# Difference between revisions of "Decimal"

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− | A decimal is a [[number]] that is not an [[integer]] | + | ''This article focuses on the non-integer real-numbers. Decimal can also refer to [[base numbers|base ten]] number system.'' |

+ | |||

+ | A '''decimal''' is a [[number]] that is not an [[integer]] and is expressed in neither fraction or percent form. The [[whole number]] portion is separated from the fractional portion by a [[decimal point]] (.) which looks identical to a period. | ||

+ | |||

+ | ==Converting Decimals== | ||

+ | |||

+ | When converting fractions, note that the first digit after the decimal point represents tenths, the next digit after the decimal point represents hundredths, and so on. So we can convert each part accordingly. For example, 3.5 would equal 35/10 or 350%. 5.68 would equal <math>5 + \frac{6}{10} + \frac{8}{100}</math>, or <math>{5}\frac{68}{100}</math>. | ||

+ | |||

+ | For a quicker strategy, note that a tenth is 10 hundredths or 100 thousandths. Thus, we can go from <math>5.68</math> straight to <math>5 \frac{68}{100}</math>. | ||

+ | |||

+ | ==Comparing Decimals== | ||

+ | |||

+ | Comparing decimals is similar to comparing whole numbers -- start with the largest digit and compare accordingly. For instance, hundreds to hundreds, tens to tens, ones to ones, tenths to tenths, hundredths to hundredths, thousandths to thousandths, etc. | ||

+ | |||

+ | ==Operations on Decimals== | ||

+ | |||

+ | ===Addition and Subtraction=== | ||

+ | |||

+ | Addition and subtraction is similar to adding and subtracting whole numbers, but knowing place value is important. In particular, we have to make sure that tenths are added to tenths, hundredths are added to hundredths, and so on. This is done by lining up the decimal point. | ||

+ | |||

+ | ===Multiplication=== | ||

+ | |||

+ | ====Powers of 10==== | ||

+ | |||

+ | When multiplying by 10, note that the quantity of tenths become the quantity of ones, the quantity of hundredths become the quantity of tenths, and so on. Thus, multiplying by 10 means moving the decimal point one place to the right. Similarly, multiplying by 100 means moving the decimal point 2 places to the right, and multiplying by <math>10^n</math> (where <math>n</math> is an integer) means moving the decimal point <math>n</math> places to the right. | ||

+ | |||

+ | When multiplying by 0.1, note that the quantity of ones become the quantity of tenths, the quantity of tenths become the quantity of hundredths, and so on. Thus, multiplying by 0.1 means moving the decimal point one place to the left. Similarly, multiplying by 0.01 means moving the decimal point 2 places to the left, and so on. | ||

+ | |||

+ | ====General Rule==== | ||

+ | |||

+ | Consider the expression <math>2.5 \cdot 0.12</math>. | ||

+ | |||

+ | First, note that <math>2.5 = 25 \cdot 0.1</math> (25 tenths) and <math>0.12 = 12 \cdot 0.01</math> (12 hundredths). Thus, because of the [[Commutative Property]], <math>2.5 \cdot 0.12 = 25 \cdot 12 \cdot 0.1 \cdot 0.01 = 25 \cdot 12 \cdot 0.001</math>. | ||

+ | |||

+ | Then we can treat part of the problem as a multiplication problem -- <math>25 \cdot 12 = 300</math>. Thus, <math>2.5 \cdot 0.12 = 300 \cdot 0.001</math>. | ||

+ | |||

+ | Finally, we locate the decimal point accordingly, and <math>300 \cdot 0.001 = 0.3</math>. Therefore, <math>2.5 \cdot 0.12 = 0.3</math>. | ||

+ | |||

+ | In similar problems, we can extract whole numbers, multiply in a way similar to integers, and count decimal places to get the final answer. | ||

+ | |||

+ | ===Division=== | ||

+ | |||

+ | ====Powers of 10==== | ||

+ | |||

+ | When dividing by 10, note that the quantity of units become the quantity of tenths, the quantity of hundredths become the quantity of thousandths, and so on. Thus, dividing by 10 means moving the decimal point one place to the left. Similarly, dividing by 100 means moving the decimal point 2 places to the left, and dividing by <math>10^n</math> (where <math>n</math> is an integer) means moving the decimal point <math>n</math> places to the left. | ||

+ | |||

+ | When dividing by 0.1, note that a tenth is one tenth of a unit, a hundredth is one tenth of a tenth, and so on. Thus, dividing by 0.1 is the same as multiplying by 10, so the decimal point is moved one place to the right. Similarly, dividing by 0.01 means moving the decimal point 2 places to the right, and so on. | ||

+ | |||

+ | ====General Rule==== | ||

+ | |||

+ | Dividing decimals by integers is similar to dividing integers by integers -- place value needs to be kept track. In particular, when doing the long division set up, the decimal point should be lined up. | ||

+ | |||

+ | As for dividing decimals by decimals, a common tactic is moving the decimal point some places to the right for both the dividend and the divisor. This can be done because both the numerator and denominator is multiplied by a power of 10. Afterward, we can proceed like a decimal divided by an integer. | ||

+ | |||

+ | ==Problems== | ||

+ | |||

+ | ===Introductory=== | ||

+ | * Practice Problems on [https://artofproblemsolving.com/alcumus/ Alcumus] | ||

+ | ** Decimal Arithmetic (Prealgebra) | ||

+ | ** Rounding Decimals (Prealgebra) | ||

+ | ** Decimals and Fractions (Prealgebra) | ||

+ | ** Repeating Decimals (Prealgebra) | ||

+ | |||

+ | ===Intermediate=== | ||

+ | * [[2002 AIME I Problems/Problem 7]] | ||

+ | * [[2006 AIME I Problems/Problem 6]] | ||

− | |||

[[Category:Definition]] | [[Category:Definition]] |

## Latest revision as of 16:48, 12 June 2020

*This article focuses on the non-integer real-numbers. Decimal can also refer to base ten number system.*

A **decimal** is a number that is not an integer and is expressed in neither fraction or percent form. The whole number portion is separated from the fractional portion by a decimal point (.) which looks identical to a period.

## Contents

## Converting Decimals

When converting fractions, note that the first digit after the decimal point represents tenths, the next digit after the decimal point represents hundredths, and so on. So we can convert each part accordingly. For example, 3.5 would equal 35/10 or 350%. 5.68 would equal , or .

For a quicker strategy, note that a tenth is 10 hundredths or 100 thousandths. Thus, we can go from straight to .

## Comparing Decimals

Comparing decimals is similar to comparing whole numbers -- start with the largest digit and compare accordingly. For instance, hundreds to hundreds, tens to tens, ones to ones, tenths to tenths, hundredths to hundredths, thousandths to thousandths, etc.

## Operations on Decimals

### Addition and Subtraction

Addition and subtraction is similar to adding and subtracting whole numbers, but knowing place value is important. In particular, we have to make sure that tenths are added to tenths, hundredths are added to hundredths, and so on. This is done by lining up the decimal point.

### Multiplication

#### Powers of 10

When multiplying by 10, note that the quantity of tenths become the quantity of ones, the quantity of hundredths become the quantity of tenths, and so on. Thus, multiplying by 10 means moving the decimal point one place to the right. Similarly, multiplying by 100 means moving the decimal point 2 places to the right, and multiplying by (where is an integer) means moving the decimal point places to the right.

When multiplying by 0.1, note that the quantity of ones become the quantity of tenths, the quantity of tenths become the quantity of hundredths, and so on. Thus, multiplying by 0.1 means moving the decimal point one place to the left. Similarly, multiplying by 0.01 means moving the decimal point 2 places to the left, and so on.

#### General Rule

Consider the expression .

First, note that (25 tenths) and (12 hundredths). Thus, because of the Commutative Property, .

Then we can treat part of the problem as a multiplication problem -- . Thus, .

Finally, we locate the decimal point accordingly, and . Therefore, .

In similar problems, we can extract whole numbers, multiply in a way similar to integers, and count decimal places to get the final answer.

### Division

#### Powers of 10

When dividing by 10, note that the quantity of units become the quantity of tenths, the quantity of hundredths become the quantity of thousandths, and so on. Thus, dividing by 10 means moving the decimal point one place to the left. Similarly, dividing by 100 means moving the decimal point 2 places to the left, and dividing by (where is an integer) means moving the decimal point places to the left.

When dividing by 0.1, note that a tenth is one tenth of a unit, a hundredth is one tenth of a tenth, and so on. Thus, dividing by 0.1 is the same as multiplying by 10, so the decimal point is moved one place to the right. Similarly, dividing by 0.01 means moving the decimal point 2 places to the right, and so on.

#### General Rule

Dividing decimals by integers is similar to dividing integers by integers -- place value needs to be kept track. In particular, when doing the long division set up, the decimal point should be lined up.

As for dividing decimals by decimals, a common tactic is moving the decimal point some places to the right for both the dividend and the divisor. This can be done because both the numerator and denominator is multiplied by a power of 10. Afterward, we can proceed like a decimal divided by an integer.

## Problems

### Introductory

- Practice Problems on Alcumus
- Decimal Arithmetic (Prealgebra)
- Rounding Decimals (Prealgebra)
- Decimals and Fractions (Prealgebra)
- Repeating Decimals (Prealgebra)