2.6 Chapter summary

• The numbers \(1\); \(2\); \(3\); \(4\), etc. are called the natural numbers. The natural numbers and \(0\) are called the whole numbers. The negatives of the natural numbers together with the whole numbers are called integers.

• The set of integers is defined as follows:

  \[\mathbb{Z} = \{\ldots; - 3; - 2; - 1;0;1;2;3;\ldots\}\]
• The number line helps us to arrange integers in order. The distance from the negative number on the left of \(0\) is the same as the distance to the opposite positive number on the right of \(0\).

• You can calculate the distance by subtracting the smaller number from the larger number. This will also give you the numbers of steps between the two numbers:

  distance = larger number – smaller number

• The symbol, \(\infty\), on each side of the number line is called an infinity symbol. This means that the number line will continue in the negative \(( - \infty)\) and in the positive \((\infty)\) direction infinitely, or forever.
• To locate or place a number on the number line, count the number of places from zero. If the number is negative, you count to the left of zero. If the number is positive, you count to the right of zero.
• On the number line, smaller numbers are further to the left, and larger numbers are further to the right. Therefore, to order the numbers from smallest to largest, we must write them as they appear from left to right on the number line:

• To place the numbers in ascending order is to place them from the smallest to the largest number.
• To place the numbers in descending order is to place them from the largest to the smallest number.
• To arrange the numbers in numerical order is to place them from smallest to largest. Numerical order is the order of numbers on the number line from left to right.
• Adding a negative number has the same effect as subtracting a positive number.
  For example: \(20 + ( - 15) = 20 - 15 = 5\).
• Adding positive numbers means going to the right, while subtracting positive numbers means going to the left.
• When you multiply or divide with negative numbers, there are two rules you must remember:
  • If the numbers have the same signs: the answer is positive.
  • If the numbers have opposite signs: the answer is negative.
• The square of an integer is always a positive number.
• The cube of an integer is positive if the integer is positive, and negative if the integer is negative.

• It is possible to take a cube root of the negative integer, but it is not possible to take a square root of a negative integer.

  \(\sqrt[3]{- 8} = - 2\) but \(\sqrt{- 16} = \text{undefined}\)

• In any numeric or algebraic sentence, the terms are separated by addition and subtraction symbols, and joined into one by multiplication symbols, division symbols, and brackets. To simplify or evaluate an expression, we must:
  • separate it into terms
  • simplify each term (if needed)
  • add or subtract from left to right.
• To simplify a term, we must first simplify any parts within brackets. Then we must apply the exponents. If there are no brackets, then the exponent only applies to the base immediately before it, and we must simplify the power first.
• Multiplication of integers is associative. This means that in a product with several factors, the factors can be placed in any sequence, and the calculations can be performed in any sequence. For example, \(2 \times ( - 10) = - 10 \times 2 = - 20\).
• Addition with integers is associative. When rearranging integers in addition, you need to be careful not to leave the minus sign behind. For example, \(20 - 30 = 20 + ( - 30) = - 30 + 20 = - 10\).

• An additive inverse is a number such that when added to the original number results in zero.

  \[a + ( - a)\ = \ 0\]

• The multiplicative inverse of a number is a number that when multiplied by the original number gives the product of \(1\).

  \[a \times \frac{1}{a} = 1\]
• For multiplicative inverses, one number is called the reciprocal of the other number. For example, \(- \frac{1}{2}\) is the reciprocal of \(- 2\).