Maths for CS

Key Values for n Bits: The Mathematical Muck

Let's talk about bits. Not those little bits of fluff you find in your pockets, but binary digits. We're gonna see how many different numbers you can make with a set number of bits.

Take two bits, for example. The smallest number you can make is zero. Nothing. Nada. And the biggest? Three. That's four different values in total.

0 + 0 = 0

20 = 1

21 = 2

2 + 1 = 3

We'll show you how to work out the smallest, largest, and total number of different values. It's all about powers of two, which is just a fancy way of saying 'multiply by two a bunch of times'.

2n - 1. That's the magic formula. Where 'n' is the number of bits.

The Three Steps to Binary Brilliance

Right, this is important. There are three steps to follow when working out the smallest number, largest number, and the total number of different values in binary.

  1. Step 1: The smallest value is 0. Simple as that. Zero is the starting point.
  2. Step 2: The largest value is 2n - 1. Where 'n' is the number of bits. It's like our magic formula, but written out properly.
  3. Step 3: The number of different values is 2n. Again, 'n' is the number of bits.

Let's try it with an example. Say we've got two bits.

  1. Step 1: The smallest value is 0. As we said, it's always zero.
  2. Step 2: The largest value is 2n - 1. Since 'n' is 2, we get 22 - 1 = 4 - 1 = 3.
  3. Step 3: The number of different values is 2n. So, 22 = 4.

There you have it. Follow these three steps and you'll be a binary whiz in no time.

Worked Examples Icon Worked examples

Example 1: Determine the smallest, largest, and number of distinct values storable with 6 bits.

Step 1: The minimum value is always 0.

Step 2: The maximum value is calculated as 2n - 1, where n is the number of bits. Thus, 26 - 1 = 64 - 1 = 63.

Step 3: The total number of unique values is 2n. So, 26 = 64.

Therefore, with 6 bits, the smallest value is 0, the largest is 63, and there are 64 unique values.

Example 2: Find the smallest, largest, and total unique values for 10 bits.

Step 1: The smallest value is 0.

Step 2: The largest value is 2n - 1. In this case, 210 - 1 = 1024 - 1 = 1023.

Step 3: The total number of unique values is 2n. So, 210 = 1024.

Example 3: Calculate the smallest, largest, and number of distinct values for 3 bits.

Step 1: The smallest possible value is 0.

Step 2: The largest value is found with 2n - 1. Here, 23 - 1 = 8 - 1 = 7.

Step 3: The number of unique values is 2n, which is 23 = 8.

Guide Icon Guided Practice

For each question, determine the smallest value, largest value, and the number of different values that can be stored with the given number of bits. Show your workings.

5 bits

Step 1: The smallest value is always 0.

Step 2: The largest value is 2bits - 1. In this case, 25 - 1 = 32 - 1 = 31

Step 3: The number of values is 2bits. In this case, 25 = 32

Answer: Smallest: 0, Largest: 31, Values: 32

4 bits

Step 1: The smallest value is always 0.

Step 2: The largest value is 2bits - 1. In this case, 24 - 1 = 16 - 1 = 15

Step 3: The number of values is 2bits. In this case, 24 = 16

Answer: Smallest: 0, Largest: 15, Values: 16

9 bits

Step 1: The smallest value is always 0.

Step 2: The largest value is 2bits - 1. In this case, 29 - 1 = 512 - 1 = 511

Step 3: The number of values is 2bits. In this case, 29 = 512

Answer: Smallest: 0, Largest: 511, Values: 512

Task List Icon Independent Practice

  1. Give the smallest value, largest value, and the number of different values that can be stored with 3 bits.

  2. Give the smallest value, largest value, and the number of different values that can be stored with 11 bits.

  3. Give the smallest value, largest value, and the number of different values that can be stored with 14 bits.

  4. Give the smallest value, largest value, and the number of different values that can be stored with 16 bits.

Answers

  1. 0, 7, 8
  2. 0, 2047, 2048
  3. 0, 16383, 16384
  4. 0, 65535, 65536