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Number systems
Visualizing Decimal and Binary
We count in base 10 because
people started by counting on
their fingers
Base 10 is a number system that uses 10 digits: 0-9
All of our numbers are made with these 10 digits:
532,699 4,887,012
956 14
60,044,117,005
How many fingers do you see?



Did you have to count them all? (I hope not)
How did you know there were 21?
We count how many groups of 10, and then add the
single fingers. 2 groups of 10 and 1 extra finger.
Numbers are easy to read because
of place value
2,514
•The ones place has the lowest value and is
on the right.
•Each place has a value that is 10 times
greater than the one to the right of it.
•Ones
•Tens
•Hundreds
•Thousands… etc.
To find the value of a number, multiply
each digit times its place value.
347
3 hundreds
 4 tens
 7 ones

To write the number
Three thousand Five hundred…
Put a 3 in the thousands place and a 5 in the
hundreds place.
Which is correct?


a)
b)
c)
d)
e)
f)
35
350
0035
3050
3500
35000
e) But Why ??
Because the PLACE where a
number appears determines its
value.
1
0
0
1
1
Binary is a number system with
only 2 digits
0
0
1
1
1
0
0
1
1
1
1
Zero and One
1
0
0
1
0

This works for things like:
 Light
bulbs or switches that are either on or off
 Digital computers that store information using many
on/off switches.
1
0
0
1
0
1
1
1
1
16
8
4
2
1
It’s easy to see that each place has twice the number of the one to it’s right.
What will be the place value of the number to the left of 16?
Hopefully you said 32.
It’s fairly easy to double numbers.
Binary Numbers are also easy to
read because of place value
1011
•The ones place has the lowest value and is
on the right.
•Each place has a value that is 2 times
greater than the one to the right of it.
•Ones ( 20)
•Twos ( 21)
•Fours ( 22)
•Eights ( 23) … etc.
To show a number in binary,
write a 1 if the card is showing,
write a zero if the card is not showing.
001 = 1
010 = 2
011 = 3
100 = 4
0
1
1
Which cards are needed to make the number 12?
8 + 4 = 12
What is the binary number for 12?
01100
Or
1100
0
0
0
0
What number is this?
1
0
1
4+1 =5
How is 5 written in binary?
There is nothing in the 16’s place
There is nothing in the 8’s place
There is 1 four
There is nothing in the 2’s place
There is 1 one
SO, the answer is 00101
Or, since extra zeros on the left don’t
count 101
1
1
0
Which cards are needed to make the number 27?
16 + 8 + 2 + 1 = 27
What is the binary number for 27?
11011
1
1
In conclusion:
Decimal is our ordinary base 10 number
system using the digits 0-9
 Binary is the base 2 number system using
only the digits 0-1
 All number systems have place value that
determines the value of the digit in each
place.
