Number Systems & Binary How to count

How do we represent numbers? Early systems: – Actual count : ||||| = 5 – Roman numers : XI = 11 Hard to do math: MCXXIII + CLXXIV = ????

What we are used to – Digits that have different meaning based on place: = 1 * * * * * 1 = Place Value Systems ten thousands thousandshundredstensones 12059

Decimal Numbers Base 10 : Each place increases by a power of 10 ten thousands 10 4 thousands 10 3 hundreds 10 2 tens 10 1 ones

Why Base 10? Only one thing special about base 10: Other systems have been used: – Mayans : Base 20 – Babylonians : Base 60

Other Bases Base determines the multiplier from one column to next Base 20 = 1 * * * * * 1 = thousands 20 4 eight- thousands 20 3 four- hundreds 20 2 twentys 20 1 ones

Base 2 Base two – Only 1 and 0 – Places are powers of 2 = 1 * * * * * * 1 = = 45 thirty-twos 2 5 sixteens 2 4 eights 2 3 fours2 twos 2 1 ones

Other Bases Base also determines how many digits needed: Base 10 : 10 digits : 0-9 Base 2 : 2 digits : 0-1 Base 8 : 8 digits : 0-7

Notation Subscript used to indicate base:  101 in base 2 (numeric value of 5)  101 in base 10 (numeric value of 101)

More digits = bigger numbers 4 digits we can represent values : 8 digits can represent digits can represent n digits can represent 0 to (2 n -1)

Leading 0's Leading 0's do not matter: 101 = 0101 =

Leading 0's Leading 0's do not matter: 101 = 0101 =

Counting to ten in binary In binary you generally show leading 0's Values for 1-10 (decimal) in binary:

Binary -> Decimal with Table 1. Make a table like this one 2. Write in the binary number 3. Add up all the numbers from the top that have a 1 under them: =

Decimal -> Binary with Table 1. Make a table like this one 2. Say we are converting 83: Look for largest number that is smaller than our value Look for largest number that is smaller than our value

Decimal -> Binary with Table 1. Make a table like this one 2. Say we are converting 83: Look for largest number that is smaller than our value Look for largest number that is smaller than our value We have 1 of those (0 of anything bigger) We have 1 of those (0 of anything bigger) Subtract that from our number 83 – 64 = still have 19 to make Subtract that from our number 83 – 64 = still have 19 to make

Decimal -> Binary with Table 1. Make a table like this one 2. Say we are converting 83: Look for largest number that is smaller than our value Look for largest number that is smaller than our value We have 1 of those (0 of anything bigger) We have 1 of those (0 of anything bigger) Subtract that from our number 83 – 64 = still have 19 to make Subtract that from our number 83 – 64 = still have 19 to make Look for largest value smaller than that… Now have 19 – 16 = 3 left Look for largest value smaller than that… Now have 19 – 16 = 3 left

Decimal -> Binary with Table 1. Make a table like this one 2. Say we are converting 83: Look for largest number that is smaller than our value Look for largest number that is smaller than our value We have 1 of those (0 of anything bigger) We have 1 of those (0 of anything bigger) Subtract that from our number 83 – 64 = still have 19 to make Subtract that from our number 83 – 64 = still have 19 to make Look for largest value smaller than that… Now have 19 – 16 = 3 left Look for largest value smaller than that… Now have 19 – 16 = 3 left Continue… need a 2, have one left… Continue… need a 2, have one left…

Decimal -> Binary with Table 1. Make a table like this one 2. Say we are converting 83: means =

Conversion With Division/ Multiplication Binary works in powers of 2 Multiplying/dividing by 2 shifts digits

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ 2521

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ 25211

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ ÷ 2

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ ÷

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ ÷ ÷

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ ÷ ÷ ÷ ÷ ÷ ÷

Decimal -> Binary with Division Step 1: Start with a blank answer and the number your are converting Step 2: Divide your number by 2 to make a quotient and a remainder Step 3: Place your remainder on the left side of your answer Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2 Convert 105 to binary: Number ÷ 2QR Answer 105 ÷ ÷ ÷ ÷ ÷ ÷ ÷ = 105

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer 11010

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2 + 1

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2 + 1

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2 + 0

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer * 2

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer _6 * 2 + 1

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer _13

Binary -> Decimal with Multiplication Step 1: Start with the number your are converting and the answer of 0 Step 2: Multiply your answer by 2 Step 3: Remove the leftmost digit of number and add it to your answer Step 4: If number has no more digits, you are done Otherwise, go back to step 2 Convert to decimal: Number Answer _ = 13