Worksheet01: Dickson 001
Current Work (editable)
First Course in The Theory of Equations CHAPTER I Complex Numbers 1. Square Roots. If p is a positive real number, the symbol $${p} is used to denote the positive square root of p. It is most easily computed by logarithms. We shall express the square roots of negative numbers in terms of the symbol i such that the relation i^{2} = -1 holds. Consequently we denote the roots of x^{2} = -1 by i and -i. The roots of x^{2} = -4 are written in the form ±2i in preference to ±$${-4}. In general, if p is positive, the roots of x^{2} = -p are written in the form ±$${p} i in preference to ±$${-p}. The square of either root is thus ($${p})^{2} i^{2} = -p. Had we used the less desirable notation ±$${-p} for the roots of x^{2} = -p, we might be tempted to find the square of either root by multiplying together the values under the radical sign and conclude erroneously that $${-p} $${-p} = $${p^{2}} = +p. To prevent such errors we use $${p} i and not $${-p}. 2. Complex Numbers. If a and b are any two real numbers and i^{2} = -1, a + bi is called a complex number[1] and a - bi its conjugate. Either is said to be zero if a = b = 0. Two complex numbers a + bi and c + di are said to be equal if and only if a = c and b = d. In particular, a + bi = 0 1 Complex numbers are essentially couples of real numbers. For a treatment from this standpoint and a treatment based upon vectors, see the author's Elementary Theory of Equations, p. 21, p. 18.}
Answer Key (non-editable)
% [** Attn] First Course in The Theory of Equations \Chapter{I}{Complex Numbers} \Par{1. Square Roots.} If $p$ is a positive real number, the symbol $\sqrt{p}$ is used to denote the positive square root of $p$. It is most easily computed by logarithms. We shall express the square roots of negative numbers in terms of the symbol $i$ such that the relation $i^{2} = -1$ holds. Consequently we denote the roots of $x^{2} = -1$ by $i$ and $-i$. The roots of $x^{2} = -4$ are written in the form $±2i$ in preference to $±\sqrt{-4}$. In general, if $p$ is positive, the roots of $x^{2} = -p$ are written in the form $±\sqrt{p} i$ in preference to $±\sqrt{-p}$. \begin{Remark} The square of either root is thus $(\sqrt{p})^{2} i^{2} = -p$. Had we used the less desirable notation $±\sqrt{-p}$ for the roots of $x^{2} = -p$, we might be tempted to find the square of either root by multiplying together the values under the radical sign and conclude erroneously that \[ \sqrt{-p} \sqrt{-p} = \sqrt{p^{2}} = +p. \] To prevent such errors we use $\sqrt{p} i$ and not $\sqrt{-p}$. \end{Remark} \Par{2. Complex Numbers.} If $a$ and $b$ are any two real numbers and $i^{2} = -1$, $a + bi$ is called a \emph{complex number}\footnote {Complex numbers are essentially couples of real numbers. For a treatment from this standpoint and a treatment based upon vectors, see the author's \textit{Elementary Theory of Equations}, p.~21, p.~18.} and $a - bi$ its \emph{conjugate}. Either is said to be \emph{zero} if $a = b = 0$. Two complex numbers $a + bi$ and $c + di$ are said to be \emph{equal} if and only if $a = c$ and $b = d$. In particular, $a + bi = 0$
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