In binary: 13 = □₂

1101. Split 13 into 8, 4, 2 and 1, each once at most: 1 where one is used, 0 where not, 1101₂.

1110₂ = 7 True or false?

False. From the right the place values are 1, 2, 4, 8, so the leftmost is 8, not 1: 1110₂ = 14.

As a decimal number: 110000₂ = □

48. From the right the place values are 1, 2, 4, 8, 16, 32: add those under the 1s, 48.

Laws of integral indices · Ages 13–14

Binary numbers

Place values 1, 2, 4, 8, 16, …; expand to get the decimal.

How to do it

Method

Place values 1, 2, 4, 8, 16, …; expand to get the decimal

1011₂ = 8 + 0 + 2 + 1 = 11

Another way

Divide by 2 again and again

13 ÷ 2 = 6 r 1, 6 ÷ 2 = 3 r 0, 3 ÷ 2 = 1 r 1, 1 ÷ 2 = 0 r 1: 1101₂

In full

Our everyday decimal numbers have place values that are powers of 10: 1, 10, 100, …. Binary uses only the digits 0 and 1, and its place values are powers of 2: from the right, 1, 2, 4, 8, 16, 32, …. Binary to decimal: expand it, 1011₂ = 1 × 2³ + 0 × 2² + 1 × 2¹ + 1 × 2⁰ = 8 + 0 + 2 + 1 = 11. Decimal to binary: divide by 2 again and again, noting each remainder, until the quotient is 0, then read the remainders from the bottom up: 13 ÷ 2 = 6 r 1, 6 ÷ 2 = 3 r 0, 3 ÷ 2 = 1 r 1, 1 ÷ 2 = 0 r 1, so 13 = 1101₂. Another way: take away the biggest power of 2 that fits, again and again. 13 = 8 + 4 + 1, so write 1 in the 8, 4 and 1 places and 0 in the 2 place: 1101₂.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
In binary: 13 = ₂
Show the answer and working
Answer: 1101
Split 13 into 8, 4, 2 and 1, each once at most: 1 where one is used, 0 where not, 1101₂.
Example 2
1110₂ = 7
True or false?
Show the answer and working
Answer: False
From the right the place values are 1, 2, 4, 8, so the leftmost is 8, not 1: 1110₂ = 14.
Example 3
As a decimal number: 110000₂ =
Show the answer and working
Answer: 48
From the right the place values are 1, 2, 4, 8, 16, 32: add those under the 1s, 48.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.1100₂ = 12
True or false?
Answer:
2.As a decimal number: 1110₂ =
Answer:
3.As a decimal number: 100001₂ =
Answer:
4.Divide by 2 again and again: 13 ÷ 2 = 6 r , 6 ÷ 2 = 3 r , 3 ÷ 2 = 1 r , 1 ÷ 2 = 0 r 1. So 13 = ₂
Fill in each blank
Answer:
5.In binary: 60 = ₂
Answer:

All the answers

1. True
2. 14
3. 33
4. 1, 0, 1, 1101
5. 111100

Free worksheet: Binary numbersWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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