[Part 1] Unit 2.1 - Binary Numbers

Summary

This educational script delves into binary representation of integers, explaining how combinations of 0s and 1s can form a vast array of values. It compares binary to decimal systems, illustrating positional values and powers of 2. The script guides through converting decimal to binary and vice versa, highlighting the fixed range of numbers representable with a given number of bits, and setting the stage for future discussions on arithmetic operations and negative numbers in binary form.

Takeaways

• 😎 Bits are the fundamental units in computing, with only two states: 1 and 0.
• 🔍 By combining bits, you can create more complex representations, with 2^n possibilities where n is the number of bits.
• 📘 Binary numbers are represented using powers of 2, similar to how decimal numbers use powers of 10.
• 🔢 The value of a binary number is determined by summing the powers of 2 corresponding to each bit's position.
• 👀 The rightmost bit in a binary sequence represents the 1s place, the next represents the 2s place, and so on.
• 💡 To convert a decimal number to binary, find the largest power of 2 that fits into the number and work downwards, marking 1s for included powers and 0s for excluded ones.
• 🚀 The maximum number representable with k bits is 2^k - 1, which is the sum of all powers of 2 from 2^0 to 2^(k-1).
• 🤔 In computers, a fixed number of bits is usually allocated, limiting the range of representable numbers.
• 🔑 With 8 bits, you can represent 256 different states, but typically half are reserved for negative numbers, leaving 0 to 127 for positive integers.
• 🛠 Understanding binary representation is crucial for performing arithmetic operations on binary numbers, which will be covered in the next unit.
• 🔄 The script provides a foundation for understanding how computers manipulate and represent data at a fundamental level, setting the stage for more complex topics like arithmetic and negative numbers.

Q & A

What are the two basic values used to represent information in a computer?

The two basic values used to represent information in a computer are 0 and 1, which are the binary digits.

How many possibilities can be represented by combining two bits together?

When two bits are combined, there are four possible combinations: 00, 01, 10, and 11.

What is the general formula to determine the number of possibilities when combining n bits?

The general formula to determine the number of possibilities when combining n bits is 2 to the power of n (2^n).

How is the binary representation of an integer number connected to its actual value?

The binary representation of an integer number is connected to its actual value through a positional system, similar to the decimal system, where each bit represents a power of 2.

What is the binary representation of the number 5 using a 4-bit sequence?

The binary representation of the number 5 using a 4-bit sequence is 0101.

How does the positional system in binary numbers differ from the decimal system?

In binary numbers, each position represents a power of 2 instead of a power of 10 as in the decimal system, and it only uses 0 and 1 for its digits.

What is the maximum number that can be represented with k bits?

The maximum number that can be represented with k bits is 2 to the power of k minus 1 (2^(k-1)) minus 1.

Why might we reserve some of the possible values when using a fixed number of bits in a computer?

Some of the possible values are reserved to represent negative numbers, allowing the system to handle both positive and negative integers.

How many different numbers can be represented with 8 bits, and what is the range of these numbers?

With 8 bits, 256 different numbers can be represented. If not considering negative numbers, the range is from 0 to 255.

How can you convert a decimal number to its binary representation?

To convert a decimal number to binary, find the largest power of 2 that fits into the number, then continue with the next largest power of 2 that, when added to the previous sum, does not exceed the number. Place a 1 in the binary sequence for each power of 2 that is part of the sum, and a 0 for those that are not.

What will be the focus of the next unit after discussing the representation of integer numbers in binary?

The next unit will focus on performing arithmetic operations, specifically addition, with binary numbers.

Outlines

00:00

🔢 Binary Number Representation

This paragraph introduces the concept of representing numbers using binary digits (bits), which are only 0s and 1s. It explains how combining bits can create multiple possibilities, allowing representation of various values. The analogy of the decimal system is used to explain positional values in binary, with powers of 2 determining the value of each bit. The process of converting a binary sequence to its decimal equivalent is discussed, as well as the maximum number range that can be represented with a given number of bits. The paragraph also touches on the limitations of fixed bit allocation in computers and the implications for representing both positive and negative numbers.

05:03

🤖 Converting Decimal to Binary

The second paragraph focuses on the method of converting decimal numbers into binary representation. It starts by explaining the process of identifying the largest power of 2 that fits into the given decimal number and then continues to break down the number into a sum of powers of 2. This breakdown is then directly translated into binary form by placing a 1 in the position corresponding to each power of 2 that is part of the sum, and a 0 where it is not. The example of converting the decimal number 87 into binary is provided to illustrate this process. The paragraph concludes by setting the stage for future discussions on arithmetic operations with binary numbers and the representation of negative numbers.

Keywords

💡Bits

Bits are the fundamental units of data in computing and digital communications, representing binary digits that can be either a 0 or a 1. In the script, bits are the building blocks for representing more complex information, such as numbers, and are central to the theme of binary representation.

💡Binary System

The binary system is a base-2 numeral system that uses only two symbols, typically 0 and 1, to represent all possible values. The script discusses the binary system as the foundation for representing numbers in a computer, emphasizing its simplicity and the way it scales to represent a wide range of values.

💡Positional System

A positional system is a method of representing numbers where the value of a digit depends on its position, or place, within the number. The script uses the decimal system as an analogy to explain how binary numbers work, with each position representing a power of 2 instead of a power of 10.

💡Integer Numbers

Integer numbers are whole numbers that can be positive, negative, or zero. The script focuses on how to represent integer numbers using the binary system, starting from simple numbers like 0 and 1 and progressing to more complex representations using multiple bits.

💡Binary Representation

Binary representation is the way numbers are encoded using bits in the binary system. The script explains how any sequence of bits can be converted to a decimal number by summing the powers of 2 corresponding to each bit that is set to 1.

💡Powers of 2

Powers of 2 are the results of 2 raised to an integer power, such as 2^0, 2^1, 2^2, and so on. In the script, powers of 2 are crucial for understanding binary representation, as each bit in a binary number corresponds to a specific power of 2.

💡Decimal Numbers

Decimal numbers are numbers in the base-10 numeral system, which is the most common system used by humans for counting and arithmetic. The script contrasts decimal numbers with binary numbers to illustrate the process of converting between the two systems.

💡Bit Sequence

A bit sequence is a string of bits that can represent data in binary form. The script discusses how different bit sequences can represent different integer numbers, with each bit's position determining its contribution to the overall value.

💡Arithmetic Operations

Arithmetic operations are the basic mathematical operations of addition, subtraction, multiplication, and division. The script mentions that after understanding binary representation, the next unit will cover performing arithmetic operations on binary numbers.

💡Negative Numbers

Negative numbers are numbers less than zero. The script briefly mentions that half of the possible bit representations are reserved for negative numbers in computers, which will be discussed in more detail in a future unit.

💡Fixed Number of Bits

In computers, a fixed number of bits is allocated for representing data, which limits the range of values that can be represented. The script explains that with a fixed number of bits, such as 8 bits, there is a maximum number of unique values that can be represented.

Highlights

Introduction to bit manipulation and the binary system's foundation with only 1s and 0s.

Explaining how combining bits can create more possibilities, such as 2 bits offering 4 possibilities.

General formula for representing possibilities with n bits as 2 to the power of n.

Representing numbers in binary, starting with simple numbers like 0 and 1, and moving to more complex representations.

The concept of positional value in binary, similar to the decimal system but with powers of 2.

Binary representation of the number 5 as an example, demonstrating positional value calculation.

General method for converting any binary sequence to its integer value using positional values.

Explanation of the maximum number representable with k bits and the calculation of this range.

The fixed number of bits in computers and the implications for the range of representable numbers.

The practical example of 8-bit representation, explaining the range from 0 to 255.

Discussion on reserving part of the 8-bit possibilities for negative numbers in computers.

Conversion of decimal numbers to binary representation, using the example of the number 87.

Methodology for determining the largest power of 2 that fits into a given decimal number for binary conversion.

Technique for converting decimal numbers to binary by summing powers of 2 and marking bits accordingly.

The significance of the rightmost bit in binary representation and its relation to the 1s place.

General approach to converting any decimal number to binary, emphasizing the process of summing powers of 2.

Conclusion of the unit on representing integer numbers in the binary system and a preview of upcoming topics on arithmetic operations with binary numbers.

Transcripts

Last week, we learned how to manipulate bits in a computer.

We have yes, we have no, 1, 0, and we know how to manipulate them.

But what can we do with only 1 and 0?

Can we represent more sophisticated things with them?

Of course we, we must, if we want to use a computer to do useful things.

But what can you do with only two values, 0 and 1?

Well, you definitely can put two of them together, and

then you get four possibilities.

Three of them together, you get eight possibilities.

In general, if you have n of them together,

you have 2 to the n possibilities and now you could pre,

represent any 2 to the n different things that you may want to.

For example, you can represent numbers.

And this is what we're going to be interested in.

How are you going to represent integer numbers?

Well, 0 is easy.

0, 1 is easy.

1.

2 is already the first problem, basically because we don't have yet

another possibility, so we have to use two bits.

0, 10.

Three is 11.

For the next one, we already need another bit, 110 for 4.

And in general,

it seems we can represent any number that we want with some sequence of bits.

Now how did we co, connect the sequence of bits,

the binary representation of an integer number, to the integer number?

What's going on?

What's the general system?

In order to understand that, we have to go back to second grade.

When we learned about decimal numbers.

When you see 789, what does that mean?

Where, what does the digit 7, 8 and 9 have to do with the value of the number 789?

Well, we learned that we have the positional system, where the right-most

digit is the ones, the next one is the tens, the next one is the 100s.

So we know that 789 is really 9 plus eight 10 plus seven 100s, and so on.

In general, the case position from the right is 10 to the k.

So now exactly the same thing that was going to happen also with binary numbers,

but it's going to be much simpler because we only have 0 and

1 rather than all the numbers, digits between 0 and 9.

Thus in binary notation, the different positions will be powers of 2.

1, 2, 4, 8 and so on.

Suppose you want to know, for example, what is a value of 1 0 1,

in binary notation?

Well, we know the rightmost bit is basically the 1s.

The next one, the 0, correspond to 2s.

And the third, the left-most bit here, is the 4s.

Altogether, we have 4 plus 2 plus 2 times 0, plus 1 times 1.

Altogether, that's 5, so our number is 5 and in general, what do we do in general?

Exactly the same thing.

You have any sequence of bits and we are going to number them from the right.

Most bit, which is going to be b 0, the next one b 1, and so

on, all the way to bn, if we have n plus 1 bits.

And the value of this thing is going to be b0 times 2 to

the 0 plus d1 times 2 to the 1, plus b2 time 2 to the 2,

all the way until the end of it, which we have, where we have bn time 2 to the n.

And, this is going to be a number.

And, that's how we can covert any sequence of bits,

any binary representation of a number to the value of a number.

One thing that we should note at this point that if we look at the maximum

number we can represent with k bits.

Well we sum up from 2 to the 0 all the way to the k minus 1.

Remember that the case bit if we start counting from 0,

the last bit is not counted as index k, k minus 1.

So we have the sum of 1 plus 2 plus 4, all the way to the 2k minus 1.

All together, we have 2 to the k of that minus 1.

And that's a range of bits, we can actually represent with k.

The range of numbers we re, can represent with k bits.

Now, so far, we assumed,

we have an arbitrary length number of bits that we can use to represent.

And of course, if we want to represent an arbitrary long number,.

We will need an arbitrary number of bits.

In computers, you usually have a fixed number of bits that is allocated.

And then of course you can only be able to represent a fixed number,

a fixed range of integer numbers.

So for example, eh, if we only have eight bits, what are the,

what are the possible numbers that you can represent?

Well you can represent that something starting with 000 eight times.

000 ending with a 1 all the way up to eight 1s.

The smallest number is, we have all together 256, 256 such possibilities.

The first one is index 0, the last one is index 255.

Now, that's not exactly true.

Really, when we have eight bits, usually we want to, to, want to reserve

part of these, all the possibilities to actually represent negative numbers.

We're not going to actually talk about this now, but rather in a unit from now.

But in general, half of the indi, half of the 256 possibilities are going to be re,

be reserved for negative numbers, and we're only going to be able to use

the numbers between 1, between 0 and 127, and these are the positive numbers

that we're going to be able to use out of these 256 possibilities of 8 bits.

So far we saw, basically, if you better or

a string of bits, how can you convert it to decimal numbers?

Now we are going to do the opposite thing,

suppose you were given a number in decimal, 87 for example.

How can you represent it as a sequence of bits?

This is also something which we should be able to do, if we're going back and

forth between decimal notation and binary numbers.

That they're actually going to be eh, represented as in the computer.

Well remember that we know that really the way that we get the decimal from

the binary is by summing up powers of 2.

So we start by figuring out,

figuring out what is the largest power of 2 that fits into our 87 number.

And that is going to be 64.

And then, what is the next one?

After we have 64, what is the next power of 2 that we can add to 64 and

we still remain under 87?

That turns out to be 16.

And so on, we can keep on going, and we write the number 87 as a sum

of binary pow, of powers of 2.

And it turns out that 87 is exactly 64 plus 16 plus 4 plus 2 plus 1 and once we

have that, from this representation of the decimal number, as a sum of powers of 2,

we can quite directly actually get the binary representation.

How do you do it?

Well every time,

we have a power that the peer is in the sum we need to put a 1 in the bit there.

And whenever is a power is not part of the sum, we put a 0 there.

So for example, look at the right-most bit, that corresponds to the 1s.

We do have a 1, so that's going to be a 1.

On the other hand,

if you look at that third bit from the right, which is correspond to the 8s.

Since we don't have 8 in the sum there, we're going to have 0s there.

And that's basically a gener, general way how you can take any number and

convert it to binary.

So this concludes this unit where basically we discussed

how you can represent integer numbers within a binary system.

In the next unit,

we're going to actually discuss how can we actually perform arithmetic operations.

In particular, addition and these represented binary numbers.

Once we get that under our belt, we will go back and

discuss the issue of negative numbers in unit 3.