Problem 29 - Thinking Mathematically

Question

Multiply in the indicated base. $$ \begin{array}{r} 543_{\text {seven }} \\ \times \quad 5_{\text {seven }} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result of the multiplication \(543_{seven} \times 5_{seven}\) is \(4011_{seven}\).

Step 1: Multiplying each digit of 543 by 5

Starting from the rightmost digit 3, multiply it by 5: \(3_{seven} \times 5_{seven} = 15_{ten}\). Because we're working in base seven, this equates to 21_{seven}. Write down the 1 and carry the 2 to the next digit.

Step 2: Continue the process with the second digit

Next, multiply 4_{seven} by 5_{seven}, then add the carry from the previous step: \(4_{seven} \times 5_{seven} + 2_{ten} = 22_{ten}\), which equates to 31_{seven}. Write down the 1 and carry the 3 to the next digit.

Step 3: Continue the process with the third digit

Finally, multiply 5_{seven} by 5_{seven}, then add the carry from the previous step: \(5_{seven} \times 5_{seven} + 3_{ten} = 28_{ten}\), which equates to 40_{seven}. Write down the final result, which includes the carry.

Step 4: Write down the final result

Reading the result from bottom to top, you get \(543_{seven} \times 5_{seven} = 4011_{seven}\).

Key Concepts

Base Seven Multiplication

When we talk about base seven multiplication, it's similar to multiplication in base ten, but you only use the digits 0 through 6. This is because in a base seven system, each place value represents a power of seven. So, to multiply numbers like \(543_{seven}\) and \(5_{seven}\), we handle each digit separately.
Just like in base ten, you start with the rightmost digit and work your way left. Here's a quick look at the steps:

• Multiply the rightmost digit, carry over values greater than six.
• Move to the next digit, including any carry from the previous multiplication.
• Repeat till all digits are multiplied and accounted for.

Understanding this step-by-step can be crucial because the carrying rules differ due to fewer digits allowed in base seven.

Number Bases in Mathematics

Number bases in mathematics are fundamental for understanding various numbering systems. While our everyday counting system is base ten (decimal), mathematicians often use other bases like base two (binary), base eight (octal), base sixteen (hexadecimal), and base seven.
Each base uses a different set of digits:

• Base ten uses 0-9.
• Base two uses 0 and 1.
• Base seven uses 0-6.

The base of a number dictates how numbers are represented and calculated. Base conversion is often required when you want to transition between different bases. Each number in a lower base has higher value in a higher base because each place value represents a power of the base. For example, in base seven, number 10 is actually \(1 \times 7^1 + 0 \times 7^0\), or ten in decimal. Understanding number bases allows you to apply arithmetic procedures accurately in environments other than decimal.

Carrying in Multiplication

Carrying in multiplication is crucial, especially in non-decimal bases. When multiplying numbers in bases other than ten, the concept of carrying over becomes vital because multiplication results can exceed the maximum digit of that base.
For example, in base seven, numbers range only from 0 to 6. Suppose your multiplication results in a number like 9 (in base ten). In base seven terms, this is more than a single digit, so you can't just place "9." Instead, you divide 9 by 7, which results in 1 with a remainder of 2. This translates to writing down "2" and carrying over "1" to the next digit.

• Step 1: Multiply two digits.
• Step 2: If the product exceeds the base maximum, find remainder and carry.
• Step 3: Add carry to next multiplication or next column.

Repeat this method throughout all the digits. Carrying ensures accuracy in multi-digit multiplication scenarios across any number base.