Convert the following numbers between number systems: 1) 630 to base 10, 2) DXIX to Hindu-Arabic numerals, 3) 496 to Roman numerals, 4) 111 base two to base ten.

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Asked on May 12, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-12T10:06:48.832Z

Step 1: Calculate the product of 6 × 100. 6 × 100 = 600 Step 2: Calculate the product of 3 × 10. 3 × 10 = 30 Step 3: Add the results from Step 1 and Step 2. 600 + 30 = 630 The number is 630. 10. Write DXIX in Hindu Arabic numerals. Step 1: Identify the value of each Roman numeral. D represents 500. X represents 10. I represents 1. Step 2: Combine the numerals, noting that IX is a subtractive notation. IX means 10 - 1 = 9. Step 3: Add the values. DXIX = D + IX = 500 + 9 = 509 The Hindu Arabic numeral is 509. 11. Write 496 in Roman numerals. Step 1: Break down the number 496 into its place values for Roman numeral conversion. 400 is represented as CD (500 - 100). 90 is represented as XC (100 - 10). 6 is represented as VI (5 + 1). Step 2: Combine the Roman numeral representations. 496 = CD + XC + VI = CDXCVI The Roman numeral is CDXCVI. 12. Convert 111_two to base ten. Step 1: Write the binary number in expanded form using powers of 2. 111_two = (1 × 2^2) + (1 × 2^1) + (1 × 2^0) Step 2: Calculate the powers of 2. 2^2 = 4 2^1 = 2 2^0 = 1 Step 3: Substitute the power values and sum the terms. 111_two = (1 × 4) + (1 × 2) + (1 × 1) = 4 + 2 + 1 = 7 The base ten equivalent is 7. 13. Convert 23_ten to binary. Step 1: Divide 23 by 2 and record the remainder. 23 ÷ 2 = 11 remainder 1 Step 2: Divide the quotient (11) by 2 and record the remainder. 11 ÷ 2 = 5 remainder 1 Step 3: Continue dividing the quotient by 2 until the quotient is 0. 5 ÷ 2 = 2 remainder 1 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Step 4: Read the remainders from bottom to top to get the binary number. 23_ten = 10111_two 14. If 23_n = 15_ten. Find n. Step 1: Convert the number 23_n from base n to base 10. 23_n = (2 × n^1) + (3 × n^0) = 2n + 3 Step 2: Set up the equation by equating the base 10 representation to 15_ten. 2n + 3 = 15 Step 3: Solve the linear equation for n. 2n = 15 - 3 2n = 12 n = (12)/(2) n = 6 The value of n is 6. 15. Add: 101_two + 11_two Step 1: Add the rightmost digits (least significant bits). 1_two + 1_two = 10_two. Write down 0, carry over 1. Step 2: Add the next column of digits, including the carry-over. 0_two + 1_two + 1_carry = 10_two. Write down 0, carry over 1. Step 3: Add the leftmost digits, including the carry-over. 1_two + 0_two + 1_carry = 10_two. Write down 10. & 1 \ 0 \ 1_two \\ + & 1 \ 1_two \\ & 1 \ 0 \ 0 \ 0_two The sum is 1000_two. 16. Subtract: 100_two - 11_two Step 1: Set up the subtraction. & 1 \ 0 \ 0_two \\ - & 1 \ 1_two \\ Step 2: Subtract the rightmost digits. 0 - 1 requires borrowing. Borrow from the 2^1 place, which is also 0. So, borrow from the 2^2 place. The 1 in the 2^2 place becomes 0. The 0 in the 2^1 place becomes 10_two (which is 2_ten). Now, borrow from the 10_two in the 2^1 place. It becomes 1_two. The 0 in the 2^0 place becomes 10_two. & 1^0 \ 0^1 \ 0_two \\ & 0 \ 10^1 \ 10_two \\ - & 1 \ 1_two \\ Step 3: Perform the subtraction column by column. 2^0 column: 10_two - 1_two = 1_two. 2^1 column: 1_two - 1_two = 0_two. 2^2 column: 0_two - 0_two = 0_two. & 1 \ 0 \ 0_two \\ - & 1 \ 1_two \\ & 1_two The difference is 1_two. Send me the next one 📸

Accepted Answer · 2026-05-12T10:06:48.832Z

Step 1: Calculate the product of 6 × 100. 6 × 100 = 600 Step 2: Calculate the product of 3 × 10. 3 × 10 = 30 Step 3: Add the results from Step 1 and Step 2. 600 + 30 = 630 The number is 630. 10. Write DXIX in Hindu Arabic numerals. Step 1: Identify the value of each Roman numeral. D represents 500. X represents 10. I represents 1. Step 2: Combine the numerals, noting that IX is a subtractive notation. IX means 10 - 1 = 9. Step 3: Add the values. DXIX = D + IX = 500 + 9 = 509 The Hindu Arabic numeral is 509. 11. Write 496 in Roman numerals. Step 1: Break down the number 496 into its place values for Roman numeral conversion. 400 is represented as CD (500 - 100). 90 is represented as XC (100 - 10). 6 is represented as VI (5 + 1). Step 2: Combine the Roman numeral representations. 496 = CD + XC + VI = CDXCVI The Roman numeral is CDXCVI. 12. Convert 111_two to base ten. Step 1: Write the binary number in expanded form using powers of 2. 111_two = (1 × 2^2) + (1 × 2^1) + (1 × 2^0) Step 2: Calculate the powers of 2. 2^2 = 4 2^1 = 2 2^0 = 1 Step 3: Substitute the power values and sum the terms. 111_two = (1 × 4) + (1 × 2) + (1 × 1) = 4 + 2 + 1 = 7 The base ten equivalent is 7. 13. Convert 23_ten to binary. Step 1: Divide 23 by 2 and record the remainder. 23 ÷ 2 = 11 remainder 1 Step 2: Divide the quotient (11) by 2 and record the remainder. 11 ÷ 2 = 5 remainder 1 Step 3: Continue dividing the quotient by 2 until the quotient is 0. 5 ÷ 2 = 2 remainder 1 2 ÷ 2 = 1 remainder 0 1 ÷ 2 = 0 remainder 1 Step 4: Read the remainders from bottom to top to get the binary number. 23_ten = 10111_two 14. If 23_n = 15_ten. Find n. Step 1: Convert the number 23_n from base n to base 10. 23_n = (2 × n^1) + (3 × n^0) = 2n + 3 Step 2: Set up the equation by equating the base 10 representation to 15_ten. 2n + 3 = 15 Step 3: Solve the linear equation for n. 2n = 15 - 3 2n = 12 n = (12)/(2) n = 6 The value of n is 6. 15. Add: 101_two + 11_two Step 1: Add the rightmost digits (least significant bits). 1_two + 1_two = 10_two. Write down 0, carry over 1. Step 2: Add the next column of digits, including the carry-over. 0_two + 1_two + 1_carry = 10_two. Write down 0, carry over 1. Step 3: Add the leftmost digits, including the carry-over. 1_two + 0_two + 1_carry = 10_two. Write down 10. & 1 \ 0 \ 1_two \\ + & 1 \ 1_two \\ & 1 \ 0 \ 0 \ 0_two The sum is 1000_two. 16. Subtract: 100_two - 11_two Step 1: Set up the subtraction. & 1 \ 0 \ 0_two \\ - & 1 \ 1_two \\ Step 2: Subtract the rightmost digits. 0 - 1 requires borrowing. Borrow from the 2^1 place, which is also 0. So, borrow from the 2^2 place. The 1 in the 2^2 place becomes 0. The 0 in the 2^1 place becomes 10_two (which is 2_ten). Now, borrow from the 10_two in the 2^1 place. It becomes 1_two. The 0 in the 2^0 place becomes 10_two. & 1^0 \ 0^1 \ 0_two \\ & 0 \ 10^1 \ 10_two \\ - & 1 \ 1_two \\ Step 3: Perform the subtraction column by column. 2^0 column: 10_two - 1_two = 1_two. 2^1 column: 1_two - 1_two = 0_two. 2^2 column: 0_two - 0_two = 0_two. & 1 \ 0 \ 0_two \\ - & 1 \ 1_two \\ & 1_two The difference is 1_two. Send me the next one 📸

Convert the following numbers between number systems: 1) 630 to base 10, 2) DXIX to Hindu-Arabic numerals, 3) 496 to Roman numerals, 4) 111 base two to base ten.

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Convert the following numbers between number systems: 1) 630 to base 10, 2) DXIX to Hindu-Arabic numerals, 3) 496 to Roman numerals, 4) 111 base two to base ten.

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