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1. Shard Calculation

Query:

Response:

Calculated Shard: 150 (from laksa145)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa150.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://davidwees.com/content/multiplication-tables-binary/\')), getAhrefsUnparsedNoserviceFromURL(\'https://davidwees.com/content/multiplication-tables-binary/\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/davidwees.com\/content\/multiplication-tables-binary\/","crawl_time":1775422080,"first_indexed_time":1477750447,"http_code":200,"src_unparsed":"com,davidwees!\/content\/multiplication-tables-binary\/ s443","src_root_hash":"15832476877428726150","history_drop_reason":null,"meta_title":"Multiplication tables in binary – The Reflective Educator","meta_descriptions":[],"attrs_boilerpipe_text":"A\nbinary number\nis a number written in base 2 format, like 101010101111.  The binary number system is handy because it can be easily related to logical operators used in circuitry, and so\nalmost\nall modern computers use this format for communication.\nWe use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands. \nTo convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 2\n2\n+ 2\n0\nwhich is the same as\n1\n×2\n2\n+\n0\n×2\n1\n+\n1\n×2\n0\n. In binary, we write 5 as 101, since those are the coefficients of the powers of 2 (\nTry out this application\nwhich lets you switch between binary and decimal numbers).\nHere is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits).\n×\n0\n1\n0\n0\n0\n1\n0\n1\nCompare this to the traditional 10 by 10 multiplication table for decimal numbers.\n(Image credit:\nvalilouve\n)\nIf you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this:\nIf you want to double check, 10101 is the same as\n1\n×2\n4\n+\n0\n×2\n3\n+\n1\n×2\n2\n+\n0\n×2\n1\n+\n1\n×2\n0\n= 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 =\n1\n×2\n6\n+\n1\n×2\n5\n+\n0\n×2\n4\n+\n1\n×2\n3\n+\n0\n×2\n2\n+\n0\n×2\n1\n+\n1\n×2\n0\n= 64 + 32 + 8 + 1 = 105.\nSee this website\nfor a more detailed example of binary multiplication.\nThe point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. \nCheck this page\nout for more information on binary number operations.\nIf all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.","attrs_markdown":"Menu Close\n\n[The Reflective Educator](https:\/\/davidwees.com\/)\n\nEducation ∪ Math ∪ Technology\n\n- [Home](https:\/\/davidwees.com\/)\n- [Archives](https:\/\/davidwees.com\/archives\/)\n  - [Instructional Routines](https:\/\/davidwees.com\/instructional-routines\/)\n  - [Technology](https:\/\/davidwees.com\/technology\/)\n- [Presentations](https:\/\/davidwees.com\/presentations\/)\n- [Services](https:\/\/davidwees.com\/services\/)\n- [Contact](https:\/\/davidwees.com\/contact\/)\n- [Privacy Policy](https:\/\/davidwees.com\/privacy-policy\/)\n\n- [Home](https:\/\/davidwees.com\/)\n- [Archives](https:\/\/davidwees.com\/archives\/)\n  - [Instructional Routines](https:\/\/davidwees.com\/instructional-routines\/)\n  - [Technology](https:\/\/davidwees.com\/technology\/)\n- [Presentations](https:\/\/davidwees.com\/presentations\/)\n- [Services](https:\/\/davidwees.com\/services\/)\n- [Contact](https:\/\/davidwees.com\/contact\/)\n- [Privacy Policy](https:\/\/davidwees.com\/privacy-policy\/)\n\n[August 20, 2011](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/) \/ [1 Comment](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/#comments)\n\n# Multiplication tables in binary\n\nA [**binary number**](http:\/\/en.wikipedia.org\/wiki\/Binary_numeral_system) is a number written in base 2 format, like 101010101111. The binary number system is handy because it can be easily related to logical operators used in circuitry, and so [**almost**](http:\/\/www.newscientist.com\/article\/dn17575-ditching-binary-will-make-quantum-computers-more-powerful.html) all modern computers use this format for communication.\n\nWe use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands.\n\nTo convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 22 + 20 which is the same as **1**×22 + **0**×21 + **1**×20. In binary, we write 5 as 101, since those are the coefficients of the powers of 2 ([**Try out this application**](http:\/\/acc6.its.brooklyn.cuny.edu\/~gurwitz\/core5\/nav2tool.html) which lets you switch between binary and decimal numbers).\n\nHere is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits).\n\n|   |   |   |\n|---|---|---|\n| × | 0 | 1 |\n| 0 | 0 | 0 |\n| 1 | 0 | 1 |\n\nCompare this to the traditional 10 by 10 multiplication table for decimal numbers.\n\n![Decimal multiplication table](https:\/\/farm1.static.flickr.com\/208\/512264816_1b31d5cfba.jpg)  \n (Image credit: [**valilouve**](http:\/\/www.flickr.com\/photos\/8471692@N07\/))\n\nIf you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this:\n\n![10101 times 101 = 1101001](https:\/\/davidwees.com\/sites\/default\/files\/example10101x101.png)\n\nIf you want to double check, 10101 is the same as **1**×24 + **0**×23 + **1**×22 + **0**×21 + **1**×20 = 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 = **1**×26 + **1**×25 + **0**×24 + **1**×23 + **0**×22 + **0**×21 + **1**×20 \\= 64 + 32 + 8 + 1 = 105. [**See this website**](http:\/\/www.binarymath.info\/multiplication-division.php) for a more detailed example of binary multiplication.\n\nThe point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. [**Check this page**](http:\/\/www.math.grin.edu\/~rebelsky\/Courses\/152\/97F\/Readings\/student-binary) out for more information on binary number operations.\n\nIf all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.\n\n- [Comments](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/)\n- [Post info](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/)\n- [Author info](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/)\n\n## 1 Comment\n#### [Add yours →](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/#respond)\n\n1. ![](https:\/\/secure.gravatar.com\/avatar\/c941e8e90b10fd66852b99fe8d34467d5c28742adb4ed47d5d9e943986fd72ed?s=150&r=g)\n   \n   #### [R. Wright](http:\/\/sucolex.blogspot.com\/) says:\n   *If all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.*\n   \n   That’s a great analogy. But if binary multiplication feels arbitrary and bizarre to someone, I would venture to say that they never really understood decimal multiplication.\n   \n   [August 20, 2011 — 6:37 pm](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/#comment-5421)\n   \n   [Reply](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/?replytocom=5421#respond)\n### Leave a Reply [Cancel reply](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/#respond)\nThis site uses Akismet to reduce spam. [Learn how your comment data is processed.](https:\/\/akismet.com\/privacy\/)\n\n[David Wees](https:\/\/davidwees.com\/content\/author\/davidwees\/ \"Posts by David Wees\")\n\n[August 20, 2011 4:12 pm](https:\/\/davidwees.com\/content\/multiplication-tables-binary\/)\n\nUncategorized\n\n[Next post Math in the real world: Sound](https:\/\/davidwees.com\/content\/math-real-world-sound\/)\n***\n[Previous post Group for Canadian educators on LinkedIn](https:\/\/davidwees.com\/content\/group-canadian-educators-linkedin\/)\n\n[![](https:\/\/secure.gravatar.com\/avatar\/dc737d4fa55f1ac387021565300e12b9fed5554fcf2f565b36f4de910a945d96?s=256&r=g)](https:\/\/davidwees.com\/content\/author\/davidwees\/)\n\n### [David Wees](https:\/\/davidwees.com\/content\/author\/davidwees\/ \"Posts by David Wees\")\nAdministrator\n\n- [Bluesky](https:\/\/bsky.app\/profile\/davidwees.bsky.social)\n- [LinkedIn](https:\/\/www.linkedin.com\/in\/davidwees\/)\n- [TikTok](https:\/\/www.tiktok.com\/@davidwees)\n- [Instagram](https:\/\/www.instagram.com\/davidwees\/)\n\n## Popular posts\n- [Why Is a Negative Times a Negative Positive?](https:\/\/davidwees.com\/content\/why-is-a-negative-times-a-negative-positive\/) 139k views\n- [What is Conceptual Understanding?](https:\/\/davidwees.com\/content\/what-is-conceptual-understanding\/) 57\\.8k views\n- [Teachers are made, not born](https:\/\/davidwees.com\/content\/teachers-are-made-not-born\/) 52\\.7k views\n- [Online Learning Recommendations](https:\/\/davidwees.com\/content\/online-learning-recommendations\/) 42\\.9k views\n- [The difference between instrumental and relational understanding](https:\/\/davidwees.com\/content\/difference-between-instrumental-and-relational-understanding\/) 32k views\n- [Paper use in schools](https:\/\/davidwees.com\/content\/paper-use-schools\/) 29\\.1k views\n- [Philosophy of Educational Technology](https:\/\/davidwees.com\/content\/philosophy-educational-technology\/) 24k views\n- [Some problems with ebooks in schools](https:\/\/davidwees.com\/content\/some-problems-ebooks-schools\/) 19\\.2k views\n- [20 things every teacher should do](https:\/\/davidwees.com\/content\/20-things-every-teacher-should-do\/) 18k views\n- [Patterns and Algebra](https:\/\/davidwees.com\/content\/patterns-and-algebra\/) 17\\.6k views\n\n### Archived posts\nArchived posts\n\n### Calendar\n| M | T | W | T | F | S | S |\n|---|---|---|---|---|---|---|\n|   | 1 | 2 | 3 | 4 | 5 |   |\n| 6 | 7 | 8 | 9 | 10 | 11 | 12 |\n| 13 | 14 | 15 | 16 | 17 | 18 | 19 |\n| 20 | 21 | 22 | 23 | 24 | 25 | 26 |\n| 27 | 28 | 29 | 30 |   |   |   |\n\n[« Mar](https:\/\/davidwees.com\/content\/2026\/03\/)\n\n### Recent Posts\n- [Teacher Evaluation or Teacher Growth? 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All rights reserved.\n\nTheme by [Anders Norén](https:\/\/andersnoren.se\/).","attrs_readable_markdown":"A [**binary number**](http:\/\/en.wikipedia.org\/wiki\/Binary_numeral_system) is a number written in base 2 format, like 101010101111. The binary number system is handy because it can be easily related to logical operators used in circuitry, and so [**almost**](http:\/\/www.newscientist.com\/article\/dn17575-ditching-binary-will-make-quantum-computers-more-powerful.html) all modern computers use this format for communication.\n\nWe use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands.\n\nTo convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 22 + 20 which is the same as **1**×22 + **0**×21 + **1**×20. In binary, we write 5 as 101, since those are the coefficients of the powers of 2 ([**Try out this application**](http:\/\/acc6.its.brooklyn.cuny.edu\/~gurwitz\/core5\/nav2tool.html) which lets you switch between binary and decimal numbers).\n\nHere is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits).\n\n|   |   |   |\n|---|---|---|\n| × | 0 | 1 |\n| 0 | 0 | 0 |\n| 1 | 0 | 1 |\n\nCompare this to the traditional 10 by 10 multiplication table for decimal numbers.\n\n![Decimal multiplication table](https:\/\/farm1.static.flickr.com\/208\/512264816_1b31d5cfba.jpg)  \n (Image credit: [**valilouve**](http:\/\/www.flickr.com\/photos\/8471692@N07\/))\n\nIf you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this:\n\n![10101 times 101 = 1101001](https:\/\/davidwees.com\/sites\/default\/files\/example10101x101.png)\n\nIf you want to double check, 10101 is the same as **1**×24 + **0**×23 + **1**×22 + **0**×21 + **1**×20 = 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 = **1**×26 + **1**×25 + **0**×24 + **1**×23 + **0**×22 + **0**×21 + **1**×20 \\= 64 + 32 + 8 + 1 = 105. [**See this website**](http:\/\/www.binarymath.info\/multiplication-division.php) for a more detailed example of binary multiplication.\n\nThe point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. [**Check this page**](http:\/\/www.math.grin.edu\/~rebelsky\/Courses\/152\/97F\/Readings\/student-binary) out for more information on binary number operations.\n\nIf all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.","meta_canonical":null}

3. Robots.txt Check

Query:

Response:

4. Spam/Ban Check

Query:

Response:

5. Seen Status Check

ℹ️ Skipped - page is already crawled

📄

INDEXABLE

✅

CRAWLED

3 days ago

🤖

ROBOTS ALLOWED

Page Info Filters

Filter	Status	Condition	Details
HTTP status	PASS	`download_http_code = 200`	HTTP 200
Age cutoff	PASS	`download_stamp > now() - 6 MONTH`	0.1 months ago
History drop	PASS	`isNull(history_drop_reason)`	No drop reason
Spam/ban	PASS	`fh_dont_index != 1 AND ml_spam_score = 0`	ml_spam_score=0
Canonical	PASS	`meta_canonical IS NULL OR = '' OR = src_unparsed`	Not set

Page Details

Property	Value
URL	https://davidwees.com/content/multiplication-tables-binary/
Last Crawled	2026-04-05 20:48:00 (3 days ago)
First Indexed	2016-10-29 14:14:07 (9 years ago)
HTTP Status Code	200
Meta Title	Multiplication tables in binary – The Reflective Educator
Meta Description	null
Meta Canonical	null
Boilerpipe Text	A binary number is a number written in base 2 format, like 101010101111. The binary number system is handy because it can be easily related to logical operators used in circuitry, and so almost all modern computers use this format for communication. We use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands. To convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 2 2 + 2 0 which is the same as 1 ×2 2 + 0 ×2 1 + 1 ×2 0 . In binary, we write 5 as 101, since those are the coefficients of the powers of 2 ( Try out this application which lets you switch between binary and decimal numbers). Here is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits). × 0 1 0 0 0 1 0 1 Compare this to the traditional 10 by 10 multiplication table for decimal numbers. (Image credit: valilouve ) If you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this: If you want to double check, 10101 is the same as 1 ×2 4 + 0 ×2 3 + 1 ×2 2 + 0 ×2 1 + 1 ×2 0 = 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 = 1 ×2 6 + 1 ×2 5 + 0 ×2 4 + 1 ×2 3 + 0 ×2 2 + 0 ×2 1 + 1 ×2 0 = 64 + 32 + 8 + 1 = 105. See this website for a more detailed example of binary multiplication. The point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. Check this page out for more information on binary number operations. If all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.
Markdown	Menu Close [The Reflective Educator](https://davidwees.com/) Education ∪ Math ∪ Technology - [Home](https://davidwees.com/) - [Archives](https://davidwees.com/archives/) - [Instructional Routines](https://davidwees.com/instructional-routines/) - [Technology](https://davidwees.com/technology/) - [Presentations](https://davidwees.com/presentations/) - [Services](https://davidwees.com/services/) - [Contact](https://davidwees.com/contact/) - [Privacy Policy](https://davidwees.com/privacy-policy/) - [Home](https://davidwees.com/) - [Archives](https://davidwees.com/archives/) - [Instructional Routines](https://davidwees.com/instructional-routines/) - [Technology](https://davidwees.com/technology/) - [Presentations](https://davidwees.com/presentations/) - [Services](https://davidwees.com/services/) - [Contact](https://davidwees.com/contact/) - [Privacy Policy](https://davidwees.com/privacy-policy/) [August 20, 2011](https://davidwees.com/content/multiplication-tables-binary/) / [1 Comment](https://davidwees.com/content/multiplication-tables-binary/#comments) # Multiplication tables in binary A [binary number](http://en.wikipedia.org/wiki/Binary_numeral_system) is a number written in base 2 format, like 101010101111. The binary number system is handy because it can be easily related to logical operators used in circuitry, and so [almost](http://www.newscientist.com/article/dn17575-ditching-binary-will-make-quantum-computers-more-powerful.html) all modern computers use this format for communication. We use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands. To convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 22 + 20 which is the same as 1×22 + 0×21 + 1×20. In binary, we write 5 as 101, since those are the coefficients of the powers of 2 ([Try out this application](http://acc6.its.brooklyn.cuny.edu/~gurwitz/core5/nav2tool.html) which lets you switch between binary and decimal numbers). Here is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits). \| \| \| \| \|---\|---\|---\| \| × \| 0 \| 1 \| \| 0 \| 0 \| 0 \| \| 1 \| 0 \| 1 \| Compare this to the traditional 10 by 10 multiplication table for decimal numbers. ![Decimal multiplication table](https://farm1.static.flickr.com/208/512264816_1b31d5cfba.jpg) (Image credit: [valilouve](http://www.flickr.com/photos/8471692@N07/)) If you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this: ![10101 times 101 = 1101001](https://davidwees.com/sites/default/files/example10101x101.png) If you want to double check, 10101 is the same as 1×24 + 0×23 + 1×22 + 0×21 + 1×20 = 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 = 1×26 + 1×25 + 0×24 + 1×23 + 0×22 + 0×21 + 1×20 \= 64 + 32 + 8 + 1 = 105. [See this website](http://www.binarymath.info/multiplication-division.php) for a more detailed example of binary multiplication. The point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. [Check this page](http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary) out for more information on binary number operations. If all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication. - [Comments](https://davidwees.com/content/multiplication-tables-binary/) - [Post info](https://davidwees.com/content/multiplication-tables-binary/) - [Author info](https://davidwees.com/content/multiplication-tables-binary/) ## 1 Comment #### [Add yours →](https://davidwees.com/content/multiplication-tables-binary/#respond) 1. ![](https://secure.gravatar.com/avatar/c941e8e90b10fd66852b99fe8d34467d5c28742adb4ed47d5d9e943986fd72ed?s=150&r=g) #### [R. Wright](http://sucolex.blogspot.com/) says: If all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication. That’s a great analogy. But if binary multiplication feels arbitrary and bizarre to someone, I would venture to say that they never really understood decimal multiplication. [August 20, 2011 — 6:37 pm](https://davidwees.com/content/multiplication-tables-binary/#comment-5421) [Reply](https://davidwees.com/content/multiplication-tables-binary/?replytocom=5421#respond) ### Leave a Reply [Cancel reply](https://davidwees.com/content/multiplication-tables-binary/#respond) This site uses Akismet to reduce spam. 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Readable Markdown	A [binary number](http://en.wikipedia.org/wiki/Binary_numeral_system) is a number written in base 2 format, like 101010101111. The binary number system is handy because it can be easily related to logical operators used in circuitry, and so [almost](http://www.newscientist.com/article/dn17575-ditching-binary-will-make-quantum-computers-more-powerful.html) all modern computers use this format for communication. We use the decimal system for communication in our day to day lives because it is related to our original numbering systems, and this entire system was developed because of we have ten fingers between our two hands. To convert a decimal number to a binary number, we want to rewrite the decimal number as a sum of powers of 2. For example, the number 5 is equal to 4 + 1 or 22 + 20 which is the same as 1×22 + 0×21 + 1×20. In binary, we write 5 as 101, since those are the coefficients of the powers of 2 ([Try out this application](http://acc6.its.brooklyn.cuny.edu/~gurwitz/core5/nav2tool.html) which lets you switch between binary and decimal numbers). Here is the basic multiplication table for binary, which only includes 0 and 1, since those are the only digits you have to multiply in binary (in a decimal system, you need a much larger multiplication table, since you need to be able to multiply each of 10 different digits, 0 – 9 by each of 10 different digits). \| \| \| \| \|---\|---\|---\| \| × \| 0 \| 1 \| \| 0 \| 0 \| 0 \| \| 1 \| 0 \| 1 \| Compare this to the traditional 10 by 10 multiplication table for decimal numbers. ![Decimal multiplication table](https://farm1.static.flickr.com/208/512264816_1b31d5cfba.jpg) (Image credit: [valilouve](http://www.flickr.com/photos/8471692@N07/)) If you want to multiply numbers in binary, you could use some similar strategies to regular decimal multiplication. For example, 10101 times 101 looks like this: ![10101 times 101 = 1101001](https://davidwees.com/sites/default/files/example10101x101.png) If you want to double check, 10101 is the same as 1×24 + 0×23 + 1×22 + 0×21 + 1×20 = 16 + 0 + 4 + 0 + 1 = 21 and 101 is 5 (as we noted before) so this multiplication in decimal is 21×5, which is 105. 1101001 = 1×26 + 1×25 + 0×24 + 1×23 + 0×22 + 0×21 + 1×20 \= 64 + 32 + 8 + 1 = 105. [See this website](http://www.binarymath.info/multiplication-division.php) for a more detailed example of binary multiplication. The point of this activity is that you have taken something which is hard to do (memorizing a 10 by 10 times table) and switched it to something which is conceptually more difficult, but easier to memorize. For smaller numbers, it is faster to multiply directly in decimal, but for larger numbers, it will actually take less time to convert them to binary, do the multiplication, and convert back. You may notice that the multiplication step itself is much easier than decimal multiplication, since it’s just a matter of remembering 2 facts (0×0 = 0 and 0×1 = 1) and lining up the numbers correctly so that the place value matches. [Check this page](http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary) out for more information on binary number operations. If all of this feels arbitrary and bizarre to you, now you know what many 3rd graders feel like when they are first introduced to multiplication.
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