Why is 1 << 3 equal to 8 and not 6?

Question

In C I have this enum:

enum {
    STAR_NONE =     1 << 0, // 1
    STAR_ONE =      1 << 1, // 2
    STAR_TWO =      1 << 2, // 4
    STAR_THREE =    1 << 3  // 8
};

Why is 1 << 3 equal to 8 and not 6?

You input `3 << 1` into that bitshift calculator website, not `1 << 3`. — , Feb 08 '15 at 15:56
Left shifting a number by x bits is nothing but multiplying the number by (2^x) and right shifting by x bits is dividing by (2^x)! Additional info: That's why many times bit shifting is used instead of Multiplication and Division for faster operation! — Swanand, Feb 08 '15 at 16:02
@Swanand That formula made everything a thousand times more clear, thanks — , Feb 08 '15 at 16:03
@LightnessRacesinOrbit Maybe this sounds stupid but I thought because 3 << 1 = 6, the other way around too — , Feb 08 '15 at 17:46
@YassineHoussni: You're right; it does ;) It's best to _research_ and _understand_ what things do, rather than wildly guessing. — Lightness Races in Orbit, Feb 08 '15 at 17:50
@LightnessRacesinOrbit lol! Yeah, I am completely new to bitshifting, I just started researchen, I've never used it before but now I took the step to learn about it. — , Feb 08 '15 at 17:55

score 7 · Accepted Answer · edited Feb 08 '15 at 19:02

Shifting a number to the left is equivalent to multiplying that number by 2ⁿ where n is the distance you shifted that number.

To see how is that true lets take an example, suppose we have the number 5 so we shift it 2 places to the left, 5 in binary is 00000101 i.e.

0×2⁷ + 0×2⁶ + 0×2⁵ + 0×2⁴ + 0×2³ + 1×2² + 0×2¹ + 1×2⁰ = 1×2² + 1×2⁰ = 4 + 1 = 5

now, 5 << 2 would be 00010100 i.e.

0×2⁷ + 0×2⁶ + 0×2⁵ + 1×2⁴ + 0×2³ + 1×2² + 0×2¹ + 0×2⁰ = 1×2⁴ + 1×2² = 16 + 4 = 20

But we can write 1×2⁴ + 1×2² as

1×2⁴ + 1×2² = (1×2² + 1×2⁰)×2² = 5×4 → 20

and from this, it is possible to conclude that 5 << 2 is equivalent to 5×2², it would be possible to demonstrate that in general

k << m = k×2^m

So in your case 1 << 3 is equivalent to 2³ = 8, since

1 << 3 → b00000001 << 3 → b00001000 → 2³ -> 8

If you instead did this 3 << 1 then

3 << 1 → b00000011 << 1 → b00000110 → 2² + 2¹ → 4 + 2 → 6

Sometimes [terseness](http://stackoverflow.com/a/28396041/2410359) beats completeness - like your fine complete answer. — chux - Reinstate Monica, Feb 08 '15 at 16:47
@chux not this time. Thanks iharob for the answer, this is exactly what I needed to get a clear view about shifting. — , Feb 08 '15 at 17:36

SMA · Answer 2 · 2015-02-08T16:00:17.893

3

1 in binary is 0000 0001

by shifting to left by 3 bits (i.e. 1 << 3) you get

0000 1000

Which is 8 in decimal and not 6.

edited Feb 08 '15 at 16:00

answered Feb 08 '15 at 15:57

SMA

score 1 · Answer 3 · answered Feb 08 '15 at 15:56

1

Because two to the power of three is eight.

answered Feb 08 '15 at 15:56

Robert Harvey

score 1 · Answer 4 · answered Feb 08 '15 at 15:57

1

Think in binary. You are actually doing this. 0001 shifted 3 times to the left = 1000 = 8

answered Feb 08 '15 at 15:57

JDWK

4 Answers4