# Complex Numbers and Quadratic Equations

## If a and b are real numbers, then a number z = a + ib is defined to be a complex number. For any two complex numbers z1 and z2, z1 + z2 and z1 z2 are complex numbers. Complex numbers hold the commutative law of addition and commutative law of multiplication. Complex numbers hold the associative law of addition and associative law of multiplication. For every complex number z , there exists the complex number 0 = 0 + i 0, called the additive identity or the zero complex number, such that, z + 0 = z = 0 + z. For every complex number z = a + ib, there exists a complex number –z = – a + i(– b),called the additive inverse or negative of z such that z + (–z) = 0 = (–z) + z. The complex number 1 = 1 + i0 is the identity element. Existence of multiplicative inverse: For any non-zero complex number z = a + ib, there exists z1 = x + iy such that z(z1) = 1 = (z1) z. Modulus of the complex number x + iy, is the distance of point P(x, y) from the origin O(0,0). A quadratic equation is of the form ax2 + bx + c = 0, where a, b, c are real coefficients and a is not zero. Fundamental theorem of Algebra: Every polynomial equation has at least one root, real or imaginary. Every polynomial equation of positive degree n has exactly n roots real or imaginary. Key Words: Principal argument, conjugate of a complex number, Properties of Complex Numbers

To Access the full content, Please Purchase
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "https://www.extramarks.com/ncert-solutions/cbse-class-11/mathematics-complex-numbers-and-quadratic-equations", "fetch_time": 1603981115000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2020-45/segments/1603107904287.88/warc/CC-MAIN-20201029124628-20201029154628-00504.warc.gz", "warc_record_offset": 709658733, "warc_record_length": 11970, "token_count": 370, "char_count": 1440, "score": 3.703125, "int_score": 4, "crawl": "CC-MAIN-2020-45", "snapshot_type": "latest", "language": "en", "language_score": 0.8589398860931396, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00012-of-00064.parquet"}