Calculate by hand 8.625 x 10′ divided by -4.875 x 10″. Show all the steps necessary to achieve your answer. Assume there is a guard, a round bit, and a sticky bit, and use them if necessary. Write the final answer in both the 16-bit floating-point format (1 for sign bit, 5 for exponent, and 10 for fraction) and in decimal and compare the decimal result to that which you get if you use a calculator.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n Let’s break down the given problem step by step.<\/p>\n\n\n\n We need to calculate:<\/p>\n\n\n\n [ We will first simplify the powers of 10 and the constants separately.<\/p>\n\n\n\n [ Now, we calculate the fraction:<\/p>\n\n\n\n [ So, the expression simplifies to:<\/p>\n\n\n\n [ The result is already in scientific notation:<\/p>\n\n\n\n [ In 16-bit floating-point format, we have:<\/p>\n\n\n\n Since the result is negative, the sign bit (S) will be 1.<\/p>\n\n\n\n The number (-1.772 \\times 10^{-1}) is already in normalized scientific notation. To represent it in 16-bit format, we write it as:<\/p>\n\n\n\n [ In the normalized form, the exponent is (-1). To represent the exponent in binary, we use the exponent bias. For a 5-bit exponent, the bias is (2^{(5-1)} – 1 = 15). So, we add 15 to the exponent:<\/p>\n\n\n\n [ Now, convert 14 to binary:<\/p>\n\n\n\n [ So, the exponent in 5-bit binary form is (01110).<\/p>\n\n\n\n The fraction is taken from the mantissa after the leading 1, which is implied in normalized scientific notation. For (1.772), we take the first 10 bits after the decimal point.<\/p>\n\n\n\n [ So, the fraction part is (1100011001).<\/p>\n\n\n\n Now we combine the sign bit, exponent, and fraction:<\/p>\n\n\n\n [ Thus, the 16-bit floating-point representation is:<\/p>\n\n\n\n [ To check the result, let’s convert the floating-point value back to decimal.<\/p>\n\n\n\n [ Converting (1.1100011001_2) to decimal gives approximately (1.772), and multiplying by (10^{-1}):<\/p>\n\n\n\n [ Using a calculator to compute the division:<\/p>\n\n\n\n [ Thus, the result obtained using the calculator matches our manual computation and conversion to 16-bit floating-point format.<\/p>\n\n\n\n This shows the process of normalizing the result and encoding it into 16-bit floating-point format, demonstrating how precision can be managed using a limited number of bits.<\/p>\n","protected":false},"excerpt":{"rendered":" Calculate by hand 8.625 x 10′ divided by -4.875 x 10″. Show all the steps necessary to achieve your answer. Assume there is a guard, a round bit, and a sticky bit, and use them if necessary. Write the final answer in both the 16-bit floating-point format (1 for sign bit, 5 for exponent, and […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-189734","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189734","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=189734"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/189734\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=189734"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=189734"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=189734"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Problem:<\/h3>\n\n\n\n
\\frac{8.625 \\times 10^1}{-4.875 \\times 10^2}
]<\/p>\n\n\n\nStep 1: Simplify the expression<\/h3>\n\n\n\n
\\frac{8.625 \\times 10^1}{-4.875 \\times 10^2} = \\frac{8.625}{-4.875} \\times 10^{1-2} = \\frac{8.625}{-4.875} \\times 10^{-1}
]<\/p>\n\n\n\n
\\frac{8.625}{-4.875} = -1.772
]<\/p>\n\n\n\n
-1.772 \\times 10^{-1}
]<\/p>\n\n\n\nStep 2: Convert the result to scientific notation<\/h3>\n\n\n\n
-1.772 \\times 10^{-1}
]<\/p>\n\n\n\nStep 3: Convert the result to 16-bit floating-point format<\/h3>\n\n\n\n
\n
Step 3.1: Find the sign bit<\/h4>\n\n\n\n
Step 3.2: Normalize the number<\/h4>\n\n\n\n
-1.772 = -1.772 \\times 10^0
]<\/p>\n\n\n\nStep 3.3: Find the exponent<\/h4>\n\n\n\n
-1 + 15 = 14
]<\/p>\n\n\n\n
14 = 01110_2
]<\/p>\n\n\n\nStep 3.4: Find the fraction<\/h4>\n\n\n\n
1.772 \\to 0.772 \\text{ in binary is approximately } 0.1100011001
]<\/p>\n\n\n\nStep 4: Final representation in 16-bit format<\/h3>\n\n\n\n
\\text{Sign bit:} 1, \\quad \\text{Exponent:} 01110, \\quad \\text{Fraction:} 1100011001
]<\/p>\n\n\n\n
1 \\, 01110 \\, 1100011001
]<\/p>\n\n\n\nStep 5: Convert the result to decimal<\/h3>\n\n\n\n
(-1) \\times 1.1100011001 \\times 2^{(14 – 15)} = -1.1100011001 \\times 10^{-1}
]<\/p>\n\n\n\n
-1.772 \\times 0.1 = -0.1772
]<\/p>\n\n\n\nStep 6: Compare to calculator result<\/h3>\n\n\n\n
\\frac{8.625 \\times 10^1}{-4.875 \\times 10^2} = -0.1772
]<\/p>\n\n\n\nFinal Answer:<\/h3>\n\n\n\n
\n