What is the equivalent decimal value of binary number 101110 ?

Correct — A, (a) 46. In binary each digit is worth twice the digit to its right, so the place values of a six-digit binary number, read from the right, are 1, 2, 4, 8, 16 and 32. Line the digits up against them: 1 0 1 1 1 0 gives 32 + 0 + 8 + 4 + 2 + 0 = 46. The rightmost digit is 0, which is a useful sanity check on its own — a binary number ending in 0 is even, so any odd option can be discarded before any addition is done. A second route is the doubling method, which needs no place-value table: start with 0 and read the digits left to right, doubling the running total and adding the digit each time. That gives 1, 2, 5, 11, 23, 46 — the same answer, and it is the method to use when the string is long enough that counting powers of two becomes error-prone. A third check: group the digits in fours from the right, 10 and 1110, which are the hexadecimal digits 2 and E, and 2 × 16 + 14 = 46.

Why the others are wrong

• (b)56 — 56 is what you get by reading 101110 as an OCTAL number after grouping the digits in threes: 101 is 5 and 110 is 6, so the string does correspond to octal 56 — but octal 56 is 5 × 8 + 6 = 46 in decimal, the same value written in a different base. Treating the two grouped digits as the decimal numeral 'fifty-six' confuses the notation with the quantity. Grouping in threes is a genuine and useful shortcut for converting binary to octal; it just does not give a decimal answer.
• (c)64 — 64 is 2 to the power 6 — the COUNT of different patterns six binary digits can form, not the value of any one of them. Those patterns run from 000000 to 111111, that is from 0 to 63, so 64 is one more than the largest six-digit binary number can reach. This option can be eliminated on sight, without converting anything, and the same reasoning kills option (d).
• (d)65 — 65 lies outside the range six binary digits can express at all: the maximum is 111111 = 32 + 16 + 8 + 4 + 2 + 1 = 63. It is also odd, and 101110 ends in 0, so it is even. Two independent checks — the range and the parity of the last digit — reject this option before any arithmetic. Building both reflexes is worth more than speed at the addition itself.

A positional number system fixes a base and gives each column a place value equal to a power of that base, counting from zero at the right. Decimal uses ten digits and powers of ten; binary uses two digits and powers of two. So 101110 in binary means 1×32 + 0×16 + 1×8 + 1×4 + 1×2 + 0×1 = 46. Binary matters to computing because a physical circuit distinguishes two states — on and off, charged and uncharged — far more reliably than ten, so every value inside a machine is ultimately a pattern of bits. Two facts follow directly and are asked constantly. First, n binary digits express exactly 2 to the power n different values, running from 0 to 2 to the power n minus 1: eight bits give 256 values, 0 to 255, which is why a byte counts that far. Second, because 8 and 16 are themselves powers of two, binary converts to octal in groups of three digits and to hexadecimal in groups of four, with no arithmetic at all — which is why programmers write long bit patterns in hex. Going the other way, decimal to binary, is repeated division by 2 with the remainders read bottom upwards: 46 gives remainders 0, 1, 1, 1, 0, 1, which read upwards is 101110.

Key facts

• Place values in binary, from the right: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
• 101110 = 32 + 8 + 4 + 2 = 46.
• n binary digits express 2 to the power n values, from 0 to 2 to the power n minus 1; six digits therefore stop at 63.
• A binary number is even exactly when its last digit is 0, and odd exactly when it is 1.
• Doubling method: read left to right, double the running total and add the digit — 1, 2, 5, 11, 23, 46.
• Decimal to binary is repeated division by 2, reading the remainders from the bottom up.
• Binary groups into octal three digits at a time (101 110 = octal 56) and into hexadecimal four digits at a time (10 1110 = hex 2E); both equal 46 in decimal.
• Binary uses the digits 0 and 1 only; a string containing any other digit is not a binary numeral.

Common traps

• Assigning place values from the left instead of the right, which reverses the number.
• Reading a binary string as though it were an octal or hexadecimal grouping and reporting the grouped digits as a decimal answer — the source of 56 here.
• Confusing 2 to the power n, the number of patterns, with the largest value the patterns reach, which is one less.
• Losing track of the zeros in the middle of a long string; writing the place values above the digits before adding prevents it.
• Forgetting the parity shortcut — a binary numeral ending in 0 cannot equal an odd decimal number.

• practice — not a real PYQ

  What is the equivalent decimal value of the binary number 110101 ?

  • (a)45
  • (b)51
  • (c)53
  • (d)61

  Answer(c) 53

• practice — not a real PYQ

  Which one of the following is the largest decimal value that can be represented using six binary digits ?

  • (a)32
  • (b)63
  • (c)64
  • (d)127

  Answer(b) 63