliff, we can write: angle 1 < angle 2 < angle 3 and angle 4 > angle 5 > angle 6. Below are the properties of inequalities. 1. The "Three Possibilities" Property Either a > b, a = b, or a <b. 2. The Transitive Property If a > b and b > c, then a > c. 3. The Addition Property If a > b, then a + c > b + c. 4. The Subtraction Property If a > b, then a  c > b  c. 5. The Multiplication Property If a > b and c > 0, then ac > bc. 6. The Division Property If a > b and c > 0, then a/c > b/c. 7. The Addition Theorem of Inequality If a > b and c > d, then a + c > b + d. 8. The "Whole Greater Than Part" Theorem If a > 0, b >0, and a + b = c, then c > a and c > b.
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