Why 58 can be rounded to the nearest hundred even though there is not a digit in the hundreds place?

Answer 1

Simple answer:

When rounding to the nearest hundred, it is rounding to which of the numbers {0, 100, 200, 300, ...} is closest* to the number.

Is 58 closer to 0 or 100?

It is 58 away from "0" but only 42 away from "100"; 42 is less than 58 so it is closer to "100" and so rounds to 100 to the nearest hundred.

*If a number is exactly halfway between two of the numbers, convention has decided that it rounds up (to the higher number).

More technical answer:

Every place value column can contain the digits 0 to 9 and the place value of the digit is its face value multiplied by the place value column:

• The place immediately to the left of the decimal point has a place value of '1';
• each column to the left has a place value 10 times the place value of the column to its right; and
• each column to the right has a place value one tenth (1/10) the place value of the column to its left.

Every number is built up by adding together all the digits of the number multiplied by their place values, eg 58 = 50 + 8 = 5 x 10 + 8 x 1.

But what about an empty place value?

Take the digits 5 and 8, written as 58 they mean 58; or do they? What if the 5 was in the hundreds column and the 8 is in the ones column? How do you know the tens column is empty? With old number systems which used multiple symbols to represent numbers, eg the Roman numbers DVIII (= 508) and LVIII (= 58), this was not a problem as all the symbols were added together (DVIII = D + V + I + I + I = 500 + 5 + 1 + 1 = 508). To solve this problem a mark was used to indicate an empty column, which over [a short] time became the zero (0). Now 508 and 58 were as distinguishable as their old versions (Roman: DVIII and LVIII).

Today we interpret a zero in the place value system as zero times the column value which is zero; it adds 0 x place_value = 0 to the number.

But in the number 58 there is an empty hundreds place value column. So it can be marked with a zero: 058. But does writing the zero change the value of the number? 58 = 5 x 10 + 8 x 1, but 058 = 0 x 100 + 5 x 10 + 8 x 1 = 58; so no it does not change the value of the number.

The decision over whether to write a zero in a place value column or not is simple: if the value of a number [can] changes without the zero written it must be written; if it does not change the value, it may be written, but it is usually not written. For example 58.0 is the same as 58, but 508 is NOT the same as 58.

So with 58, the hundreds place value is empty and could be written with a zero (to indicate no hundreds (0 x 100) are present) and so does have a digit in the hundreds place value. Now the "rules" of rounding whereby "if the next digit is 5 or more add one to the place" can be done: The next digit after the hundreds is the tens digit which is 5, which is 5 or more, so 1 is added to the hundreds digit which is zero and so becomes 0 + 1 = 1.

Answer 2

58.0