Answer:
110011
Step-by-step explanation:
The tedious way to do this is to divide by 2 until the quotient is 0, noting remainders at each step:
- 51/2 = 25 r 1
- 25/2 = 12 r 1
- 12/2 = 6 r 0
- 6/2 = 3 r 0
- 3/2 = 1 r 1
- 1/2 = 0 r 1
Taken from bottom to top, the list of remainders comprises the binary number: 110011.
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Alternatively, you can convert the number to hexadecimal (base-16) or octal (base-8), then make the simple conversions from those digits to binary. In octal, we have ...
Then the number in octal is 51 = 63₈. Your familiarity with binary lets you write the binary number from memory, since you recall 6 = 110₂ and 3 = 011₂. Each octal digit must be expressed as three binary digits (or bits) in order to maintain appropriate place values.
That is, ...
63₈ = 110011₂
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<em>Comment on octal-binary conversion</em>
The binary values of the digits 0-7 are ...
- 0 = 000₂
- 1 = 001₂
- 2 = 010₂
- 3 = 011₂
- 4 = 110₂
- 5 = 101₂
- 6 = 110₂
- 7 = 111₂
Three binary bits can express numbers 0-7 as shown. So, using octal as an intermediate base in doing the conversion to binary lets you do the conversion 3 bits at a time, instead of one bit at a time.
Likewise, four binary bits can express numbers 0-15, so hexadecimal as an intermediate base lets you do the conversion 4 bits at a time: 51/16 = 3 r 3 ⇒ 0011 0011₂.
