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Number systems. Converting numbers between binary, octal, decimal, hexadecimal (the easy way). Small numbers are easy to convert. But it helps to have a system for converting larger numbers to avoid errors. 12 10 = C 16. 5 10 -> 101 2. 1100 2 = 12 10. DEMONSTRATE.

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number systems

Number systems

Converting numbers between binary, octal, decimal, hexadecimal

(the easy way)

small numbers are easy to convert
Small numbers are easy to convert
  • But it helps to have a system for converting larger numbers to avoid errors.

1210 = C16

510 -> 1012

11002 = 1210

converting from base 10 decimal to base 2 binary

DEMONSTRATE

Converting from base 10 (decimal)to base 2 (binary)

example number = 42

  • Write the powers of 2 in a row starting on the RIGHT side with a 1
  • Keep doubling (*2) until you get to something greater than your number (42)

64

32

16

8

4

2

1

This is too big

1

0

1

1

3. Write a 1 underneath if that place value is used, 0 if not.

subtract to find out what is left.

0

0

42

-32

----

10

10

- 8

----

2

2

-2

----

0

Watch

Read your answer from left to right

The number in binary is 101010

converting from base 10 decimal to base 2 binary1

DO TOGETHER

Converting from base 10 (decimal)to base 2 (binary)

example number = 7053

  • write the powers of 2 in a row until you get to something > the number

8192

4096

2048

1024

512

256

128

64

32

16

8

4

2

1

Too

big

0

1

1

0

1

1

1

0

0

0

1

1

1

7053

-4096

-------

2957

2957

-2048

-------

909

909

- 512

-------

397

397

-256

------

141

141

-128

-------

13

13

- 8

-----

5

5

-4

---

1

1

-1

---

0

Do this together

the number in binary is 1101110001101

do this one 150 10 binary

STUDENT’S TURN

Do this one 15010 binary
  • 1
  • 2
  • 4
  • 8
  • 16
  • 32
  • 64
  • 128
  • 256

1

0

Too big

1

0

0

1

0

1

Click to see each digit that is needed.

The answer is: 10010110

to convert binary to decimal
To convert binary to decimal

the number in binary is 10111001101

  • Write the powers of 2 below each digit and only add the values with a 1 above them.
  • 0 1 1 1 0 0 1 1 0 1
  • 1024 512 256 128 64 32 16 8 4 2 1

Start at the right and double each number

1024 + 256+128+64 + 8 + 4 + 1 = 1,485

Watch

your turn convert 100010011 2 to decimal
Your turn. Convert 1000100112 to decimal
  • 1 0 0 0 1 0 0 1 1
  • 256 128 64 32 16 8 4 2 1
  • 256 + 16 + 2+1 =
  • 275
  • …. And now, for more about number systems.
part 2
Part 2
  • Number Systems
quick review
Quick review
  • What’s 41 in binary?
  • 32 16 8 4 2 1
  • 1 0 1 0 0 1

The answer is: 101001

quick review binary to decimal
Quick Review: binary to decimal
  • 10011012 decimal
  • 64 + 8 + 4 + 1
  • =77
an introduction to hexadecimal
An Introduction toHexadecimal
  • 16 digits
  • Use letters when you run out of single digits
  • 0 1 2 3 4 5 6 7 8 9 A B C D E F
  • SO… 1110 = ?16
    • B16
  • 1510 = ?
    • F16
  • 1610 = ?
    • 1016
from base 10 to base 16 decimal to hexadecimal
from base 10 to base 16 (decimal tohexadecimal)

example number = 7053

  • write the powers of 16 in a row until you get to one > the number
  • divide the number by each power of 16 and write the answer and save the remainder

65,536 4,096 256 16 1

  • Too high
  • 7053/4096 = 1 R 2957
  • 2957/256 = 11 R 141
  • 141/16 = 8 R 13
  • 13 ones
  • the numbers in hex are:
    • 1 2 3 4 5 6 7 8 9 A B C D E F (A=10…. F=15)
  • So your number is 1 11 8 13 = 1B8D16

Watch

do this one
Do this one
  • 96210 hexadecimal
  • 3C216
  • This is 3*256 + C(10)*16 + 2
from hexadecimal base 16 back to decimal
from hexadecimal (base 16) back to decimal

Watch

1B8D16

  • Write the number across a row. Write the powers of 16 below it. Multiply. Then add the products.
  • 1 B 8 D
  • =(1X4096)+
  • (11*256)+
  • (8*16)+(13*1) =
  • 4096 + 2816 + 128 + 13 = 7053

4096

256

16

1

do this one1
Do this one
  • A10E16 decimal
  • 41230
octal
Octal
  • Base 8
  • Uses 8 different digits
  • 0 1 2 3 4 5 6 7
from base 10 to base 8 decimal to octal
from base 10 to base 8(decimal tooctal)

example number = 7053

  • write the powers of 8 in a row until you get to one > the number
  • divide the number by each power of 8
  • write the answer and save the remainder
  • 32768 4096 512 64 8 1
  • too high
  • 7053/4096 = 1 R 2957
  • 2957/512 = 5 R 397
  • 397/64 = 6 R 13
  • 13/8 = 1 R 5
  • = 5 ones
  • so your number in octal is 156158

Watch

do this one2
Do this one:
  • 94610 octal
  • 16628
from octal base 8 back to decimal
from octal (base 8) back to decimal

156158

  • write the number
  • write the powers of 8 below it and multiply. then add the products.
  • 1 5 6 1 5
  • 4096 512 64 8 1
  • 1 *4096 = 4096
  • 5 * 512 = 2560
  • 6 * 64 = 384
  • 1* 8 = 8
  • 5 * 1 = 5
    • added together = 7053

Watch

do this one3
Do this one
  • 20458
  • 106110
binary hex octal

0

1

10

11

100

101

110

111

1000

1001

1010

1011

1100

1101

1110

1111

Binary  hex  octal
  • If you can count from 1 to 15 in binary you have it made
binary to hexadecimal and hex to binary
Binary to hexadecimal and hex to binary

Watch

  • 4 binary digits correspond to 1 hexadecimal digit
  • Start grouping digits on the RIGHT side
  • 0000 0
  • 0001 1
  • 0010 2
  • 0011 3
  • 0100 4
  • 0101 5
  • 0110 6
  • 0111 7
  • 1000 8
  • 1001 9
  • 1010 A
  • 1011 B
  • 1100 C
  • 1101 D
  • 1110 E
  • 1111 F

To convert binary 1101011110 to hex

Binary  Hexadecimal

11 0101 1110

3 5 E

35E16

Write this down the side of your paper.

Hex  Binary

28D1

10 1000110100012

practice hex binary hex
Practice Hex  Binary  Hex
  • Convert E5816 to Binary
    • 111001011000
  • Convert 110010110 to Hexadecimal
    • 196
binary to octal and octal to binary
binary to octal and octal to binary
  • 3 binary digits correspond to 1 octal digit
  • 000 0
  • 001 1
  • 010 2
  • 011 3
  • 100 4
  • 101 5
  • 110 6
  • 111 7

Binary to octal

10110011

10 110 011

263

  • Octal to binary
  • 451
  • 100 101 001
  • 101001

Watch

practice octal binary octal
Practice Octal  Binary  Octal
  • Convert 3078 to Binary
    • 11000111
  • Convert 110010110 to Octal
    • 646
octal to hex and hex to octal
octal to hex and hex to octal.
  • Convert to binary, regroup and convert to other base.

Octal to binary to hex

4518

100 101 001

100101001

1 0010 1001

12916

Watch

practice octal hex
Practice Octal  Hex
  • Convert 3078 to Hex
    • 11 000 111 first in binary
    • 11000111
    • 1100 0111 divide into groups of 4
    • 12 7
    • C716
practice hex octal
Practice Hex  Octal
  • Convert 2B1D16 to Octal
    • 10 1011 0001 1101 first in binary
    • 10101100011101
    • 10 101 100 011 101 divide into groups of 3
    • 2 5 4 3 5
    • 254358