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Visualizing Rounded Whole Numbers

We're going to look at a way to visualize rounded whole numbers. There are only three steps to remember.

1. Pick the place value being rounded to.
2. Check the number to the right of the selected place value. If it is $4$ or lower, we will round down and not change the digit. If it is $5$ or greater, we will round up and raise the digit by 1.
3. Make all of the numbers to the right of the rounded value $0$.

We can think of this process as a scale. Let's say we are trying to round $23$ to the nearest ten.

$23$ doesn't quite make it to the middle point of the scale, so we round it down to $20$.

Rounding $27$ looks a little bit different.

$27$ is to the right of the tipping point, so we round this up to $30$.

Things get a little bit tricky with numbers ending in $5$. For instance, $25$ is exactly in the middle between $20$ and $30$ but still rounds to $30$.

A rounded number has about the same value as the number you start with, but it is less exact. Most of time, we round to powers of $10$.

Here are the general rules for rounding an integer to the nearest $10^n:$

If the $n+1$-th digit from the right of the number you are rounding is followed by $5$, $6$, $7$, $8$, or $9$, round the number up.

On the other hand, if the $n+1$-th digit from the right of the number you are rounding is followed by $0$, $1$, $2$, $3$, or $4$, round the number down.

Now, some more examples:

Example:

• $4{,}827$ rounded to the nearest ten is $4{,}830$.
• $4{,}827$ rounded to the nearest hundred is $4{,}800$.
• $4{,}827$ rounded to the nearest thousand is $5{,}000$.

All the numbers to the right of the place you are rounding to become zeros. Here are some more examples:

Example: $34$ rounded to the nearest ten is $30$.

Example: $6{,}809$ rounded to the nearest hundred is $6{,}800$.

Example: $1{,}951$ rounded to the nearest thousand is $2{,}000$.

(Video) Rounding

by Math Antics

Here's a video by mathantics that explains rounding. It goes over how you round, precision, and why it can be useful.

$\large{\text{Summary}}$

What Is Rounding?

Numbers are usually used to talk about amounts of something. How many days left until your birthday. How much money you need to buy a Playstation. How many students went to your school.

When answering any of these, you might round. You would probably say, "About $2{,}000$ kids went to my school." This is not an exact number, but an estimate. The actual number might be $1{,}863$.

Rounding is a way to give a close estimate of a number, but not its exact value.

Precision

Let's look again at the number $1{,}863$. We can round this in different ways. $$1{,}860\\ 1{,}900\\ 2{,}000$$ Each of these is a valid way to round our number, but as we move down, we're getting less precise. The rounded number is farther away from our original. We could also plot all the numbers on the number line. We would see $1{,}900$ is further from $1{,}863$ then $1{,}860$ is. Also, $2{,}000$ is further from $1{,}863$ than $1{,}900$, and so on.

Number Line Representation

We can get a better representation of rounding by looking at a number line.

If we round to the nearest ten, we have to decide whether it goes up to $1870$ or down to $1860$.

When deciding if we round up or down, we round in the direction that is the shortest. In other words, since the arrow is shortest when rounding down to $1860$, that's the direction we round.

We can do this for the nearest hundred. This time, rounding up is shorter.

Finally, we can do this for the nearest thousand.

Why?

So why would we want to make a number less precise? For starters, rounding can help make calculations easier. Adding $300$ and $500$ would be easier. $$300+500$$ Adding $312$ and $498$ would be harder to do. $$312+498$$

Also, you don't always need to be precise. If you weigh your pet, you don't need to know they weigh $55.83297\text{ kg}$; it's enough to know they weigh $55.8\text{ kg}$. Really, who cares about more than one place past the decimal point?

The Procedure

The first step in rounding a number is to locate the place value that the question wants it rounded to.

So, if we want to round $8{,}246{,}315$ to the nearest thousand, we want to locate the digit in the thousand place. $$8{,}24\color{#29A603}{6}{,}315$$

Next, we look at the digit to the right of our rounding place. In our example, this is $3$. $$8{,}24\color{#29A603}{6}{,}\color{#8925AE}{3}15$$ If this digit is $5$ or greater, we round up. If it's less than $5$ we round down. Since $3$ is less than $5$, we round down.

• Rounding down means we keep the rounding digit the same and make every digit to the right of it, zero.
• Round up means we increase the rounding digit by $1$ and make every digit to the right of it, zero.

So, since we're rounding down, our number becomes: $$8{,}24\color{#29A603}{6}{,}\color{#8925AE}{000}$$