Home > Articles

This chapter is from the book

Binary Arithmetic

At their lowest level, computers cannot subtract, multiply, or divide. Neither can calculators. The world's largest and fastest supercomputer can only add—that's it. It performs the addition at the bit level. Binary arithmetic is the only means by which any electronic digital computing machine can perform arithmetic.

Suppose you want the computer to add seven 6s together. If you asked the computer (through programming) to perform the calculation

6 + 6 + 6 + 6 + 6 + 6 + 6

the computer would zing the answer, 42, back to you before you could say bit bucket. The computer has no problem performing addition. The problems arise when you request that the computer perform another type of calculation, such as this one:

42 – 6 – 6 – 6 – 6 – 6 – 6 – 6

Because the computer can only add, it cannot do the subtraction. However, the computer can negate numbers. That is, the computer can take the negative of a number. Therefore, it can take the negative of 6 and represent (at the bit level) negative 6. Once it has done that, it can add –6 to 42 seven times. In effect, the internal calculation becomes this:

42 + (–6) + (–6) + (–6) + (–6) + (–6) + (–6) + (–6)

Adding seven –6s produces the correct result of 0. This may seem like a cop-out to you. After all, the computer is really subtracting, right? In reality, the computer is not subtracting. At its bit level, the computer can convert a number to its negative through a process known as 2's complement. A number's 2's complement is the negative of its original value at the bit level. The computer has in its internal logic circuits the capability to convert a number to its 2's complement and then carry out the addition of negatives, thereby seemingly performing subtraction.

Once the computer can add and simulate subtraction, it can simulate multiplying and dividing. To multiply 6 times 7, the computer actually adds 6 together seven times and produces 42. Therefore

6 x 7

becomes this:

6 + 6 + 6 + 6 + 6 + 6 + 6

To divide 42 by 7, the computer subtracts 7 from 42 (well, it adds the negative of 7 to 42) until it reaches zero and counts the number of times (6) it took to reach zero, like this:

42 + (–7) + (–7) + (–7) + (–7) + (–7) + (–7)

The computer represents numbers in a manner similar to characters. As Table 3.2 shows, numbers are easy to represent at the binary level. Once numbers reach a certain limit (256 to be exact), the computer will use more than one byte to represent the number, taking as many memory locations as it needs to represent the number. After it is taught to add, subtract, multiply, and divide, the computer can then perform any math necessary as long as a program is supplied to direct it.

Table 3.2 All Numbers Can Be Represented as Binary Numbers

Number

Binary Equivalent

0

00000000

1

00000001

2

00000010

3

00000011

4

00000100

5

00000101

6

00000110

7

00000111

8

00001000

9

00001001

10

00001010

11

00001011

12

00001100

13

00001101

14

00001110

15

00001111

16

00010000

17

00010001

18

00010010

19

00010011

20

00010100


To see an example of what goes on at the bit level, follow this example to see what happens when you ask the computer to subtract 65 from 65. The result should be zero and, as you can see from the following steps, that is exactly what the result is at the binary level.

TIP

The first 255 binary numbers overlap the ASCII table values. That is, the binary representation for the letter A is 01000001, and the binary number for 65 is also 01000001. The computer knows by the context of how your programs use the memory location whether the value is the letter A or the number 65.

  1. Suppose you want the computer to calculate the following:

    65 – 65

  2. The binary representation for 65 is 01000001, and the 2's complement for 65 is 10111111 (which is –65 in computerese). Therefore, you are requesting that the computer perform this calculation:

    01000001 + 10111111

  3. Because a binary number cannot have the digit 2 (there are only 0s and 1s in binary), the computer carries 1 anytime a calculation results in a value of 2; 1 + 1 equals 10 in binary. Although this can be confusing, you can make an analogy with decimal arithmetic. People work in a base 10 numbering system. (Binary is known as base 2.) There is no single digit to represent ten; we have to reuse two digits already used to form ten: 1 and 0. In base 10, 9 + 1 is 10. Therefore, the result of 1 + 1 in binary is 10 or "0 and carry 1 to the next column."

    01000001

    +10111111

    100000000

  4. Because the answer should fit within the same number of bits as the two original numbers (at least for this example—your computer may use more bits to represent numbers), the ninth bit is discarded, leaving the zero result. This example shows that binary 65 plus binary negative 65 equals zero as it should.

TIP

The good thing about all this binary arithmetic is that you don't have to understand a bit of it (pun intended) to be an expert programmer. Nevertheless, the more you know about what is going on under the hood, the better you will understand how programming languages work and the faster you will master new ones by seeing similarities between them.

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020