基底为2：二进制数

Binary

Binary might be the second most known number system. Whereas our number system uses $10$ digits, binary uses only $2$ digits: $$0,\,1$$

"Computers use binary to do things." $\\$ -Alex on the topic of binary.

From Decimal to Binary

Convert from decimal to binary can be a little tedious. We want to take the number and continuously divide it by $2,$ rounding down for decimals, and listing the numbers out. Each even number we represent with a $0$ and each odd number we represent with $1.$

That's a lot of words. Maybe an example will make things clearer. Let's convert $42_{10}$ (the little $10$ means base-$10$) to base-$2.$

Start by dividing $42$ by $2$ until we get down to a $1.$

Looking from the bottom number up, we have the numbers: $$\begin{array}{ccc} 1&2&5&10&21&42 \end{array}$$ The final step is to put a $1$ under the odd numbers and a $0$ under the even. $$\begin{array}{ccc} 1&2&5&10&21&42\\ 1&0&1&0&1&0 \end{array}$$ So: $$42_{10}=101010_2$$

From Binary to Decimal

Luckily, going from binary to decimal isn't as difficult. Starting from the last number, we want to multiply each digit by $2$ raised to increasing powers (starting from $0$ and moving up). We then add all these up.

Let's see it in action with our newly converted base-$2$ number, $101010.$ $$\begin{align} &101010\\ &=(1\cdot2^5)+(0\cdot2^4)+(1\cdot2^3)+(0\cdot2^2)+(1\cdot2^1)+(0\cdot2^0)\\ &=(1\cdot32)+(0\cdot16)+(1\cdot8)+(0\cdot4)+(1\cdot2)+(0\cdot1)\\ &=32+0+8+0+2+0\\ &=42 \end{align}$$

So: $$101010_2=42_{10}$$

Practice

Now that you're basically a glorified computer, try some out on your own.

Solution

We want to divide $19$ by $2,$ rounding down, until we reach $1.$ $$\begin{array}{ccc} 19&\div&2&=&9.5\\ 9&\div&2&=&4.5\\ 4&\div&2&=&2\\ 2&\div&2&=&1\\ 1 \end{array}$$

Next, put a $1$ for the odd numbers and a $0$ for the even numbers. $$\begin{array}{ccc} \color{red}{1}&19&\div&2&=&9.5\\ \color{red}{1}&9&\div&2&=&4.5\\ \color{red}{0}&4&\div&2&=&2\\ \color{red}{0}&2&\div&2&=&1\\ \color{red}{1}&1 \end{array}$$

Write this number from bottom up. $$10011_2=19_{10}$$

Solution

When going from binary to decimal, we want to multiply each digit by a power of $2.$ We start with the last digit and multiply it by $2$ raised to the zero power. $$\begin{align} &100110\\ &=(1\cdot2^5)+(0\cdot2^4)+(0\cdot2^3)+(1\cdot2^2)+(1\cdot2^1)+(0\cdot2^0)\\ &=(1\cdot2^5)+(1\cdot2^2)+(1\cdot2^1)\\ &=2^5+2^2+2^1\\ &=32+4+2\\ &=38 \end{align}$$

So, $$38_{10}=100110_2$$

(Video) Convert Numbers into Binary

by Mind Your Decisions

Here's a video by MindYourDecisions explaining how to convert from base-$10$ to base-$2.$

$\large{\text{Summary}}$

To convert from $10$ to $2,$ divide the number by $2$ (ignoring the remainder) until you reach $1.$ Then any odd number gets a $1$ below it and any even gets a $0.$ The following string of $1$'s and $0$'s is your new binary number. To go from base-$2$ to base-$10$ you use a similar method explained in the base-$3$ section but this time multiply by $2^n$ instead of $3^n$

Think about a decimal (base $10$) number: $7.$ The binary number representation of $7$ is $111.$ The base $2$ representation (or equivalently, binary representation) of $21$ is $10101.$

Let's analyze this: $21 = 2 \times 10 + 1$. In base $10,$ each digit represents a power of $10.$ Now going back to $7,$ it is $7 = 1 \times 2^2 +1 \times 2 + 1 \times 1$. Here, the digits are just powers of the base.

There are many equivalent ways to represent a number.

For example, twelve can be represented as $2\times6,9+3,$ or just $12$.

Normally, we represent numbers by using the digits $0$ through $9$. This is known as a base-ten number. A base-ten number is equal to $1$ times its first digit (from the right), plus $10$ times its second digit, plus $100$ times its third digit, and so on. For example, the base-ten number $12$ is equal to $2\times1+1\times10$.

Another way to represent numbers is by using the digits $0$ and $1$. This is known as a binary number. A binary number is equal to $1$ times its first digit (from the right), plus $2$ times its second digit, plus $4$ times its third digit, and so on. For example, the binary number $1100_{2}$ is equal to $0\times1+0\times2+1\times4+1\times8$. Although $12$ and $1100_{2}$ look nothing alike, they are actually equivalent numbers!

When we write out binary numbers, we append a subscripted 2 to the end of the number. This is to differentiate a binary number from a base-ten number. We do not need to add a subscripted 10 to the end of base-ten numbers because they are the default.

Now that you have some idea of how binary numbers work, try to do some of the examples!