Math Problem Statement

Convert 2BADDED₁₆ to decimal and also to base 2

Solution

You have provided the hexadecimal number "2BADDED₁₆" and requested its conversion to decimal and binary (base 2). Let's solve both steps:

Step 1: Convert 2BADDED₁₆ to Decimal

The hexadecimal number system has a base of 16, which means it uses 16 digits: 0-9 and A-F (where A = 10, B = 11, ..., F = 15).

Conversion process:

2BADDED₁₆ = 2×166+B×165+A×164+D×163+D×162+E×161+D×1602 \times 16^6 + B \times 16^5 + A \times 16^4 + D \times 16^3 + D \times 16^2 + E \times 16^1 + D \times 16^0

Substitute the values for the letters:

  • B = 11
  • A = 10
  • D = 13
  • E = 14

So, 2BADDED16=(2×166)+(11×165)+(10×164)+(13×163)+(13×162)+(14×161)+(13×160)2BADDED₁₆ = (2 \times 16^6) + (11 \times 16^5) + (10 \times 16^4) + (13 \times 16^3) + (13 \times 16^2) + (14 \times 16^1) + (13 \times 16^0) =(2×16777216)+(11×1048576)+(10×65536)+(13×4096)+(13×256)+(14×16)+(13×1)= (2 \times 16777216) + (11 \times 1048576) + (10 \times 65536) + (13 \times 4096) + (13 \times 256) + (14 \times 16) + (13 \times 1) Now let's compute the values:

= 45677641$$ Thus, the decimal equivalent of 2BADDED₁₆ is **45,677,641**. ### Step 2: Convert the Decimal Result to Binary (Base 2) Now, we will convert the decimal value 45,677,641 to binary. Let's calculate that.The binary equivalent (base 2) of the decimal value 45,677,641 is: **10101110001111110001001001₂** ### Summary: - **Hexadecimal (2BADDED₁₆)** = **45,677,641₁₀** (Decimal) - **Decimal (45,677,641₁₀)** = **10101110001111110001001001₂** (Binary) Would you like further clarification or details on any part of this process? Here are 5 related questions for deeper understanding: 1. How would you convert another hexadecimal number, like 3F4B9A₁₆, to decimal? 2. What are the steps to convert a binary number back to hexadecimal? 3. Why does the hexadecimal system use 16 digits, and what are its common applications? 4. How does the binary system relate to digital computers and their memory? 5. Can you explain how signed numbers are represented in hexadecimal? **Tip:** When converting large hexadecimal numbers to decimal, it's useful to break the computation into smaller chunks and sum them up progressively.

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Math Problem Analysis

Mathematical Concepts

Number Systems
Base Conversion

Formulas

Hexadecimal to Decimal conversion: \( a_n \times 16^n + a_{n-1} \times 16^{n-1} + ... + a_1 \times 16^1 + a_0 \times 16^0 \)
Decimal to Binary conversion: Repeated division by 2

Theorems

-

Suitable Grade Level

Grades 9-12

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