Decimals

This course on decimals is *ingenius and amazingly simple*.

Decimals are introduced as *standard form of fractions*.

Fractions are part-of-whole, with different place-values specified as denominators. The Decimals are standard form of fractions.

• place-values standardized to $1/10$, $1/100$, etc.

• decimal point and position of digits after decimal point speficies the standarized place-value.

In this, learn the following.

• Decimals as standard form of fractions

• Decimals as directed-numbers (positive and negative)

• Decimal arithmetics : first principles

• Decimal arithmetics : simplified procedures

Standard form of Fractions: Decimals

Decimals are introduced as "standard form of fractions".

Fractions are part-of-whole, with different place-values specified as denominators. The Decimals are standard form of fractions with standard form for

• place-values : $1/10$, $1/100$, etc.

• position of digits : decimal point and order of digits after decimal point.

Decimals : Expanded form and number-line

This page provides a short-and-to-the-point overview of expanded form of decimals, ordinal property of decimals, and placing decimals in number line

Conversion of Fraction and Decimals

This page provides a simple overview of conversion between fractions and decimals.

A fraction can be converted to the equivalent decimal by long division method.

A decimal can be converted into its equivalent fraction by introducing the denominator based on the number of digits after decimal points.

Conversion of Repeating Decimals to Fractions

This page gives a brief overview of conversion of numbers having repeating digits after decimal point into equivalent fractions.

Decimals: Not repeating and not ending

This page provides an overview of numbers with digits in decimal place not repeating and not ending. These are called irrational numbers.

This introduction on irrational numbers is *amazingly simple*. All students should go through this once to understand irrational numbers.

Comparing Decimals

Comparing two decimals is explained.

Comparison in first principles -- matching two quantities to find one as smaller, equal, or larger than the other -- is extended for decimals (fractions of standard place values).

Based on this a simplified procedure by place value is explained.

Decimal Addition and Subtraction

Addition or Subtraction of decimals is explained. The following are explained for decimals.

• addition in first principles -- combining two quantities and measuring the combined

• subtraction in first principles -- taking away a part of a quantity and measuring the remaining quantity.

• Simplified procedure : Sign property of Addition and Addition by place value

• Simplified procedure : Sign property of Subtraction and Subtraction by place value

Decimal Multiplication

Decimals are fractions with standard place values. In this page, the following for decimals are explained.

• multiplication in first principles -- repeatedly combining a quantity and measuring the combined and

• simplified procedure : Sign property of multiplication and Multiplication by Place value for Decimals.

Decimal Division

Decimals are fractions with standard place values. In this page, the following for decimals are explained.

• Division in first principles -- splitting a quantity into a number of parts and measuring one part

• simplified procedure : Sign property of division and long division method for Decimals.

Decimals: Simplification of Expressions

In this page, handling of decimals, both positive and negative decimals, in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.

Decimals: Precedence Order PEMA

Redefining the precedence order in arithmetics with PEMA or BOMA. As part of fractions and decimals, the revised version is in handling division -- handle division as inverse of multiplication.

eg: $x\xf7y\xf7x$ This cannot be simplified using PEMDAS / BODMAS, instead, it can be simplified as $x\times \left(\frac{1}{y}\right)\times \left(\frac{1}{x}\right)$ and applying properties of arithmetics, it is $\left(\frac{1}{y}\right)\times x\times \left(\frac{1}{x}\right)=\frac{1}{y}$