In learning mathematics, we regularly encounter state that are not always clear. Example of such term is the word undefined. What do we average by undefined?


The native undefined might slightly different in definition depending ~ above the context. In level language, it way something which has no wise meaning. Because that instance, throughout the time when the an adverse numbers were no yet invented, the number expression 5 – 8 has no meaning. In our time, we can say that 5 – 8 is undefined in the collection of confident integers.

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Below room some instances of the different contexts wherein the different interpretations of “undefined” have the right to be drawn.

Numbers

Square root of negative Numbers. The

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is undefined in the set of actual numbers. This means that no real number exist that once multiply the by itself, the product is same to -1. Note, however, that part operations might be unknown under some sets, yet defined in other sets. We recognize from high school math that square root of – 1 amounts to i in the collection of facility numbers.

Algebra

Division by 0. Because we carry out not recognize the prize if a number split by 0, allow us examine the quotient of numbers when divided by number close to 0. To make it simple, let us shot 1/0.


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Figure 1 – The worth of the 1/x together x ideologies 0.


As we have the right to see, as x philosophies 0 native the right, the quotient the 1/x is gaining larger and also larger. ~ above the other hand, as x viewpoints 0 from the left, 1/x is gaining smaller and smaller. Together a consequence, over there is no solitary number the 1/x approaches as x approaches 0. Therefore we have the right to say the 1/0 is undefined.

Geometry

Intersection of two lines. In Euclidean Geometry, if us talk around the intersection of 2 lines, we deserve to have three cases: intersecting lines have actually one intersection, coinciding lines have actually infinitely countless intersections, and also parallel lines have no intersection.


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Figure 2 – The three cases, in terms of intersection, 2 lines have the right to be locations on a plane.


We deserve to say that if two lines room parallel, no intersection exists. Algebraically, the equipment to the mechanism of equations the the 2 lines is their intersection. Hence, the equipment of the two equipment of direct equations that parallel lines as graphs is undefined.

Matrices

Matrices with various Sizes. If A is a 2 by 2 matrix and also B is a 3 x 3 matrix, then A + B has no an interpretation since five of the entries of matrix B have no equivalent entries in procession A. We deserve to say that the sum of matrix A and procession B is not well-defined.

Sets

Intersections that Sets. If E is the collection of even integers and also O it is in the set of odd inteers, then there is no usual value to both sets.


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Figure 3 – The Venn diagram of the intersection of even and also odd numbers.


In set theory, we contact the usual values the intersection, and in this example, the intersection is the empty set. As a consequence, we can say the the intersection of set E and also O is undefined.

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Although the word “undefined” has actually different interpretations depending top top the context, by now, friend would have actually realized the the paragraph “does not exist”, “without judicious meaning” and also “cannot be determined” are somewhat synonymous to it. If the an outcome of an procedure yields no value at every (note that 0 is a value and also is no the very same to no value), climate it is more likely the it is undefined.