Convert The Number { 1456_{10} $}$ To Base 4.Provide Your Answer In Both Lower Bases And Higher Bases.

Introduction

In mathematics, converting numbers from one base to another is an essential skill that helps us understand the representation of numbers in different number systems. In this article, we will focus on converting the decimal number $1456_{10}$ to base 4. We will provide the answer in both lower and higher bases, making it easier for readers to understand the concept.

What is Base 4?

Base 4 is a number system that uses four distinct symbols or digits to represent numbers. The four digits used in base 4 are 0, 1, 2, and 3. In base 4, each digit position represents a power of 4, starting from the rightmost digit, which represents $4^0$, then $4^1$, $4^2$, and so on.

Converting Decimal to Base 4

To convert a decimal number to base 4, we need to divide the decimal number by 4 and keep track of the remainders. The remainders will form the digits of the base 4 number, starting from the rightmost digit.

Step 1: Divide the Decimal Number by 4

We start by dividing the decimal number $1456_{10}$ by 4.

$1456 \div 4 = 364$ with a remainder of 0.

Step 2: Divide the Quotient by 4

Next, we divide the quotient 364 by 4.

$364 \div 4 = 91$ with a remainder of 0.

Step 3: Divide the Quotient by 4

We continue by dividing the quotient 91 by 4.

$91 \div 4 = 22$ with a remainder of 3.

Step 4: Divide the Quotient by 4

Next, we divide the quotient 22 by 4.

$22 \div 4 = 5$ with a remainder of 2.

Step 5: Divide the Quotient by 4

We continue by dividing the quotient 5 by 4.

$5 \div 4 = 1$ with a remainder of 1.

Step 6: Divide the Quotient by 4

Next, we divide the quotient 1 by 4.

$1 \div 4 = 0$ with a remainder of 1.

Step 7: Write the Remainders in Reverse Order

Now that we have the remainders, we write them in reverse order to get the base 4 representation of the decimal number $1456_{10}$.

The base 4 representation of $1456_{10}$ is $21100_4$.

Higher Base Representation

To represent the base 4 number $21100_4$ in higher bases, we can convert it to base 8 (octal) and base 16 (hexadecimal).

Base 8 (Octal) Representation

To convert the base 4 number $21100_4$ to base 8, we need to divide the base 4 number by 8 and keep track of the remainders.

$21100_4 \div 8 = 5260_4$ with a remainder of 4.

$5260_4 \div 8 = 1301_4$ with a remainder of 4.

$1301_4 \div 8 = 321_4$ with a remainder of 1.

$321_4 \div 8 = 80_4$ with a remainder of 1.

$80_4 \div 8 = 20_4$ with a remainder of 0.

$20_4 \div 8 = 2_4$ with a remainder of 4.

$2_4 \div 8 = 0_4$ with a remainder of 2.

The base 8 representation of $21100_4$ is $24411410_8$.

Base 16 (Hexadecimal) Representation

To convert the base 4 number $21100_4$ to base 16, we need to divide the base 4 number by 16 and keep track of the remainders.

$21100_4 \div 16 = 5260_4$ with a remainder of 4.

$5260_4 \div 16 = 1301_4$ with a remainder of 4.

$1301_4 \div 16 = 321_4$ with a remainder of 1.

$321_4 \div 16 = 80_4$ with a remainder of 1.

$80_4 \div 16 = 20_4$ with a remainder of 0.

$20_4 \div 16 = 2_4$ with a remainder of 0.

$2_4 \div 16 = 0_4$ with a remainder of 2.

The base 16 representation of $21100_4$ is $2A4$.

Conclusion

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Q: What is the process of converting a decimal number to base 4?

A: The process of converting a decimal number to base 4 involves dividing the decimal number by 4 and keeping track of the remainders. The remainders will form the digits of the base 4 number, starting from the rightmost digit.

Q: How do I convert a decimal number to base 4?

A: To convert a decimal number to base 4, follow these steps:

1. Divide the decimal number by 4 and note the remainder.
2. Divide the quotient by 4 and note the remainder.
3. Repeat step 2 until the quotient is 0.
4. Write the remainders in reverse order to get the base 4 representation of the decimal number.

Q: What is the base 4 representation of the decimal number 1456?

A: The base 4 representation of the decimal number 1456 is 21100.

Q: How do I convert a base 4 number to a higher base, such as base 8 or base 16?

A: To convert a base 4 number to a higher base, such as base 8 or base 16, you can divide the base 4 number by the base you want to convert to and keep track of the remainders. The remainders will form the digits of the higher base number, starting from the rightmost digit.

Q: What is the base 8 representation of the base 4 number 21100?

A: The base 8 representation of the base 4 number 21100 is 24411410.

Q: What is the base 16 representation of the base 4 number 21100?

A: The base 16 representation of the base 4 number 21100 is 2A4.

Q: Why is it important to understand how to convert decimal numbers to base 4?

A: Understanding how to convert decimal numbers to base 4 is important because it helps you to understand the representation of numbers in different number systems. It also helps you to perform calculations and conversions between different number systems.

Q: Can I use a calculator to convert decimal numbers to base 4?

A: Yes, you can use a calculator to convert decimal numbers to base 4. However, it's also important to understand the process of converting decimal numbers to base 4 by hand, as it helps you to develop your problem-solving skills and understand the underlying mathematics.

Q: How do I convert a base 4 number to a decimal number?

A: To convert a base 4 number to a decimal number, you can multiply each digit of the base 4 number by the corresponding power of 4 and add the results together.

Q: What is the decimal representation of the base 4 number 21100?

A: The decimal representation of the base 4 number 21100 is 1456.

Q: Can I convert a base 4 number to a base 8 or base 16 number using a calculator?

A: Yes, you can use a calculator to convert a base 4 number to a base 8 or base 16 number. However, it's also important to understand the process of converting base 4 numbers to higher bases by hand, as it helps you to develop your problem-solving skills and understand the underlying mathematics.