![]() ![]() As pointed out when introducing the concept of the point at infinity, even innocuous functions such as f ( z) = z have singularities at infinity we now know that this is a property of every entire function that is not simply a constant. ![]() Apart from the trivial constant functions then, singularities are a fact of life, and we must learn to live with them. Hence f ( z) = a 0.Ĭonversely, the slightest deviation of an analytic function from a constant value implies that there must be at least one singularity somewhere in the infinite complex plane. If now we choose to let r approach ∞, we may conclude that for all n > 0, | a n| = 0. In fact, if | f ( z)| ≤ M for all z, then Cauchy's inequality Eq. (11.37), is Liouville's theorem: If f ( z) is analytic and bounded in the entire complex plane it is a constant. Īn immediate consequence of the inequality, Eq. This is an instance of the following fairly obvious theorem, whose proof we skip. Glide reflection, since it is the composition of a reflection that switches sense and a translation that does not, must also switch sense. (Draw a picture to illustrate this.)īy contrast, translation and rotation clearly preserve the sense of any sequence of three points. Reflection switches the sense of any sequence of three points. Imagine that we apply a reflection to ABC in order to get A′ B′ C′. Will the two ordered triangles and have the same sense? Not necessarily. Suppose A′ B′ C′ is a triangle that is congruent to ABC, with the correspondence A→ A′, B→ B′, C→ C′. If we take the same set of points but put them into a sequence in another way, say,, then we may get a sequence with the opposite sense to the sense of. Notice that the idea of sense depends on the order in which we list the points, not just on the points themselves. If we travel clockwise, then we say the sequence has clockwise sense.įor example, in Figure 4.23, has clockwise sense while has counterclockwise sense. Given three points A, B, C not lying on the same line, if we travel in a counterclockwise direction as we go in order from A to B and then to C, then we say has counterclockwise sense. There is no real rule or reason that one particular direction should be used over the other, and it is mostly a matter of custom.Walter Meyer, in Geometry and Its Applications (Second Edition), 2006 DEFINITION This can depend on area however, and some regions and countries tend to have turns or gameplay proceed in the counterclockwise direction. In certain games where players take turns, such as board games, the turn order often proceeds in a clockwise direction. In coordinate planes, angles are usually measured counterclockwise, starting from 0°, which is on the right-hand end of the coordinate plane. There are special cases however, where clockwise rotation loosens, and counterclockwise rotation tightens. This is because more people are right-handed than left-handed, and it is generally easier for a right-handed person to tighten a screw clockwise than counterclockwise. They are usually designed to be tightened when turned in a clockwise direction with respect to the person looking at the object, and loosened when turned counterclockwise. For example, many objects such as jar lids, screws, bolts, bottle caps, nuts, and more, need to be turned a specific direction to be tightened or loosened. Other usagesĪlthough the term clockwise originated with the use of clocks, it is also used in other areas. The rotation of the Earth is clockwise if viewed from above the South Pole, but counterclockwise if viewed from above the North Pole, where "above" means farther away from the center of the Earth. If there were someone else on the other side of the wall who could see the same rotation that we were looking at, from their perspective, a clockwise rotation for us would be counterclockwise for them.Īnother example is the rotation of the Earth. Another possible direction is a person standing on the other side of the wall. For example, looking straight at a clock on a wall is one possible direction. Refer to the time page to learn more about various aspects of time or about how to read an analog clock.īoth the terms clockwise and counterclockwise are defined based on looking at the clock from a specified direction. The opposite direction of motion is described with the term counterclockwise. In the image below, the arrows indicate a clockwise direction: With reference to the numbers on a clock, they are passed in order from least to greatest, starting from 12, which is the 0 position of a clock. Looking at a clock from the front or above, the hand of a clock moves right and around in a circular motion. The term clockwise describes the direction in which the hands of a clock move. Home / primary math / time / clockwise Clockwise ![]()
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