Number systems

1
Number systems

• Visualizing Decimal and Binary

2
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
3
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.

4
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.

5
To find the value of a number, multiply each
digit times its place value.
347

• 3 hundreds
• 4 tens
• 7 ones

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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?
• 35
• 350
• 0035
• 3050
• 3500
• 35000

e) But Why ??
Because the PLACE where a number appears
determines its value.
7
Binary is a number system with only 2 digits
1 0 0 1 1 0 1 1 1 0 0 1 0 1 1
1 0 0 1 1 0 1 1 1 0 0 1 0 1 1
Zero and One

• 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.

8
1
4
8
16
2
Its easy to see that each place has twice the
number of the one to its right.
What will be the place value of the number to the
left of 16?
Hopefully you said 32. Its fairly easy to double
numbers.
9
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.

10
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
11
0 1 1 0 0
Which cards are needed to make the number 12?
8 4 12
What is the binary number for 12?
01100 Or 1100
12
0 0 1 0 1
What number is this?
41 5
How is 5 written in binary?
There is nothing in the 16s placeThere is
nothing in the 8s placeThere is 1 fourThere is
nothing in the 2s placeThere is 1 one SO, the
answer is 00101 Or, since extra zeros on the left
dont count 101
13
1 1 0 1 1
Which cards are needed to make the number 27?
16 8 2 1 27
What is the binary number for 27?
11011
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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.