If you notice, the expressions for the least button clicks also give the Indian place value notation of the numbers. Think about why this is so.

Question

Homi · Accepted Answer

Both methods break down numbers using their place values.

Step 1 — Understanding Place Value

Our number system is a place value system. Each digit's value depends on its position. For example, in the number 543, the digit 5 is in the hundreds place. This means its value is 500. The digit 4 is in the tens place. Its value is 40. The digit 3 is in the ones place. Its value is 3. We can write any number as a sum of these place values. This is called the expanded form.

Let us write 543 in expanded form.

$543 = 5 \times 100 + 4 \times 10 + 3 \times 1$

$= 500 + 40 + 3$

$\boxed{543}$

Step 2 — Understanding Least Button Clicks

Imagine we have buttons for different values. We have buttons for 1, 10, 100, 1,000, and so on. To make a number using the fewest button clicks, we must be smart. We should always use the largest possible value button first. For example, to make 543, we do not press the 1 button 543 times. That would be many clicks. Instead, we press the 100 button 5 times. Then we press the 10 button 4 times. Finally, we press the 1 button 3 times. This method uses the largest value units first. This gives us the least number of clicks.

Let us find the least clicks for 543.

Number of 100s buttons = 5 Number of 10s buttons = 4 Number of 1s buttons = 3 Total clicks = $5 + 4 + 3$

$\boxed{12 \text{ clicks}}$

Step 3 — Connecting the Ideas

The expressions for the least button clicks are the expanded form of the number. For 543, this expression is $5 \times 100 + 4 \times 10 + 3 \times 1$ . The Indian place value notation also describes a number using its expanded form. For example, 5,43,210 is "Five lakh forty-three thousand two hundred ten". This means 5 times 1,00,000, plus 4 times 10,000, and so on. This is the same as the expanded form:

$5,43,210 = 5 \times 1,00,000 + 4 \times 10,000 + 3 \times 1,000 + 2 \times 100 + 1 \times 10 + 0 \times 1$

Both methods rely on the same basic idea. We break down a number into its place values. We use the largest possible place value units first. This is how our number system works. Each digit's position uniquely determines its value. So, the "least button clicks" expression is simply the number's expanded form. This expanded form is what the Indian place value notation describes.

Answer

Both the "least button clicks" expressions and the Indian place value notation use the expanded form of a number. This means they break down a number into a sum of its digits multiplied by their respective place values (powers of ten). To achieve the "least button clicks," we must use the largest possible place value units first, which is exactly how our place value system (including the Indian system) represents numbers.

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