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# Math: Approximation and Estimation

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by

Tweet## Yun Xi Lau

on 14 September 2012#### Transcript of Math: Approximation and Estimation

Approximation, Estimation and Truncation System Macro-concepts Scale Change Follows RULES No longer accurate as number is changed, but number is close (approximation) It is a form of measurement Rounding off to the nearest Concepts Rounding off

Truncation

Estimation

Significant Figures when rounding off a digit, there is a digit for consideration, which is the digit on the RIGHT of the digit to be rounded off. if the digit is 5 and above: increase the previous digit by 1 and Use zeroes to keep the place values as necessary ` ≈ ≈ the phrase "is approximately equal to" can be represented by this symbol Approximation Truncation Limit the number of digits by discarding the less significant digits double tilde! :D after you have the intended number of significant digits, just discard the rest

WITHOUT ROUNDING UP OR DOWN Real life Application Mainly with computers, truncation usually occur when a decimal number is typecast as an integer; it is truncated to zero decimal digits because integers cannot exist as decimals.

However, it can also occur when a number is not fully represented due to memory limitations (again in computers) if the digit is 4 and below: replace that digit with 0 and Examples of what the required accuracy may be:

ones

tens

hundreds

tenths

hundredths

etc. (: Remember to state what the accuracy of the number is correct to:

eg. correct to 1 decimal place/ 3 significant figures etc. Significant Figures Rules:

1. All non-zero digits are significant.

(eg. 123 has 3 s.f.)

2. All zeroes between non-zero digits are significant. (eg. 101 has 3 s.f.)

3. Trailing zeroes may or may not be significant, depending on the extent of accuracy a number is corrected to (ones/tens/hundreds etc).

(eg. 1200 - could be 2/3 s.f when the number is correct to the hundreds or the tens place respectively) IN WHOLE NUMBERS IN DECIMAL NUMBERS Rules:

1. All non-zero digits are significant.

2. All zeroes between non-zero digits are significant.

3. Trailing zeroes are significant. When a decimal number has been rounded off to a certain number of significant figures, it may result in trailing zeroes.

4. All leading zeroes in numbers LESS THAN ONE are not significant. ESTIMATION Estimation often involves rounding off This can be done by either rounding a number to a specified decimal place or rounding a number to a specific number of significant figures. Estimation is an educated guess. An educated guess is based on common sense, while trying to guess for an unknown quantity or outcome as close to the number's actual value We need estimation in our daily life. In situations such as...

Estimating the amount of money you need for a week (so that you can ask for the amount from your dad)

Estimating the packets of juice you need for a year

Estimating the time you will take to do a task

Estimating the amount of revenue you need to start up a business

and many many other situations!! In some cases, truncating would yield the same result as rounding, but truncation does not round up or down the digits; it merely cuts off at a specified digit (e.g. 3 decimal digits).

Therefore, whenever you truncate, you do not round like how you would round normally in rounding. You just simply get rid of the decimal numbers behind. :) EXAMPLES 1. 358 (3 s.f.)

2. 1002 (4 s.f.) Examples! a)105.097 mm = 11cm First, we convert 105.097mm into 10.5097 cm. Then, we look at the tenths place. Since it is 5 and above, we round it up. Thus it would be 11 cm. Examples: 1. pi (3.1415 etc,)

truncated to 3.14 required accuracy ` thankYOU :D done by:

Valerie (1)

Chelsea (4)

Yun Xi (16)

Rachel (29) Numbers can be rounded off according to 1.A given place value2.A given number of decimal places3. A given number of significant figures Truncation means that the decimal places are cut off, after a specified number of places, no values are changed. MAIN CONCEPT:

SYSTEMS

Estimation, approximation and truncation have rules that they have to follow. For example, for rounding off, we have to identify the digit for consideration • If it is below 5, replace this digit and all other digits to its right (if any) with zero(s). •If it is 5 and above, add one to the previous digit and replace this digit and all other digits to the right (if any) with zero(s). b)226.1g = 230g (to the nearest 10g) Since we have to are to round it off to the nearest 10g, we have to look at the digit behind the tens place, which is the ones place, since it is 5 and above, we would round it up to 230g. Numbers can be rounded off according to a given number of decimal places. Round off the following examples :D a) 0.9961= 1.00 (correct to 2 decimal places) Since we are to round off the number to 2 decimal places, we are to look at the digit behind the second decimal place, the tenths place, and see if the digit is below five or 5 and above. Since the digit is six, we round it up, as we carry the digit forward, we get 1.00. (We must not leave it at 1 since they requested for 2 decimal places, we have to add in two zeros in the 2 decimal places.) a)24.0348= 24.035 (correct to 3 decimal places) Since we are to round off the number to 3 decimal places, we are to look at the digit behind the third decimal place, the thousandth place, and see if the digit is below five or 5 and above. Since the digit is eight, we round it up to 24.035. a)24.0348= 24.035 (correct to 3 decimal places) b)0.9961= 1.00 (correct to 2 decimal places) 2.36992 truncated to 2 decimal places

= 2.36 We cut off the digits after the second decimal place. We do not have to see the digit after the second decimal place. We just cut off after the specified number of places, no values are changed. 3. 4006 expressed as three significant figures is 4010.

Since the zero digits between non- zero digits are significant, there are now four significant figures. To make it to three significant figures, we round it up to the nearest tens, which we will get 4010. Since there is only one zero between two non-zero digits, there are only 3 significant figures with another zero trailing behind which is not counted to be a significant number. 3. Express 4006 in three significant figures.

Full transcriptTruncation

Estimation

Significant Figures when rounding off a digit, there is a digit for consideration, which is the digit on the RIGHT of the digit to be rounded off. if the digit is 5 and above: increase the previous digit by 1 and Use zeroes to keep the place values as necessary ` ≈ ≈ the phrase "is approximately equal to" can be represented by this symbol Approximation Truncation Limit the number of digits by discarding the less significant digits double tilde! :D after you have the intended number of significant digits, just discard the rest

WITHOUT ROUNDING UP OR DOWN Real life Application Mainly with computers, truncation usually occur when a decimal number is typecast as an integer; it is truncated to zero decimal digits because integers cannot exist as decimals.

However, it can also occur when a number is not fully represented due to memory limitations (again in computers) if the digit is 4 and below: replace that digit with 0 and Examples of what the required accuracy may be:

ones

tens

hundreds

tenths

hundredths

etc. (: Remember to state what the accuracy of the number is correct to:

eg. correct to 1 decimal place/ 3 significant figures etc. Significant Figures Rules:

1. All non-zero digits are significant.

(eg. 123 has 3 s.f.)

2. All zeroes between non-zero digits are significant. (eg. 101 has 3 s.f.)

3. Trailing zeroes may or may not be significant, depending on the extent of accuracy a number is corrected to (ones/tens/hundreds etc).

(eg. 1200 - could be 2/3 s.f when the number is correct to the hundreds or the tens place respectively) IN WHOLE NUMBERS IN DECIMAL NUMBERS Rules:

1. All non-zero digits are significant.

2. All zeroes between non-zero digits are significant.

3. Trailing zeroes are significant. When a decimal number has been rounded off to a certain number of significant figures, it may result in trailing zeroes.

4. All leading zeroes in numbers LESS THAN ONE are not significant. ESTIMATION Estimation often involves rounding off This can be done by either rounding a number to a specified decimal place or rounding a number to a specific number of significant figures. Estimation is an educated guess. An educated guess is based on common sense, while trying to guess for an unknown quantity or outcome as close to the number's actual value We need estimation in our daily life. In situations such as...

Estimating the amount of money you need for a week (so that you can ask for the amount from your dad)

Estimating the packets of juice you need for a year

Estimating the time you will take to do a task

Estimating the amount of revenue you need to start up a business

and many many other situations!! In some cases, truncating would yield the same result as rounding, but truncation does not round up or down the digits; it merely cuts off at a specified digit (e.g. 3 decimal digits).

Therefore, whenever you truncate, you do not round like how you would round normally in rounding. You just simply get rid of the decimal numbers behind. :) EXAMPLES 1. 358 (3 s.f.)

2. 1002 (4 s.f.) Examples! a)105.097 mm = 11cm First, we convert 105.097mm into 10.5097 cm. Then, we look at the tenths place. Since it is 5 and above, we round it up. Thus it would be 11 cm. Examples: 1. pi (3.1415 etc,)

truncated to 3.14 required accuracy ` thankYOU :D done by:

Valerie (1)

Chelsea (4)

Yun Xi (16)

Rachel (29) Numbers can be rounded off according to 1.A given place value2.A given number of decimal places3. A given number of significant figures Truncation means that the decimal places are cut off, after a specified number of places, no values are changed. MAIN CONCEPT:

SYSTEMS

Estimation, approximation and truncation have rules that they have to follow. For example, for rounding off, we have to identify the digit for consideration • If it is below 5, replace this digit and all other digits to its right (if any) with zero(s). •If it is 5 and above, add one to the previous digit and replace this digit and all other digits to the right (if any) with zero(s). b)226.1g = 230g (to the nearest 10g) Since we have to are to round it off to the nearest 10g, we have to look at the digit behind the tens place, which is the ones place, since it is 5 and above, we would round it up to 230g. Numbers can be rounded off according to a given number of decimal places. Round off the following examples :D a) 0.9961= 1.00 (correct to 2 decimal places) Since we are to round off the number to 2 decimal places, we are to look at the digit behind the second decimal place, the tenths place, and see if the digit is below five or 5 and above. Since the digit is six, we round it up, as we carry the digit forward, we get 1.00. (We must not leave it at 1 since they requested for 2 decimal places, we have to add in two zeros in the 2 decimal places.) a)24.0348= 24.035 (correct to 3 decimal places) Since we are to round off the number to 3 decimal places, we are to look at the digit behind the third decimal place, the thousandth place, and see if the digit is below five or 5 and above. Since the digit is eight, we round it up to 24.035. a)24.0348= 24.035 (correct to 3 decimal places) b)0.9961= 1.00 (correct to 2 decimal places) 2.36992 truncated to 2 decimal places

= 2.36 We cut off the digits after the second decimal place. We do not have to see the digit after the second decimal place. We just cut off after the specified number of places, no values are changed. 3. 4006 expressed as three significant figures is 4010.

Since the zero digits between non- zero digits are significant, there are now four significant figures. To make it to three significant figures, we round it up to the nearest tens, which we will get 4010. Since there is only one zero between two non-zero digits, there are only 3 significant figures with another zero trailing behind which is not counted to be a significant number. 3. Express 4006 in three significant figures.