Binary arithmetic

Densely packed decimal

Densely packed decimal (DPD) is an efficient method for binary encoding decimal digits. The traditional system of binary encoding for decimal digits, known as binary-coded decimal (BCD), uses four bits to encode each digit, resulting in significant wastage of binary data bandwidth (since four bits can store 16 states and are being used to store only 10), even when using packed BCD. Densely packed decimal is a more efficient code that packs three digits into ten bits using a scheme that allows compression from, or expansion to, BCD with only two or three hardware gate delays. The densely packed decimal encoding is a refinement of Chen–Ho encoding; it gives the same compression and speed advantages, but the particular arrangement of bits used confers additional advantages: * Compression of one or two digits (into the optimal four or seven bits respectively) is achieved as a subset of the three-digit encoding. This means that arbitrary numbers of decimal digits (not only multiples of three digits) can be encoded efficiently. For example, 38 = 12 × 3 + 2 decimal digits can be encoded in 12 × 10 + 7 = 127 bits – that is, 12 sets of three decimal digits can be encoded using 12 sets of ten binary bits and the remaining two decimal digits can be encoded using a further seven binary bits. * The subset encoding mentioned above is simply the rightmost bits of the standard three-digit encoding; the encoded value can be widened simply by adding leading 0 bits. * All seven-bit BCD numbers (0 through 79) are encoded identically by DPD. This makes conversions of common small numbers trivial. (This must break down at 80, because that requires eight bits for BCD, but the above property requires that the DPD encoding must fit into seven bits.) * The low-order bit of each digit is copied unmodified. Thus, the non-trivial portion of the encoding can be considered a conversion from three base-5 digits to seven binary bits. Further, digit-wise logical values (in which each digit is either 0 or 1) can be manipulated directly without any encoding or decoding being necessary. (Wikipedia).

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Multiplying decimals

This is a short video tutorial on multiplying decimals. For interactive applets, worksheets, and more videos go to http://www.mathvillage.info

From playlist All about decimals

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In this video, you’ll learn more about decimals. Visit https://www.gcflearnfree.org/decimals/ for our interactive text-based tutorial. This video includes information on: • Reading decimals • Comparing decimals We hope you enjoy!

From playlist Math Basics

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Dividing decimals

This is a short video tutorial on dividing decimals. For interactive applets, worksheets, and more videos go to http://www.mathvillage.info

From playlist All about decimals

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Multiplying Decimals

This video explains how to multiply decimals. http://mathispower4u.wordpress.com/

From playlist Number Sense - Decimals, Percents, and Ratios

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Dividing a whole number by a decimal

This is a short video tutorial on dividing a whole number by a decimal. For interactive applets, worksheets, and more videos go to http://fearlessmath.net ▶︎ ▶︎ ▶︎ FOLLOW ME ON TWITTER ◀︎ ◀︎ ◀︎ http://twitter.com/dhabecker ▶︎ ▶︎ ▶︎ All About Decimals playlist ◀︎ ◀︎ ◀︎ https://youtube.com/

From playlist All about decimals

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Putting LiPO in my TI Programmer Calculator

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From playlist Vintage Equipment

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Q is dense in R

Rational numbers are dense in the real numbers In this video, I present a classic proof that the rational numbers are dense in the real numbers. Enjoy! Archimedean Property: https://youtu.be/DssLr2xHWNg Check out my Real Numbers Playlist: https://www.youtube.com/playlist?list=PLJb1qAQIr

From playlist Real Numbers

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Locating decimals and fractions on a number line

A short tutorial on how to locate decimals and fractions on a number line. For more videos and interactive applets, please visit http://www.MathVillage.info

From playlist All about decimals

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From playlist Introduction to Materials Science and Engineering Fall 2016

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Real Analysis Ep 4: The Archimedean Property

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From playlist Math 3371 (Real analysis) Fall 2020

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Lattices: from geometry to cryptography - Oded Regev

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From playlist Mathematics

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From playlist Analysis and Beyond

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Expansion in Linear Groups and Applications - Jean Bourgain

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EEVblog #1333 - Nuclear Diamond Self-Charging Battery DEBUNKED!

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Day 13 metal crystal structures

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From playlist Introduction to Materials Science and Engineering Fall 2017

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Episode 2: The Irrationals Are a Dense Set Within the Reals [#MathChops]

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Geometry and arithmetic of sphere packings - Alex Kontorovich

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From playlist Mathematics

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Math Mornings: Chaos on the Circle, by Taylor McAdam

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Decimals with patterns can be irrational

There is a major difference between decimals with repeating patterns and decimals with patterns

From playlist Arithmetic

Related pages

Binary data | Decimal32 floating-point format | Decimal | Binary integer decimal | Decimal128 floating-point format | Decimal64 floating-point format | Hertz encoding | Chen–Ho encoding | IEEE 754-2008 | Binary-coded decimal | Prefix code