The. Binary. Number System

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1 The Binary Number System

2 Why is Binary important? Everything on a computer (or other digital device) is represented by Binary Numbers

3 One to Five in various systems I II III IV V

4 Decimal System Deci means Ten Base Ten System Ten symbols:

5 Decimal System Example: a decimal number

6 Decimal System Digits occupy Places

7 Decimal System The places have names Ten- Thousand s Thousand s Hundred s Ten s One s

8 Decimal System Place names as numbers ,000 1, (See a pattern in the place names?)

9 Decimal System Place determines the meaning of a digit ,000 1, means three thousand 3 means three tens

10 Decimal System Place names as exponents of (See a pattern in the place names?)

11 Bi means Two Base Two System Two symbols: 0 1 Understanding the Decimal system, replace Ten with Two

12 Place names as exponents of

13 Place names expressed in decimal numbers

14 Place names expressed in English Sixteen s Eight s Four s Two s One s

15 Interpreting the bits Where there is a 1: add the place name Where there is a 0: DON T add

16 Interpreting the bits = EIGHTEEN (18 in decimal)!

17 Conversion Shortcut: know the pattern! 1.Start with 1 as the right-most digit 2.Multiply by 2 each time, right to left:

18 Converting Binary to Decimal Example: Convert to Decimal: Write the pattern beneath each digit: Add the pattern number where there is a 1: = 157

19 Converting Decimal to Binary: Official Method Repeatedly divide by 2 recording the remainders (1 or 0) until the Dividend is 0. The answer is the list of remainders, last to first.

20 Converting Decimal to Binary: Official Method Example: convert 87 to Binary Dividend Answer: Quotient Remainder <-- right-most <-- left-most

21 Converting Decimal to Binary: Official Method

22 Converting Decimal to Binary: Other Method 1. Let LeftOver = the number to convert 2. Write the pattern, right to left, until the place number is greater than LeftOver. Discard that digit. 3. Repeat left to right, for each bit: - If LeftOver >= place number: o Write 1 in the bit o Subtract the place number from LeftOver - If LeftOver < place number: o Write 0 in the bit

23 Converting Decimal to Binary: Other Method Example: Convert 87 to Binary Step 1: Let LeftOver = 87 Step 2: we don t need bit 128 (because 87 < 128), so start with bit 64 XX

24 Converting Decimal to Binary: Other Method Is 87 >= 64? Yes LeftOver = = 23 Is 23 >= 32? No LeftOver left at 23

25 Is 23 >= 16? Yes LeftOver = = 7 Is 7 >= 8? No LeftOver left at 7 Is 7 >= 4? Yes LeftOver = 7 4 = 3

26 Is 3 >= 2? Yes LeftOver = 3 2 = 1 Is 1 >= 1? Yes LeftOver = 1 1 = 0 Answer:

27 Converting Decimal to Binary: Padding Sometimes a problem may state give the answer in a certain number of bits usually 4,8,16, etc. bits. If your answer contains less than the required number, simply add zeros on the left. Just as, in decimal, 48 = 0048, in Binary 1101 =

28 Addition: Adding 2 binary numbers Once again, it works similar to decimal - Starting on the right, going right to left: - Add the two digits (plus the carry if there is one) - Write the one digit result, or - If there is a two digit result, write down the right digit and carry the 1.

29 Addition: Adding 2 binary numbers Decimal Example: _ _ _

30 Addition: Adding 2 binary numbers Adding in binary is even easier, since for any column, the highest sum is three (carry bits in red): _0 + _1 +_1 +_

31 Addition: Adding 2 binary numbers Example: ( = 26) _1 1 0_ _1 1 0_ _1 1 0_ _1 1 0_

32 Fractions Fractions in binary work as expected, following the pattern of how the Decimal system works the exponent on the base number is subtracted by 1 for each digit to the right _1_ 10 _1_ 100 _1_

33 Fractions _1_ 2 _1_ 4 _1_ Convert : = Note: the point is called a radix point rather than a decimal point

34 Summary - for the test: - Know how to convert Binary to Decimal maximum 8 bit integer maximum 2 bits past the radix point - Know how to convert Decimal to Binary integers 0 to 255 fractions of.00,.25,.50, and.75 only - Know how to add two Binary integers - Vocabulary: Radix Point

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