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The absolute value |z| = Va² +b² represents the distance of z from 0, and more generally, u – v| represents the distance between u and v. When combined with the distributive law, u(v – w) = uv – uw, a geometric property of multiplication comes to light. 1.2.4 Deduce, from the distributive law and multiplicative absolute value, that |uv – uw| : lu||v – w|. Explain why this says that multiplication of the whole plane of complex numbers by u multiplies all distances by |u|.

The absolute value |z| = Va² +b² represents the distance of z from 0, and more generally, u – v| represents the distance between u and v. When combined with the distributive law, u(v – w) = uv – uw, a geometric property of multiplication comes to light. 1.2.4 Deduce, from the distributive law and multiplicative absolute value, that |uv – uw| : lu||v – w|. Explain why this says that multiplication of the whole plane of complex numbers by u multiplies all distances by |u|.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 30E
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Please help with 1.2.4 (highlighted) for Modern ALgebra

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1.2.1 Derive the two-square identity from the multiplicative property of det. 1.2.2 Write 5 and 13 as sums of two squares, and hence express 65 as a sum of two squares using the two-square identity. 1.2.3 Using the two-square identity, express 372 and 37ª as sums of two nonzero squares. The absolute value |z| = va² +b² represents the distance of z from 0, and more generally, |u – v| represents the distance between u and v. When combined with the distributive law, Z. u(v – w) = uv – Uw, a geometric property of multiplication comes to light. 1.2.4 Deduce, from the distributive law and multiplicative absolute value, that |uv – uw = |u||v – w|. Explain why this says that multiplication of the whole plane of complex numbers by u multiplies all distances by |u|. 1.2.5 Deduce from Exercise 1.2.4 that multiplication of the whole plane of com- plex numbers by cos 0+ i sin 0 leaves all distances unchanged. A map that leaves all distances unchanged is called an isometry (from the Greek for "same measure"), so multiplication by cos 0 +isin 0 is an isometry of the plane. (In Section 1.1 we defined the corresponding rotation map Re as a linear map that moves 1 and i in a certain way; it is not obvious from this definition that a rotation is an isometry.)
Transcribed Image Text:1.2.1 Derive the two-square identity from the multiplicative property of det. 1.2.2 Write 5 and 13 as sums of two squares, and hence express 65 as a sum of two squares using the two-square identity. 1.2.3 Using the two-square identity, express 372 and 37ª as sums of two nonzero squares. The absolute value |z| = va² +b² represents the distance of z from 0, and more generally, |u – v| represents the distance between u and v. When combined with the distributive law, Z. u(v – w) = uv – Uw, a geometric property of multiplication comes to light. 1.2.4 Deduce, from the distributive law and multiplicative absolute value, that |uv – uw = |u||v – w|. Explain why this says that multiplication of the whole plane of complex numbers by u multiplies all distances by |u|. 1.2.5 Deduce from Exercise 1.2.4 that multiplication of the whole plane of com- plex numbers by cos 0+ i sin 0 leaves all distances unchanged. A map that leaves all distances unchanged is called an isometry (from the Greek for "same measure"), so multiplication by cos 0 +isin 0 is an isometry of the plane. (In Section 1.1 we defined the corresponding rotation map Re as a linear map that moves 1 and i in a certain way; it is not obvious from this definition that a rotation is an isometry.)
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