3b Converting Between Number Bases
- Understand how binary can be used to represent whole numbers.
- Understand how hexadecimal can be used to represent whole numbers.
- Be able to convert in both directions between:
- binary and decimal
- binary and hexadecimal
- decimal and hexadecimal
Numbers in Binary
- Ten digits (fingers)
- 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
- After nine, move left a place:
- 10, 11, 12, ...
- After ninety-nine, move left a place:
- 100, 101, ...
Converting Denary to Binary
To convert a denary (decimal) number to a binary number:
- Divide by two and write any remainder.
- Repeat until the answer is zero with or without a remainder.
- Write down the remainders in reverse order.
- Add enough zeroes on the left hand side to make the number eight digits.
Converting Denary to Binary
Convert the denary (decimal) number 153 to a binary number.
- Divide by two and write any remainder.
- Repeat until the answer is zero with or without a remainder.
- Write down the remainders in reverse order.
- Add enough zeroes on the left hand side to make the number eight digits.
\(1\)
\(0\)
\(0\)
\(0\)
\(1\)
\(1\)
\(0\)
\(1\)
\(153 \div 2 = 76 \ r1\)
\(76 \div 2 = 38 \ r0\)
\(38 \div 2 = 19 \ r0\)
\(19 \div 2 = 9 \ r1\)
\(9 \div 2 = 4 \ r1\)
\(4 \div 2 = 2 \ r0\)
\(2 \div 2 = 1 \ r0\)
\(1 \div 2 = 0 \ r1\)
Converting Denary to Binary
Convert the denary (decimal) number 63 to a binary number.
- Divide by two and write any remainder.
- Repeat until the answer is zero with or without a remainder.
- Write down the remainders in reverse order.
- Add enough zeroes on the left hand side to make the number eight digits.
\(63 \div 2 = 31 \ r1\)
\(31 \div 2 = 15 \ r1\)
\(15 \div 2 = 7 \ r1\)
\(7 \div 2 = 3 \ r1\)
\(3 \div 2 = 1 \ r1\)
\(1 \div 2 = 0 \ r1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(0\)
\(0\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
\(1\)
Converting Denary to Binary
Convert these denary numbers to binary numbers.
- Divide by two and write any remainder.
- Repeat until the answer is zero with or without a remainder.
- Write down the remainders in reverse order.
- Add enough zeroes on the left hand side to make the number eight digits.
- 168
- 47
- 255
- 1
- 0
\(1010 \ 1000\)
\(0010 \ 1111\)
\(1111 \ 1111\)
\(0000 \ 0001\)
\(0000 \ 0000\)
Converting Binary to Denary
To convert a number from binary to denary:
- Write the number out.
- Put the place value above each digit.
- Wherever there is a '1' add up the place value.
Converting Binary to Denary
Convert the binary number 0110 1101 to a denary number.
- Write the number out.
- Put the place value above each digit.
- Wherever there is a 1 add up the place value.
128's
64's
32's
4's
16's
8's
2's
1's
64 + 32 + 8 + 4 + 1 = 109
\(0\)
\(1\)
\(1\)
\(1\)
\(0\)
\(1\)
\(0\)
\(1\)
Converting Binary to Denary
Convert the following binary numbers to denary numbers.
- Write the number out.
- Put the place value above each digit.
- Wherever there is a 1 add up the place value.
128 + 32 + 16 + 8 = 184
- 1011 1000
- 0010 1111
- 0000 0100
- 1111 1111
32 + 8 + 4 + 2 + 1 = 47
4
128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255
Base Sixteen - Hexadecimal
- Sixteen digits (represents 4 bits)
-
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F -
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15 - Usually written as a pair of digits:
-
00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 0A, 0B, 0C, 0D, 0E, 0F
Converting to Hexadecimal
A = 10
B = 11
C = 12
D = 13
E = 14
F = 15
- Convert to 8-bit binary first.
- Split into two four-bit segments.
- Convert each segment back into denary.
- Convert each of the two segments into hexadecimal.
Converting to Hexadecimal
Convert the denary (decimal) number 153 to a hexadecimal number.
- Convert to 8-bit binary first.
- Split into two four-bit segments.
- Convert each segment back into denary.
- Convert each of the two segments into hexadecimal.
\(153_{10} = 1001 \ 1001_2\)
\(153_{10} = 99_{16}\)
\(8 + 1 = 9_{10}\)
\(9_{10} = 9_{16}\)
8's
4's
2's
1's
\(1\)
\(0\)
\(0\)
\(1\)
\(8 + 1 = 9_{10}\)
\(9_{10} = 9_{16}\)
8's
4's
2's
\(0\)
1's
\(1\)
\(0\)
\(1\)
Converting to Hexadecimal
Convert the denary (decimal) number 63 to a binary number.
\(63_{10} = 0011 \ 1111_{2}\)
- Convert to 8-bit binary first.
- Split into two four-bit segments.
- Convert each segment back into denary.
- Convert each of the two segments into hexadecimal.
\(8 + 4 + 2 + 1 = 15_{10}\)
\(15_{10} = F_{16}\)
8's
4's
2's
1's
\(1\)
\(1\)
\(1\)
\(1\)
\(2 + 1 = 3_{10}\)
\(3_{10} = 3_{16}\)
8's
4's
2's
\(0\)
1's
\(0\)
\(1\)
\(1\)
\(63_{10} = 3F_{16}\)
Converting to Hexadecimal
Convert these denary numbers to hexadecimal numbers.
- 168
- 47
- 255
- 1
- 0
\(A8\)
\(2F\)
\(FF\)
\(01\)
\(00\)
- Convert to 8-bit binary first.
- Split into two four-bit segments.
- Convert each segment back into denary.
- Convert each of the two segments into hexadecimal.
Converting Hexadecimal to Denary
To convert a hexidecimal number to a denary number.
- Convert each digit to binary.
- Add the place value above each binary digit.
- Add the place values for every '1' digit.
Converting Hexadecimal to Denary
Convert the hexadecimal number 6D to a denary number.
128's
64's
32's
4's
16's
8's
2's
1's
64 + 32 + 8 + 4 + 1 = 109
\(0\)
\(1\)
\(1\)
\(1\)
\(0\)
\(1\)
\(0\)
\(1\)
- Convert each digit to binary.
- Add the place value above each binary digit.
- Add the place values for every '1' digit.
616 = 01102
D16 = 11012
6D16 = 0110 11012
Converting Hexadecimal to Denary
Convert the following hexadecimal numbers to denary numbers.
1011 01002 = 18010
- Convert each digit to binary.
- Add the place value above each binary digit.
- Add the place values for every '1' digit.
- B4
- 3F
- FF
- 0F
0011 11112 = 6310
1111 11112 = 25510
0000 11112 = 1510