Transcript The Binary Number System

The Binary Number
System
Emily Beck and Susan Cantrell
Purpose
Through this instruction you will
become familiar with the binary
number system, how to make
conversions, and how we are using
binary today.
Objectives
• 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
How do we count?
• When you were young you were taught to
count using the decimal number system.
• The word decimal means ten.
• How many ways can you symbolize the
number 10?
Decimal System
• There 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?
• An infinite amount!
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?
• 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.
• Now we must move to the next numeral in the ten’s
column to make the number 20.
• This process continues forever!
What if you only had two
numerals?
• The word binary means two.
• The binary number system has two
symbols: 0 and 1.
• With just these two symbols you can also
count forever.
Binary System
• 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.
• 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.
• Our new column is the four’s column. We have 100,
101, 110 and 111.
• This process continues forever too!
What do these
numbers mean to
us?
Each number in the binary
system corresponds to a
number in our traditional
decimal system.
• Decimal
numbers 1-15
with their
corresponding
binary number
conversion.
Number in
Decimal
Number in Binary
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
10
1010
11
101
12
110
13
1101
14
1110
15
1111
Decimal to Binary
• 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: 20 is
one, 21 is two 22 is four, and so on.
Converting a Binary Number into a
Decimal Number
• 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?
• That’s right, they represent:
– 20, 21, 22, 23, 24, 25
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 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.
Integer
41/2
20/2
10/2
5/2
2/2
1/0
Quotient
20
10
5
2
1
0
Remainder
1
0
0
1
0
1
Read up!
• When the quotient goes to zero you are done.
• Read the numbers in the remainder column starting
from the bottom and going up.
• Thus 41 is 101001 in the binary system.
Converting a Decimal Number into
a Binary Number
• Convert a decimal number to binary by
finding the remainders during successive
division by 2
• Example: Convert the decimal number 41
to binary
Binary to Decimal
• We must multiply each numeral in the
binary number by whatever value its
column has.
• Example: Convert the binary number
1101 to decimal form:
1 x 2 3 + 1 x 2 2 + 0 x 21 + 1 x 2 0
=8+4+0+1
=13
Binary Uses
• Binary 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 knows
• By combining groups of bits and manipulating
them, a computer can accomplish all the
remarkable things for which it has its reputation
So Handy
• Binary 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 data
• 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'
Questions?