Decimals

There are two primary methods for converting a fraction to a decimal number, when this is possible.

• Method 1: Direct Division
  Since a fraction \(\frac{a}{b}\) is equal to the quotient \(a \div b\), perform the division of the numerator by the denominator. When the division stops, it gives the decimal writing of the fraction; otherwise, the fraction is not equal to any decimal number.
• Method 2: Denominator as a Power of 10 (equal fractions)
  Find an equivalent fraction where the denominator is a power of 10 (e.g., 10, 100, 1000). This new fraction is then written as a decimal number: \(\dfrac{75}{100} = 0.75\). This method only works when the denominator can be turned into a power of \(10\) (for example \(2\), \(4\), \(5\), \(25\)). For a denominator such as \(3\), no power of \(10\) works, so use Method 1: \(\frac{1}{3} = 1 \div 3 = 0.333\ldots\) (the decimal expansion never ends).

Watch out: not every fraction equals a decimal number — \(\frac{1}{3} = 0.333\ldots\) never terminates. A fraction (once simplified) converts to a decimal number exactly when its denominator can be multiplied up to a power of \(10\).

The procedure for converting a terminating decimal to a fraction is as follows:

1. Write the decimal as the numerator of a fraction without the decimal point.
2. The denominator is 1 followed by as many zeros as there are decimal places in the original number.
3. Simplify the fraction to its lowest terms, if necessary.

Since \(0.1 = \dfrac{1}{10}\), \(0.01 = \dfrac{1}{100}\) and \(0.001 = \dfrac{1}{1\,000}\):

• multiplying by \(0.1\) is dividing by \(10\): the decimal point moves \(1\) place to the left;
• multiplying by \(0.01\) is dividing by \(100\): the decimal point moves \(2\) places to the left;
• multiplying by \(0.001\) is dividing by \(1\,000\): the decimal point moves \(3\) places to the left.

$$47 \times 0.1 = 47 \div 10 = 4.7 \qquad\quad 3.4 \times 0.01 = 3.4 \div 100 = 0.034 \qquad\quad 27 \times 0.001 = 27 \div 1\,000 = 0.027$$

To compute a product such as \(0.4 \times 0.3\) without any drawing:

1. Step 1: Write each decimal as a whole number times \(0.1\), \(0.01\) or \(0.001\). Here \(0.4 = 4 \times 0.1\) and \(0.3 = 3 \times 0.1\).
2. Step 2: Change the order of the factors and group them differently — in a product, this is always allowed: the whole numbers together, the \(0.1\)'s together. $$0.4 \times 0.3 = (4 \times 0.1) \times (3 \times 0.1) = (4 \times 3) \times (0.1 \times 0.1)$$
3. Step 3: Compute the two products. \(4 \times 3 = 12\) and \(0.1 \times 0.1 = 0.01\) (a tenth of a tenth). $$0.4 \times 0.3 = 12 \times 0.01 = 0.12$$

The product of two decimals is thus brought back to the product of two whole numbers, followed by a multiplication by \(0.1\), \(0.01\), \(0.001\), …

In long division, first convert the divisor to a whole number.

1. Step 1: Make the divisor whole. Move the decimal point in the divisor to the right until the divisor is a whole number. Count how many places you moved it.
2. Step 2: Move the other decimal. Move the decimal point in the dividend the same number of places to the right, adding zeros if there are no digits left (\(6 \div 0.75\) becomes \(600 \div 75\)).
3. Step 3: Divide. Now the problem is a "division by a whole number" problem. Follow the steps of that division.

A quotient does not change when the dividend and the divisor are both multiplied by the same non-zero number: there are as many times \(1.2\) cm in \(4.56\) cm as \(12\) mm in \(45.6\) mm — changing the unit multiplies both numbers by \(10\) without changing the answer. $$4.56 \div 1.2 = (4.56 \times 10) \div (1.2 \times 10) = 45.6 \div 12$$ “Moving both decimal points one place to the right” is exactly “multiplying both numbers by 10”. The method replaces the division by a decimal with an equal division whose divisor is a whole number — a division that can already be set up.