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## Number systems

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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)

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 (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 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

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 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

- Number Systems

Quick Review: binary to decimal

- 10011012 decimal
- 64 + 8 + 4 + 1
- =77

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 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

- 96210 hexadecimal
- 3C216
- This is 3*256 + C(10)*16 + 2

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 one

- A10E16 decimal
- 41230

Octal

- Base 8
- Uses 8 different digits
- 0 1 2 3 4 5 6 7

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 one:

- 94610 octal
- 16628

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 one

- 20458
- 106110

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

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

- Convert E5816 to Binary
- 111001011000
- Convert 110010110 to Hexadecimal
- 196

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

- Convert 3078 to Binary
- 11000111
- Convert 110010110 to Octal
- 646

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

- Convert 3078 to Hex
- 11 000 111 first in binary
- 11000111
- 1100 0111 divide into groups of 4
- 12 7
- C716

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

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