Wikijunior:The Book of Estimation/Approximation of numbers

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Wikijunior:The Book of Estimation
Numbers Approximation of numbers Accuracy and precision

In this long, but not too difficult chapter, we will go through different ways of getting the approximate values of numbers. Once you know how to handle them, you should be able to apply them in later chapters.

Front-end method[edit]

Front-end method is a quick way of getting the approximate value of a number. While it is not very accurate, we can use it when we don't have enough time to think. Here's how we use it.

Example 1
Question Write down the approximate value of the following numbers using the front-end method.
(a)123,456,789
(b)123.4567789
(c)0.10
Solution (a)100,000,000
(b)100
(c)0.1

Simply change all the digits to 0 except for the first one. All 0s behind the decimal points are omitted. The placeholder 0 is not counted as a real digit.

Rounding up and down[edit]

Sometimes it's much better to overestimate than underestimate something. Sometimes it's the other way around. Let's say you're trying to find the total weight of people who want to enter a lift. The lift can only support a certain maximum weight, or the cable will break and the lift will fall straight to the ground floor. You don't want that to happen, so your estimated value must be higher rather than lower than the actual weight. When you're shopping it's a different story. You'd rather take more money than less before you go shopping!

That's why the concepts of rounding up and down developed. Let's look at a few examples.

Example 2
Question Round up and down the following numbers to the nearest hundred:
(a)742,723,002,000
(b)123,456,999
Solution (a)Round up: 742,723,000,000; Round down: 742,723,000,000
(b)Round up: 123,457,000; round down: 123,456,900


The basic concepts of rounding up and down can be seen from the above example. First, determine which place you round to; in this case, the nearest hundred. Add one to the hundreds place if you want to round up; don't touch it if you want to round down. Turn all digits to the RIGHT of the hundreds place into zero. The only exception to this rule is when all the digits to the right of the hundreds place are already zero, in which case you need not do anything.

From part (b) of the example, notice that the hundreds place is also zero. This is because 9 + 1 = 10, so we must move the 1 to the place of the left, which is the thousands place.

The same rules apply for decimal numbers.

Example 3
Question Round up and down the following numbers to the nearest hundredth:
(a)0.9999999
(b)0.001
Solution (a)Round up: 1.00; Round down: 0.99
(b)Round up: 0.01; round down: 0.00


Note the importance of the 0 digit. If you are rounding to a decimal place, make sure that the place is not empty. For example, 1 is NOT CORRECT for rounding up (a). Also, make sure there is nothing after that place. For example, 1.000 is also NOT CORRECT for rounding up (a).

You will never be asked to round up or down any number to the nearest one or unit. Instead, the question will ask you the round up or down to the nearest integer.

Convert a vulgar fraction to a decimal number before rounding it up or down.

One should be more careful when dealing with negative numbers.

Example 4
Question (a)Round up and down -0.457 to the nearest hundredth
(b)Round up and down -1254 to the nearest hundred.
Solution (a)Round up: -0.45; Round down: -0.46
(b)Round up: -1200; round down: -1300


Remember that -0.45 is larger than -0.46 and -1200 is larger than -1300!

Rounding off[edit]

Note

Some people round the number down when the digit they round to is even and the digit on the left is five. We will not do this here.

Rounding off is very like rounding up and down, only whether you increase of decrease the digit depends on the size of the digit on the right.

Example 5
Question Round off 0.9999999 to the nearest thousandth.
(b)Round off 123,456,789 to the nearest hundred thousand.
(c)Round off 456.123 to the nearest integer.
Solution (a)1.000
(b)123,500,000
(c)456


If the digit on the right is 0-4, the digit to which we are rounding remains unchanged. If the digit on the left is 5-9, we add one to the digit to which we are rounding. Although this method is the most difficult of the three, it will be very useful both later on in this book and in other areas of mathematics, and even in other subjects as well as daily life.

Truncation[edit]

Truncation is like a hybrid of the front-end method and rounding up/down. With truncation, we first locate the decimal digit to which we are rounding, then we rub out all the digits after it.

Example 6
Question (a) Truncate 431.563.
(b) Truncate 243.566 to the one decimal place.
Solution (a) 431
(b) 243.5


Note that in (a), we are not told to which decimal place we should truncate, so we can assume that we should truncate to zero decimal places.

Vocabulary list[edit]

  • Rounding up
  • Rounding down
  • Rounding off
  • Front-end method
  • Truncation

Exercises[edit]

  1. Apply the front-end method to the following numbers:
    1. 999,999
    2. 200.777
  2. Round up:
    1. 134,836,143 to the nearest million
    2. -623,624,723 to the nearest ten thousand
    3. 999.99 to the nearest integer
    4. 0.132007 to the nearest ten thousandth
  3. Round down the numbers in 2.
  4. Round off the numbers in 2.

Answers:

  1. Remember not to add any extra zeros.
    1. 900,000
    2. 200.777
  2. Be careful when doing the second and third parts.
    1. 135,000,000
    2. -623,610,000
    3. 1000
    4. 0.1321
  3. Be careful when doing the second and forth parts.
    1. 134,000,000
    2. -623,620,000
    3. 999
    4. 0.1320
  4. Be careful when doing the third and forth parts.
    1. 135,000,000
    2. -623,610,000
    3. 1000
    4. 0.1320