Rounding — to the nearest, up and down

Top left is the number. The three cards on the right show it rounded to the thousands: rounded up at the top, to the nearest in the middle, rounded down at the bottom. The card bottom left picks out the hundreds digit. Changing the number updates all four cards.

Rounding to the nearest picks the closer one

34,567 lies between 34,000 and 35,000. Rounded to the thousands, those are the only two candidates.

Rounding down always takes the lower one, 34,000. Rounding up always takes the higher one, 35,000. Rounding to the nearest takes whichever is closer.

Halfway between the two is 34,500. 34,567 is above halfway, so 35,000 is closer, and that is the result.

The digit that decides is one place below

Which one is closer can be read off the digit just below the place you keep. Rounding to the nearest thousand, that is the hundreds digit.

In 34,567 the hundreds digit is 5 (bottom left on Board 1). A hundreds digit of 5 or more means the number is at least 34,500, so it goes up to 35,000. A digit of 4 or less means it is below halfway and goes down to 34,000. The tens and ones make no difference.

The formula in that card uses rem, the remainder of a division, and quot, the whole-number quotient. rem(Number, 1000) takes the last three digits, 567, and dividing that by 100 leaves the hundreds digit, 5.

A number exactly halfway, like 34,500, goes up to 35,000. That is the usual school rule ("round half up"). Some standards and programming languages instead send exact halves to the even neighbour ("round half to even", also called banker's rounding). The board's roundto rounds halves up.

The place you round to changes the result

The same number rounds differently depending on how many places you keep. Board 2 rounds 34,567 to several places.

The column on the right shows, from the top, the nearest hundred, the nearest thousand, the nearest ten thousand and two significant figures. All of them round to the nearest.

• Nearest hundred — the tens digit is 6, so 34,600
• Nearest thousand — the hundreds digit is 5, so 35,000
• Nearest ten thousand — the thousands digit is 4, so 30,000

It goes up to the nearest thousand but down to the nearest ten thousand, because a different digit decides each time.

In roundto(Number, 3), the 3 means the thousands: 1,000 is 10 to the power 3. A 2 rounds to the hundreds and a 4 to the ten thousands. −1 rounds to one decimal place and −2 to two. ceilto (round up) and floorto (round down) count places the same way.

Rounding a rounded number again

The card on the far right takes 35,000, the number already rounded to the thousands, and rounds it again to the ten thousands. The result is 40,000 — not the 30,000 from rounding the original number.

35,000 sits exactly halfway between 30,000 and 40,000, so it rounds up. The original 34,567 was below halfway, but the first rounding pushed it up to the halfway mark. Rounding from the original number avoids the difference.

Significant figures

To report numbers of different sizes at the same precision, the rule is often a number of significant figures rather than a place. 34,567 to two significant figures is 35,000, the same as the nearest thousand. The four-digit 8,765 becomes 8,800, the same as the nearest hundred. In the card's formula, log(Number) says what power of 10 the number is, and that sets the place to round to.

Up or down depends on what the number is for

Choosing between rounding up and rounding down has nothing to do with which is closer. It depends on what the number will be used for. Board 3 shows two divisions whose results have to become whole numbers.

The top row counts buses and the bottom row counts boxes. Changing the numbers on the left updates the results on the right.

Buses needed: round up. Suppose 130 people travel on buses with 45 seats each. 130 ÷ 45 is 2.888889 buses. 2.888889 buses cannot be booked, and 2 buses leave some people without a seat, so the answer rounds up to 3 buses. ceil rounds up to a whole number.

With 91 people it is 2.022222 buses. That is much closer to 2, but 2 buses are still one seat short, so it is 3 buses again. When the question is how many you need, what matters is whether it is enough, not which whole number is closer.

Full boxes: round down. Suppose 100 items are packed 12 to a box. 100 ÷ 12 is 8.333333 boxes. The boxes that are actually full number 8, the result rounded down; the 9th box holds only 4 items. floor rounds down to a whole number, and rem gives the remainder.

How many you can buy on a budget, or how many sessions fit into a period, round down in the same way. Money has its own conventions too: whether fractions of a cent in tax are rounded, dropped or rounded up is set by local rules or by each business.

Other numbers

The number on Board 1 can be stepped by 1 with the − and + buttons.

34,499 and 34,500 are only 1 apart, yet they round to 34,000 and 35,000: the hundreds digit changes from 4 to 5.

35,000 is already a whole number of thousands, so all three methods leave it alone. One more, 35,001, and only rounding up jumps to 36,000 — rounding up moves to the next thousand for anything above zero in the lower places, even 1.

In this table, rounding in two steps disagrees with rounding once for 34,567 and 1,234,567. In both, the first step lands exactly on a halfway mark (35,000 and 1,235,000). Two significant figures matches the nearest thousand for a five-digit number, the nearest hundred for four digits and the nearest hundred thousand for seven.

On Board 3, changing the number of people shows where the bus count steps up:

• 90 people — exactly 2 buses → 2
• 91 people — 2.022222 buses → 3
• 135 people — exactly 3 buses → 3
• 136 people — 3.022222 buses → 4

Once the number of people passes a multiple of 45 (90, 135) by even one, the count goes up by a bus. For the boxes, 96, 108 and 120 items give 8, 9 and 10 full boxes with 0 left over.

Read next — Division and fractions — quotient and remainder, fraction, decimal It calculates one division as a quotient and remainder, a fraction and a decimal, and shows why the remainder is always smaller than the divisor.

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