Mudança de base sem envolver bases com relação de potência da base 10 para alguma outra base

Summary

This video lesson teaches how to convert numbers from base 10 (decimal) to other numeral systems, with a focus on base 16 (hexadecimal). It covers the fundamental concepts of numeral bases, such as the limited set of digits in each base, and demonstrates how to convert both the integer and fractional parts of a decimal number into the desired base. The process includes step-by-step division for the integer part and successive multiplication for the fractional part. The video also explains how to handle negative numbers during the conversion process, ensuring comprehensive understanding of numeral base conversions.

Takeaways

• 😀 The lesson teaches how to convert a decimal number (base 10) to another base, such as base 16 (hexadecimal).
• 😀 A number written with a subscript after the parentheses indicates the base it is written in.
• 😀 Numbers without a specified base are assumed to be in base 10 by default.
• 😀 Each base has a limited set of digits, and the number of digits corresponds to the base. For example, base 2 has digits 0 and 1, while base 16 includes digits 0-9 and letters A-F.
• 😀 Digits in a number retain their value regardless of the base in which they appear. For example, '7' is always worth 7, whether in base 10, base 16, or any other base.
• 😀 As the base increases, more digits are introduced. For example, base 11 uses digits 0-9 and the letter 'A' for 10, while base 12 includes 'B' for 11, and so on.
• 😀 To convert a decimal number to another base, you separate the integer and fractional parts and apply different methods to each part.
• 😀 For the integer part, successive integer divisions by the base (16 for hexadecimal) are performed until the result is 0. The remainders are then mapped to the corresponding digits.
• 😀 For the fractional part, you multiply the fractional value by the base (16), extracting the integer part at each step, and continue until the fractional part is 0.
• 😀 After converting both the integer and fractional parts, the results are combined, with the integer part being written in reverse order of the division remainders, and the fractional part using the extracted digits.
• 😀 When converting a negative number, you perform the same steps as for positive numbers but reattach the negative sign at the end of the process.

Q & A

What is the main objective of this video lesson?

The main objective is to teach how to convert a decimal number (base 10) into another base, such as binary, hexadecimal, or any other base.

What does the subscript indicate in a number written with parentheses?

The subscript indicates the numeric base of the number. For example, if a number is written as (123)_2, it means the number 123 is in base 2 (binary).

How are digits represented in bases higher than 10?

In bases higher than 10, digits 10-15 are represented by letters A-F. For example, in hexadecimal (base 16), the digits 10-15 are represented as A, B, C, D, E, and F, respectively.

How do we convert the integer part of a decimal number into a different base?

To convert the integer part, we divide the number by the target base (e.g., 16 for hexadecimal) and record the remainder. This process is repeated until the quotient is zero, and the remainders are read in reverse order to form the new number in the target base.

What is the method for converting the fractional part of a decimal number into a different base?

To convert the fractional part, we multiply the fraction by the target base (e.g., 16 for hexadecimal) and record the integer part. This process is repeated with the fractional part of the result, and the integer parts collected form the digits of the fractional part in the new base.

What happens when converting a negative decimal number?

To convert a negative decimal number, you ignore the negative sign during the conversion process, and after obtaining the result, you add the negative sign to the final converted number.

Why is a table of values important in base conversion?

A table of values is important because it helps to map numerical values greater than 9 (in bases higher than 10) to their corresponding digit symbols. For example, in hexadecimal, the value 10 corresponds to 'A', 11 to 'B', and so on.

Can you provide an example of converting a decimal integer to hexadecimal?

Sure! To convert 2020 (decimal) to hexadecimal, we divide 2020 by 16, which gives a quotient of 126 and a remainder of 4. Then divide 126 by 16, resulting in a quotient of 7 and a remainder of 14 (which corresponds to 'E' in hexadecimal). Finally, divide 7 by 16 to get a quotient of 0 and a remainder of 7. The result in hexadecimal is 7E4.

What does 'successive divisions' mean when converting a decimal number?

Successive divisions refer to the repeated process of dividing the decimal number by the target base (e.g., 16 for hexadecimal) and recording the remainders at each step. The process stops when the quotient becomes zero.

What is the significance of reversing the order of remainders during base conversion?

The significance of reversing the order of remainders is that the first remainder corresponds to the least significant digit (rightmost) in the target base, and the last remainder corresponds to the most significant digit (leftmost). Therefore, reading the remainders in reverse order gives the correct number in the new base.