Binary Number and bases

What base are normal numbers in?

base 10 and the option of digits are 0-9

What does 0 and 1 mean in computers

0 = off

1 = on

what is a base

• A numbering system used to represent values

• the base is a numerical number represented in the subscript

• the option of digits is always 0 to n-1

• to convert the base multiply each digit by the baseⁿ starting from the ones digit where n = 0 and increases by one across the digits aka the tens digit n = 1 etc

• then add the products together to get the number in that base

What base do computers understand/work in

base 2

First method of converting to base 2

Quotient method of converting to base 2

• only pay attention to integers

• work right to left

• if the number is odd put a 1, if the number is even put a 0

• divide the original number wanting to be converted by 2, if the quotient is odd put a 1, if the number is even put a 0

  • only take the number, don’t round it and ignore the decimal

• continue till the quotient is 0, when the quotient is 0 the number is finished

Remainder method of converting to base 2

• only pay attention to integers

• work right to left

• if the number is odd put a 1, if the number is even put a 0

• divide the original number wanting to be converted by 2, put the quotient which is 0 or 1

  • only take the number, don’t round it and ignore the decimal

• continue till the quotient is 0

  • when the quotient is less than 1 multiply it byb the base and put it as 1

What is a more efficient way of converting bases with numbers of the same base but different powers

base 2 to base 8 = 2¹ to 2³ which means every 3 digits of the base 2 number is equal to 1 digit in base 8

(add 0s at the end when there are leftover digits)

what to write when number can’t be converted

error

what is hexidecimal code

base 16

How to represent digits >=10

use the alphabet starting from A=10, B=11, C=12, etc.

what is codemapping

mapping/assigning text to numbers

how are images made/stored

• made of pixels

• each pixel has a color and color is stored in binary

• for black and white, with 0s as black and 1s as white hence each pixel uses only 1 bit

• for color images each pixel has a range of 0-255 of r, g, and, b. each pixel needs 24 bits, each in base 2 but better to represent in base 16

  • #RRGGBB

Sign magnitude to store +/- numbers

• first digit shows whether they’re positive or negative,

  • 0 = positive

  • 1 = negative

• while the rest of the digits represent what the number was

• 11011= -11

1’s compliment to store +/- numbers

• first digit shows whether they’re positive or negative,

  • 0 = positive

  • 1 = negative

• flip the numbers i.e. every 0 becomes a 1 and vice versa

• find the value of the flipped number

• 11011 = -4

2’s compliment to store +/- numbers

• first digit shows whether they’re positive or negative,

  • 0 = positive

  • 1 = negative

• flip the numbers i.e. every 0 becomes a 1 and vice versa

• add 1, addition is done normally but all numbers done in binary

  • the 1 was added to fix the issue from previous methods where there was +/- 0s

• find the value of the new number

• 11011 = -5

How to convert decimal numbers into binary:

1. Do the integer part normally

2. Take the decimal part and multiply by 2

   1. When the product is greater than 1 put a 1

   2. When the product is less than 1 put a 0

3. Repeat step 2, ignore the integer part and continue the previous step with the decimal part until there is no more decimal part

4. Put each digit from left to right, starting from the decimal point

Max range of numbers in n number of bits

• Unsigned: 2^n-1

• Sign magnitude & 1s compliment: -(2^n-1 - 1) to (2^n-1 - 1)

2s compliment: -(2^n-1) to (2^n-1 - 1)

How to subtract binary numbers

A + B

1. Convert numbers to binary

2. Convert the B number into 2s compliment

3. Add A and Bs 2s compliment

4. Get the 2s compliment of the sum

how to convert decimal binary numbers back to base 10

After the decimal point the base is raised to negative numbers starting from -1 in the 10s place, -2 in the 100s place and so on