A number system is a system for representing numbers on a digital computer. Number systems are important because they provide a way for computers to represent and manipulate numerical data. The most commonly used number system in digital computers is the binary number system, which uses only two digits: 0 and 1. Other number systems that are used in computing include decimal, octal, and hexadecimal.
Binary number system
The binary number system is a number system that uses only two digits: 0 and 1. It is based on the concept of binary digits, which are the simplest form of data that a digital computer can understand and manipulate. In the binary system, each digit is known as a bit and each bit has a value of either 0 or 1. The binary number system is the foundation of all digital computing and is used by computers to represent and manipulate numerical data. It is important because it allows computers to perform arithmetic operations and to store and retrieve data in a convenient and efficient manner.
Here is an example of how the binary number system works:
- In the decimal (base-10) number system, the number 123 is represented as 1x10^2 + 2x10^1 + 3x10^0.
- In the binary (base-2) number system, the number 123 is represented as 1x2^7 + 1x2^6 + 0x2^5 + 1x2^4 + 1x2^3 + 0x2^2 + 1x2^1 + 1x2^0.
This can be written more concisely as 1111011 in binary, or 123 in decimal.
Decimal to binary conversion
To convert a decimal number to binary, you can use the following steps:
- Divide the decimal number by 2 and write down the remainder.
- Divide the result from step 1 by 2 and write down the remainder.
- Continue dividing the result from the previous step by 2 and writing down the remainder until the result is 0.
- Read the remainders from bottom to top to get the binary representation of the decimal number.
Here is an example:
- To convert the decimal number 123 to binary, we start by dividing it by 2:
123 / 2 = 61 R 1
- We then divide the result from the previous step by 2 and write down the remainder:
61 / 2 = 30 R 1
- We continue dividing the result from the previous step by 2 and writing down the remainder until the result is 0:
30 / 2 = 15 R 0
15 / 2 = 7 R 1
7 / 2 = 3 R 1
3 / 2 = 1 R 1
1 / 2 = 0 R 1
We read the remainders from bottom to top to get the binary representation of the decimal number 123:
123 in decimal = 1111011 in binary
Decimal to binary - 458, 13, 369
To convert a decimal number to binary, you need to divide the number by 2 repeatedly and take the remainder each time. Here's how the process would work for each of the numbers you provided:
For 458:
- 458 / 2 = 229 remainder 0
- 229 / 2 = 114 remainder 1
- 114 / 2 = 57 remainder 0
- 57 / 2 = 28 remainder 1
- 28 / 2 = 14 remainder 0
- 14 / 2 = 7 remainder 0
- 7 / 2 = 3 remainder 1
- 3 / 2 = 1 remainder 1
- 1 / 2 = 0 remainder 1
For 13:
- 13 / 2 = 6 remainder 1
- 6 / 2 = 3 remainder 0
- 3 / 2 = 1 remainder 1
- 1 / 2 = 0 remainder 1
For 369:
- 369 / 2 = 184 remainder 1
- 184 / 2 = 92 remainder 0
- 92 / 2 = 46 remainder 0
- 46 / 2 = 23 remainder 0
- 23 / 2 = 11 remainder 1
- 11 / 2 = 5 remainder 1
- 5 / 2 = 2 remainder 1
- 2 / 2 = 1 remainder 0
- 1 / 2 = 0 remainder 1
The binary representation of a decimal number is the sequence of remainders (starting from the bottom) in reverse order. So, the binary representation of 458 is 111001010, the binary representation of 13 is 1101, and the binary representation of 369 is 101110001.
For 37
- Divide 37 by 2 and record the remainder of 1: 37 / 2 = 18 R 1
- Divide the result of the previous division by 2 and record the remainder of 0: 18 / 2 = 9 R 0
- Divide the result of the previous division by 2 and record the remainder of 1: 9 / 2 = 4 R 1
- Divide the result of the previous division by 2 and record the remainder of 0: 4 / 2 = 2 R 0
- Divide the result of the previous division by 2 and record the remainder of 0: 2 / 2 = 1 R 0
- Divide the result of the previous division by 2 and record the remainder of 1: 1 / 2 = 0 R 1
Binary to decimal conversion
To convert a binary number to decimal, you can use the following steps:
- Write down the binary number and assign each digit a value based on its position in the number. The rightmost digit has a value of 2^0, the next digit to the left has a value of 2^1, and so on.
- Multiply each digit by its assigned value and add the results together to get the decimal representation of the binary number.
Here is an example:
- To convert the binary number 1111011 to decimal, we start by writing down the number and assigning each digit a value based on its position:
1 1 1 1 0 1 1
1 2 4 8 0 2 4
- We then multiply each digit by its assigned value and add the results together to get the decimal representation of the binary number:
1x1 + 1x2 + 1x4 + 1x8 + 0x0 + 1x2 + 1x4 = 123
1111011 in binary = 123 in decimal
Binary to decimal 1110001, 10010111, 1111
For example, to convert 1110001 to decimal, we can follow these steps:
- Write down the digits of the binary number: 1, 1, 1, 0, 0, 0, 1
- For each digit, multiply it by 2 to the power of its position:
1 * 2^0 = 1 * 1 = 1
1 * 2^1 = 1 * 2 = 2
1 * 2^2 = 1 * 4 = 4
0 * 2^3 = 0 * 8 = 0
0 * 2^4 = 0 * 16 = 0
0 * 2^5 = 0 * 32 = 0
1 * 2^6 = 1 * 64 = 64
- Add up the products to get the decimal representation of the binary number: 1 + 2 + 4 + 0 + 0 + 0 + 64 = 73
- Write down the digits of the binary number: 1, 0, 0, 1, 0, 1, 1, 1
- For each digit, multiply it by 2 to the power of its position:
1 * 2^0 = 1 * 1 = 1
0 * 2^1 = 0 * 2 = 0
0 * 2^2 = 0 * 4 = 0
1 * 2^3 = 1 * 8 = 8
0 * 2^4 = 0 * 16 = 0
1 * 2^5 = 1 * 32 = 32
1 * 2^6 = 1 * 64 = 64
1 * 2^7 = 1 * 128 = 128
- Add up the products to get the decimal representation of the binary number: 1 + 0 + 0 + 8 + 0 + 32 + 64 + 128 = 151
- Write down the digits of the binary number: 1, 1, 1, 1
- For each digit, multiply it by 2 to the power of its position:
1 * 2^0 = 1 * 1 = 1
1 * 2^1 = 1 * 2 = 2
1 * 2^2 = 1 * 4 = 4
1 * 2^3 = 1 * 8 = 8
- Add up the products to get the decimal representation of the binary number: 1 + 2 + 4 + 8 = 15
- Therefore, the decimal representation of 1110001 is 113, the decimal representation of 10010111 is 151, and the decimal representation of 1111 is 15.
100101 in decimal:
1 * 2^5 = 32
0 * 2^4 = 0
0 * 2^3 = 0
1 * 2^2 = 4
0 * 2^1 = 0
1 * 2^0 = 1
32 + 4 + 1 = 37
1001100 in decimal:
1 * 2^6 = 64
0 * 2^5 = 0
0 * 2^4 = 0
1 * 2^3 = 8
1 * 2^2 = 4
0 * 2^1 = 0
0 * 2^0 = 0
64 + 8 + 4 + 0 = 76
011011:
To get the decimal equivalent of binary 011011, we need to multiply each binary digit by the corresponding power of 2 and add them up.
0 * 2^5 = 0
1 * 2^4 = 16
1 * 2^3 = 8
0 * 2^2 = 0
1 * 2^1 = 2
1 * 2^0 = 1
16 + 8 + 2 + 1 = 27
To get the decimal equivalent of binary 100101, we need to multiply each binary digit by the corresponding power of 2 and add them up.
1 * 2^5 = 32
0 * 2^4 = 0
0 * 2^3 = 0
1 * 2^2 = 4
0 * 2^1 = 0
1 * 2^0 = 1
32 + 4 + 1 = 45
Octal number system
The octal number system, or base-8 system, is a numeral system that uses eight digits, 0 to 7, to represent numbers. It is a useful system for representing and manipulating binary data in computing and digital electronics because it provides a convenient way to group binary digits into units of 8 bits (called octets).
Here is an example of how the octal number system works:
- In the decimal (base-10) number system, the number 123 is represented as 1x10^2 + 2x10^1 + 3x10^0.
- In the octal (base-8) number system, the number 123 is represented as 1x8^2 + 2x8^1 + 3x8^0.
- This can be written more concisely as 173 in octal, or 123 in decimal.
Conversion of decimal numbers to octal
To convert a decimal number to an octal, you can use the following steps:
- Divide the decimal number by 8 and write down the remainder.
- Divide the result from step 1 by 8 and write down the remainder.
- Continue dividing the result from the previous step by 8 and writing down the remainder until the result is 0.
- Read the remainders from bottom to top to get the octal representation of the decimal number.
Here is an example:
- To convert the decimal number 123 to octal, we start by dividing it by 8:
123 / 8 = 15 R 3
- We then divide the result from the previous step by 8 and write down the remainder:
15 / 8 = 1 R 7
- We continue dividing the result from the previous step by 8 until the result is 0:
1 / 8 = 0 R 1
- We read the remainders from bottom to top to get the octal representation of the decimal number 123:
123 in decimal = 173 in octal
Decimal to octal - 2356, 2569, 3001, 526
To convert 2356 to octal, we can follow these steps:
- Divide 2356 by 8: 2356 / 8 = 294 with a remainder of 4
- Divide 294 by 8: 294 / 8 = 36 with a remainder of 6
- Divide 36 by 8: 36 / 8 = 4 with a remainder of 4
- Divide 4 by 8: 4 / 8 = 0 with a remainder of 4
To convert 342 to octal, we can follow these steps:
- 342 / 8 = 42 R 6
- 42 / 8 = 5 R 2
- 5 / 8 = 0 R 5
So, 342 in octal is 526.
To convert 2569 to octal, we can follow these steps:
- Divide 2569 by 8: 2569 / 8 = 321 with a remainder of 5
- Divide 321 by 8: 321 / 8 = 40 with a remainder of 1
- Divide 40 by 8: 40 / 8 = 5 with a remainder of 0
- Divide 5 by 8: 5 / 8 = 0 with a remainder of 5
The remainders from the division steps (5, 1, 0, 5) are written in reverse order to get the octal representation of 2569: 5015.
To convert 3001 to octal, we can follow these steps:
- Divide 3001 by 8: 3001 / 8 = 375 with a remainder of 1
- Divide 375 by 8: 375 / 8 = 46 with a remainder of 7
- Divide 46 by 8: 46 / 8 = 5 with a remainder of 6
- Divide 5 by 8: 5 / 8 = 0 with a remainder of 5
The remainders from the division steps (1, 7, 6, 5) are written in reverse order to get the octal representation of 3001: 5671
Converting from octal to decimal
To convert an octal number to a decimal, you can use the following steps:
- Write down the octal number and assign each digit a value based on its position in the number. The rightmost digit has a value of 8^0, the next digit to the left has a value of 8^1, and so on.
- Multiply each digit by its assigned value and add the results together to get the decimal representation of the octal number.
Here is an example:
- To convert the octal number 173 to decimal, we start by writing down the number and assigning each digit a value based on its position:
1 7 3
1 8 1
- We then multiply each digit by its assigned value and add the results together to get the decimal representation of the octal number:
1x1 + 7x8 + 3x1 = 123
173 in octal = 123 in decimal
To convert 1414 from octal to decimal, we need to multiply each octal digit by the corresponding power of 8 and add them up.
- 1 * 8^3 = 512
- 4 * 8^2 = 256
- 1 * 8^1 = 8
- 4 * 8^0 = 4
512 + 256 + 8 + 4 = 780, so 1414 in octal is 780 in decimal.
The octal equivalent of binary 100 is 4.
In binary, 100 is equal to 1 * 2^2 = 4 in decimal. In octal, 4 is represented by the digit 4.
Hexadecimal system
The hexadecimal number system, or base-16 system, is a numeral system that uses sixteen digits, 0 to 9 and A to F, to represent numbers. It is a useful system for representing and manipulating binary data in computing and digital electronics because it provides a convenient way to group binary digits into units of 4 bits (called nibbles) and 8 bits (called bytes).
Here is an example of how the hexadecimal number system works:
- In the decimal (base-10) number system, the number 123 is represented as 1x10^2 + 2x10^1 + 3x10^0.
- In the hexadecimal (base-16) number system, the number 123 is represented as 1x16^2 + 2x16^1 + 3x16^0.
- This can be written more concisely as 7B in hexadecimal, or 123 in decimal.
Conversion from decimal to hexadecimal
To convert a decimal number to hexadecimal, you can use the following steps:
- Divide the decimal number by 16 and write down the remainder.
- Divide the result from step 1 by 16 and write down the remainder.
- Continue dividing the result from the previous step by 16 and writing down the remainder until the result is 0.
- Read the remainders from bottom to top to get the hexadecimal representation of the decimal number. If any of the remainders is greater than 9, convert it to its corresponding hexadecimal digit (A-F) before writing it down.
Here is an example:
- To convert the decimal number 123 to hexadecimal, we start by dividing it by 16:
123 / 16 = 7 R 11
- We then divide the result from the previous step by 16 and write down the remainder:
7 / 16 = 0 R 7
- We continue dividing the result from the previous step by 16 until the result is 0:
- Since the result from the previous step is 0, there are no more remainders to write down.
- We read the remainders from bottom to top to get the hexadecimal representation of the decimal number 123:
- The remainder 11 is greater than 9, so we convert it to its corresponding hexadecimal digit (B) before writing it down:
123 in decimal = 7B in hexadecimal
To convert 154 from decimal to hexadecimal, we need to divide 154 by 16 and record the remainder until we get a quotient of 0.
- 154 / 16 = 9 R 10
- 9 / 16 = 0 R 9
The remainders 10 and 9 correspond to the hexadecimal digits A and 9, respectively. So, 154 in decimal is 99 in hexadecimal.
Conversion of hexadecimal to decimal
To convert a hexadecimal number to a decimal, you can use the following steps:
- Write down the hexadecimal number and assign each digit a value based on its position in the number. The rightmost digit has a value of 16^0, the next digit to the left has a value of 16^1, and so on.
- Convert each hexadecimal digit to its corresponding decimal value and multiply it by its assigned value. Add the results together to get the decimal representation of the hexadecimal number.
Here is an example:
- To convert the hexadecimal number 7B to decimal, we start by writing down the number and assigning each digit a value based on its position:
7 B
1 16
- We then convert each hexadecimal digit to its corresponding decimal value and multiply it by its assigned value:
7 in decimal x 1 = 7
B in decimal x 16 = 11
- We add the results together to get the decimal representation of the hexadecimal number:
7 + 11 = 123
7B in hexadecimal = 123 in decimal
Hexadecimal to decimal - AB12, 5600, AFE, 9A
To convert AB12 to decimal, we can follow these steps:
- Write down the digits of the hexadecimal number: A, B, 1, 2
- For each digit, multiply it by 16 to the power of its position:
A * 16^0 = 10 * 1 = 10
B * 16^1 = 11 * 16 = 176
1 * 16^2 = 1 * 256 = 256
2 * 16^3 = 2 * 4096 = 8192
- Add up the products to get the decimal representation of the hexadecimal number: 10 + 176 + 256 + 8192 = 43714
Similarly, to convert 5600 to decimal, we can follow these steps:
- Write down the digits of the hexadecimal number: 5, 6, 0, 0
- For each digit, multiply it by 16 to the power of its position:
5 * 16^0 = 5 * 1 = 5
6 * 16^1 = 6 * 16 = 96
0 * 16^2 = 0 * 256 = 0
0 * 16^3 = 0 * 4096 = 0
- Add up the products to get the decimal representation of the hexadecimal number: 5 + 96 + 0 + 0 = 101
And to convert AFE to decimal, we can follow these steps:
- Write down the digits of the hexadecimal number: A, F, E
- For each digit, multiply it by 16 to the power of its position:
A * 16^0 = 10 * 1 = 10
F * 16^1 = 15 * 16 = 240
E * 16^2 = 14 * 256 = 3584
- Add up the products to get the decimal representation of the hexadecimal number: 10 + 240 + 3584 = 2814
- Therefore, the decimal representation of AB12 is 43714, the decimal representation of 5600 is 21840, and the decimal representation of AFE is 2814.
To convert 9A from hexadecimal to decimal, we need to multiply each hexadecimal digit by the corresponding power of 16 and add them up.
- 9 * 16^1 = 144
- A * 16^0 = 10
144 + 10 = 154, so 9A in hexadecimal is 154 in decimal.
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