During multiplication by 10,100,etc, why is it that the digits move one or more places to the left depending on the number of zeros?what's the logic?
Ex: 24X10=240, 24X100=2400.
Think back on your days of learning multiplication, where if you are multiplying by a 3-digit number, you write down the partial answers, each row shifted over by one from the row above.
That's all this is. We work with a base-10 system, so multiplying by a power of 10 shifts the value that many places to the left.
167 =
7 x 1
+
6 x 10
+
1 x 100
When you multiply a number by 10, 100, or any power of 10, the digits move to the left because each place value in the number increases by one position to the left.
To understand this, let's break down the example you mentioned, 24 multiplied by 10:
1. The original number is 24.
2. When you multiply it by 10, you are essentially multiplying it by 10^1.
- The exponent 1 indicates that you are multiplying by a single power of 10.
- In this case, 10^1 means multiplying by 10.
3. When you multiply 24 by 10, each digit in the number shifts one place to the left.
- The digit 2 moves to the left and becomes the hundreds place, resulting in 2 being in the hundreds place, which is equivalent to 200.
- The digit 4 moves to the left, becoming the tens place and resulting in 4 being in the tens place, which is equivalent to 40.
4. Combining the shifted digits, you get 240, which is the result of 24 multiplied by 10.
This same logic applies when multiplying by any power of 10. For example, if you multiply 24 by 100 (10^2), the digits in 24 will shift two places to the left, resulting in 2400.
The reason why the digits move one or more places to the left when multiplying by 10, 100, and so on, is because of the base-10 number system that we use. This number system is also known as the decimal system.
In the decimal system, each digit's position represents a different power of 10. The rightmost digit represents the 0th power of 10 (which is 1), the next digit to the left represents the 1st power of 10 (which is 10), the next digit represents the 10th power of 10 (which is 100), and so on.
When we multiply a number by 10, we are essentially multiplying it by the next power of 10. Since each digit's position represents a power of 10, when we multiply by 10, all the digits in the number will move one place to the left. For example, in the case of 24 multiplied by 10, the digit '2' moves one place to the left and becomes the tens digit '2', while the digit '4' also moves one place to the left and becomes the units digit '4', resulting in the number 240.
Similarly, when we multiply by 100, we are multiplying by the next power of 10, which has two zeros. In this case, all the digits in the number will move two places to the left. Therefore, in the case of 24 multiplied by 100, the digit '2' moves two places to the left and becomes the hundreds digit '2', while the digit '4' also moves two places to the left and becomes the tens digit '4', resulting in the number 2400.
The logic behind this pattern is that as we multiply by higher powers of 10, we are essentially increasing the place value of each digit by one or more places to the left, which results in a greater value overall.