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Welcome to NRICH.

Binary numbers

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By Frank J Bradley (M1130) on Monday, May 28, 2001 - 09:40 pm:

Please explain how numbers are shown in binary notation. For example, take the number 36.

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By Dan Goodman (Dfmg2) on Tuesday, May 29, 2001 - 12:36 am:

Do you know what the binary notation means?
e.g. 100110 means 0×2⁰ + 1×2¹ + 1×2² + 0×2³ + 0×2⁴ + 1×2⁵=38.

Call the rightmost digit the 0^th digit, the one to the left of that the 1^st digit, to the left of that the 2^nd digit, and so on. Then to convert from binary to decimal, you add up all of the 2ⁿ where the n^th digit is 1.

The way to change a number from decimal to binary is to first find the largest power of 2 less than or equal to your number. In this case 32=2⁵. Subtract this to get 36-2⁵=36-32=4. Now try and find the highest power of 2 which is less than or equal to 4. In this case, 4 is actually a power of 2, 4=2². So 36 = 32 + 4 = 2⁵ + 2². So if we write it in binary, the 2^nd digit will be 1, the 5^th digit will be 1, and 0s elsewhere. So 36 in binary is 100100.

As an exercise, try writing 255 and 170 in binary.

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By The Editor:

It is perhaps worth comparing binary notation with decimal (ordinary) notation.

In decimal notation, the rightmost digit is the units (10⁰=1), then we have the 10s (10¹), 100s (10²), 1000s (10³), etc.

In binary, we use powers of 2 instead of 10. Binary can be called base 2, and decimal can be called base 10.

There is an article all about different number bases here.

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