2. The Binary Number System Emily Beck and Susan Cantrell

3. Purpose Through this instruction you will become familiar with the binary number system, how to make conversions, and how we are using binary today.

4. Objectives By the end of this presentation the student will be able to:By the end of this presentation the student will be able to: –Define a binary number system –Convert a decimal number to binary –Convert a binary number to decimal

5. How do we count? When you were young you were taught to count using the decimal number system.When you were young you were taught to count using the decimal number system. The word decimal means ten.The word decimal means ten. How many ways can you symbolize the number 10? How many ways can you symbolize the number 10?

6. Decimal System There are 9 numerals in the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9There are 9 numerals in the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 How many numbers can we make from these numerals?How many numbers can we make from these numerals? An infinite amount!An infinite amount!

7. How Do We Write Numbers in the Decimal System? The numerals are in the one’s column. When we run out of numerals for this column, what do we do?The numerals are in the one’s column. When we run out of numerals for this column, what do we do? We make a new column to write 10. The 1 is in the ten’s column, and the 0 is in the one’s column.We make a new column to write 10. The 1 is in the ten’s column, and the 0 is in the one’s column. We then continue with 11, 12, 13,…,17, 18, 19 until we run out of numerals in the one’s column again.We then continue with 11, 12, 13,…,17, 18, 19 until we run out of numerals in the one’s column again. Now we must move to the next numeral in the ten’s column to make the number 20.Now we must move to the next numeral in the ten’s column to make the number 20. This process continues foreverThis process continues forever!

8. What if you only had two numerals? The word binary means two.The word binary means two. The binary number system has two symbols: 0 and 1. The binary number system has two symbols: 0 and 1. With just these two symbols you can also count forever.With just these two symbols you can also count forever.

9. Binary System Now let’s imagine that we only have two numerals:Now let’s imagine that we only have two numerals: 0 and 1 Our first two numbers are 0, 1 but then we run out of numbers in our one’s column.Our first two numbers are 0, 1 but then we run out of numbers in our one’s column. Like in the decimal system we need to make a new column, this time a two’s column.Like in the decimal system we need to make a new column, this time a two’s column. Now we have 10 and 11 but again we run out of numerals in the one’s column.Now we have 10 and 11 but again we run out of numerals in the one’s column. Our new column is the four’s column. We have 100, 101, 110 and 111.Our new column is the four’s column. We have 100, 101, 110 and 111. This process continues forever too!This process continues forever too!

10. What do these numbers mean to us?

11. Each number in the binary system corresponds to a number in our traditional decimal system.

12. Decimal numbers 1-15 with their corresponding binary number conversion.Decimal numbers 1-15 with their corresponding binary number conversion. Number in Decimal Number in Binary 00 11 210 311 4100 5101 6110 7111 81000 91001 101010 11101 12110 131101 141110 151111

13. Decimal to Binary In decimal notation, each position to the left of the decimal point indicates an increased power of 10.In decimal notation, each position to the left of the decimal point indicates an increased power of 10. In binary, or base 2, each place to the left signifies an increased power of two: 2 0 is one, 2 1 is two 2 2 is four, and so on.In binary, or base 2, each place to the left signifies an increased power of two: 2 0 is one, 2 1 is two 2 2 is four, and so on.

14. Converting a Binary Number into a Decimal Number Each column in the binary number system has a name:Each column in the binary number system has a name: –one’s, two’s four’s, eight’s, thirty-two’s Notice anything special about these numbers?Notice anything special about these numbers? That’s right, they represent:That’s right, they represent: –2 0, 2 1, 2 2, 2 3, 2 4, 2 5

15. Reading Binary Numbers In the binary number system, as in the decimal system, the value of a digit is determined by where it stands in relation to the other digits in a number.In the binary number system, as in the decimal system, the value of a digit is determined by where it stands in relation to the other digits in a number. –In the decimal system, the number 1 by itself is worth 1; putting it to the left of two zeros makes the number worth 100. –In the decimal system, the number 1 by itself is worth 1; putting it to the left of two zeros makes the number worth 100. –This simple rule is the backbone of arithmetic. –Numbers to be added or subtracted are first arranged so that their place columns line up.

17. IntegerQuotientRemainder 41/2201 20/2100 10/250 5/221 2/210 1/001 Read up! When the quotient goes to zero you are done. When the quotient goes to zero you are done. Read the numbers in the remainder column starting from the bottom and going up. Read the numbers in the remainder column starting from the bottom and going up. Thus 41 is 101001 in the binary system. Thus 41 is 101001 in the binary system.

18. Converting a Decimal Number into a Binary Number Convert a decimal number to binary by finding the remainders during successive division by 2Convert a decimal number to binary by finding the remainders during successive division by 2 Example: Convert the decimal number 41 to binaryExample: Convert the decimal number 41 to binary

19. We must multiply each numeral in the binary number by whatever value its column has.We must multiply each numeral in the binary number by whatever value its column has. Example: Convert the binary number 1101 to decimal form:Example: Convert the binary number 1101 to decimal form: 1 x 2 3 + 1 x 2 2 + 0 x 2 1 + 1 x 2 0 =8+4+0+1=13 Binary to Decimal

20. Binary Uses Binary numbers are used to represent all information in the digital worldBinary numbers are used to represent all information in the digital world A "bit" (short for "binary digit") is the smallest piece of data that a computer knowsA "bit" (short for "binary digit") is the smallest piece of data that a computer knows By combining groups of bits and manipulating them, a computer can accomplish all the remarkable things for which it has its reputationBy combining groups of bits and manipulating them, a computer can accomplish all the remarkable things for which it has its reputation

21. So Handy Binary is handy because now we can easily use something physical to represent numbersBinary is handy because now we can easily use something physical to represent numbers 1’s and 0’s tell the computer “on” or “off” in coding data1’s and 0’s tell the computer “on” or “off” in coding data For instance we could use a laser -For instance we could use a laser - –When it's on you know it means '1' and when it's off you know it means '0'

22. Questions?