Brüche, Dezimalzahlen und Ganze Zahlen

A fraction represents a part of a whole. It consists of a numerator (Zähler) and a denominator (Nenner). The denominator indicates into how many equal parts the whole is divided, and the numerator indicates how many of these parts are considered. Fractions can be 'erweitert' (expanded) by multiplying the numerator and denominator by the same number, or 'gekürzt' (simplified) by dividing them by the same number. This does not change the value of the fraction.

To add or subtract fractions, they must have the same denominator (gleichnamig machen), usually by finding the least common multiple (kleinstes gemeinsames Vielfaches) of the denominators. Then, add or subtract the numerators while keeping the denominator. To multiply fractions, multiply the numerators and multiply the denominators. To divide by a fraction, multiply by its reciprocal (Kehrwert). The reciprocal of a/b is b/a.

Decimal numbers are another way to write fractions with powers of ten (10, 100, 1000, etc.) in the denominator. They use a decimal comma (Komma) to separate the whole number part from the fractional part (Zehntel, Hundertstel, Tausendstel, etc.).

To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal as a fraction with a power of ten in the denominator (e.g., 0,5 = 5/10, 0,25 = 25/100) and then simplify the fraction.

For addition and subtraction, align the decimal commas and then perform the operation as with whole numbers. For multiplication, multiply the numbers as if they were whole numbers. The number of decimal places in the product is the sum of the decimal places in the factors. For division, make the divisor a whole number by multiplying both the dividend and the divisor by a power of ten. Then perform the division.

Integers include all positive whole numbers (1, 2, 3, ...), all negative whole numbers (-1, -2, -3, ...), and zero. They do not include fractions or decimals.

Negative numbers are numbers less than zero. On a number line (Zahlenstrahl), they are located to the left of zero. They are used to represent quantities like debt, temperatures below zero, or depths below sea level.

For addition: If signs are the same, add the absolute values and keep the sign. If signs are different, subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value. For subtraction: Subtracting a number is the same as adding its opposite (Gegenzahl). For example, a - b = a + (-b) and a - (-b) = a + b.