Learning Objectives
- Calculate confidently with positive and negative integers.
- Distinguish rational and irrational numbers.
- Use reciprocals and understand division by zero.
- Place real numbers correctly on a number line.
Key Terms
- Integer
- A positive whole number, zero or a negative whole number.
- Rational number
- A number that can be written as \frac{p}{q} where p and q are integers and q\ne0.
- Irrational number
- A real number that cannot be written as a fraction of two integers.
- Real number
- Any rational or irrational number represented on the number line.
- Reciprocal
- The multiplicative inverse of a non-zero number.
- Absolute value
- The distance of a number from zero, written |x|.
Integers And The Number Line
Numbers increase as you move right on a number line and decrease as you move left. A negative number with a larger absolute value is smaller: -12<-5. The absolute value ignores direction and gives distance from zero, so |-12|=12.
Operations With Negative Numbers
For addition, moving right represents adding a positive number and moving left represents adding a negative number. Subtracting a number is the same as adding its opposite.
| Operation | Sign rule |
|---|---|
| Positive × positive | Positive |
| Positive × negative | Negative |
| Negative × positive | Negative |
| Negative × negative | Positive |
Worked Example: Integer Calculation
Question: Evaluate -18-(-7)+4(-3).
- Rewrite subtraction: -18+7-12.
- Calculate from left to right: -11-12=-23.
Answer: -23
Rational Numbers
Rational numbers include integers, fractions, terminating decimals and recurring decimals. For example, -4=-\frac{4}{1}, 0.125=\frac{1}{8} and 0.\overline{3}=\frac{1}{3}. A rational decimal either terminates or eventually repeats.
Irrational Numbers
Irrational numbers have non-terminating, non-recurring decimal expansions. Examples include \sqrt{2}, \sqrt{7} and \pi. Not every square root is irrational: \sqrt{49}=7 is rational. A square root of a positive integer is irrational only when the integer is not a perfect square after simplification.
| Number | Classification | Reason |
|---|---|---|
| -9 | Integer and rational | -9=-\frac{9}{1}. |
| \frac{11}{20} | Rational | Already a ratio of integers. |
| 0.1010010001\ldots | Irrational | The digits do not settle into a repeating block. |
| \sqrt{81} | Natural, integer and rational | \sqrt{81}=9. |
| \pi | Irrational | It cannot be written as an exact ratio of integers. |
Reciprocals
The reciprocal of a non-zero number x is \frac{1}{x}. A number multiplied by its reciprocal equals 1. The reciprocal of \frac{a}{b} is \frac{b}{a}, provided a\ne0. Zero has no reciprocal because there is no number that can be multiplied by zero to give 1.
Worked Example: Reciprocals
Question: Find the reciprocal of -2\frac{3}{5}.
- Convert to an improper fraction: -2\frac{3}{5}=-\frac{13}{5}.
- Invert numerator and denominator, keeping the negative sign.
Answer: -\frac{5}{13}
Order Of Operations With Real Numbers
Use brackets first, then powers and roots, then multiplication and division, then addition and subtraction. Multiplication and division have equal priority and are performed from left to right; the same applies to addition and subtraction.
Worked Example: Order Of Operations
Question: Evaluate 5-3^2\div(-3)+2(4-7).
- Bracket: 4-7=-3.
- Power: 3^2=9.
- Multiplication and division: 9\div(-3)=-3 and 2(-3)=-6.
- Then 5-(-3)-6=2.
Answer: 2
8th Edition Chapter Map
- Negative numbers and the number line
- Addition, subtraction, multiplication, division and combined operations
- Negative fractions and negative decimals
- Rational, irrational and real numbers
Absolute Value And Distance
The absolute value |x| is the distance of x from zero, so it is never negative. For example, |-8|=8. Distance between two positions on a number line can be calculated as the absolute difference: the distance between -6 and 11 is |11-(-6)|=17.
Sign Rules Explained
Subtraction can be interpreted as adding the opposite: a-b=a+(-b). This explains why subtracting a negative number increases a value. For multiplication and division, equal signs produce a positive result and different signs produce a negative result.
Worked Example: Combined Operations
Evaluate -18-3(-4)+24\div(-6).
- Multiplication and division first: 3(-4)=-12 and 24\div(-6)=-4.
- -18-(-12)-4=-18+12-4=-10.
Negative Fractions And Decimals
The negative sign may be written in front of a fraction, in the numerator or in the denominator: -\frac35=\frac{-3}{5}=\frac{3}{-5}. Two negative signs cancel. The same order-of-operations rules apply to fractions and decimals.
Worked Example: Negative Rational Numbers
-1.8+\frac34-\left(-\frac25\right)=-\frac95+\frac34+\frac25=-\frac75+\frac34=-\frac{28}{20}+\frac{15}{20}=-\frac{13}{20}=-0.65.
Rational And Irrational Numbers
A rational number can be written as \frac{p}{q}, where p and q are integers and q\ne0. Its decimal form terminates or recurs. An irrational number cannot be written in that form; its decimal expansion is non-terminating and non-recurring. Examples include \sqrt2, \sqrt7 and \pi. A root is irrational only when the number under the root is not an exact square (after simplification).
| Set | Examples | Relationship |
|---|---|---|
| Natural numbers | 1,2,3,\ldots | Contained in the integers |
| Integers | \ldots,-2,-1,0,1,2,\ldots | Contained in the rational numbers |
| Rational numbers | \frac58,-3,0.\overline4 | Together with irrational numbers form the real numbers |
| Irrational numbers | \sqrt3,\pi | Real but not rational |
Extended Practice
A. Evaluate -3\{4-2(-5)\}+7.
Inside brackets, 4+10=14. Therefore -3(14)+7=-42+7=-35.
B. Classify \sqrt{81},\sqrt{18},-\frac{11}{4},0.121212\ldots.
\sqrt{81}=9 is an integer and rational; \sqrt{18}=3\sqrt2 is irrational; the other two are rational.
C. The temperature rises from -7.5^\circ\text{C} to 4.8^\circ\text{C}. Find the rise.
4.8-(-7.5)=12.3^\circ\text{C}.
Examination Guidance
- Use brackets around negative numbers when substituting into powers, e.g. (-3)^2=9 but -3^2=-9.
- Classify numbers as specifically as requested; a natural number is also an integer, rational and real number.
- Do not use a calculator decimal to decide whether a number is rational unless the representation is exact.
- State that division by zero is undefined, not zero.
Common Mistakes
- Assuming every square root is irrational.
- Thinking -8 is less negative than -3; on the number line, -8<-3.
- Forgetting that (-4)^2=16 while -4^2=-16.
- Giving 0 as the reciprocal of 0.
Knowledge Check
1. Evaluate -6+4(-5)-(-9).
Answer
-17.
2. Classify 0.\overline{125}.
Answer
Rational.
3. Is \sqrt{50} rational or irrational?
Answer
Irrational, because \sqrt{50}=5\sqrt2.
4. Find the reciprocal of -\frac{7}{9}.
Answer
-\frac{9}{7}.
5. Evaluate (-2)^4-3(-5).
Answer
31.
Mixed Review With Full Solutions
1. Evaluate -4.2-\{1.7-3(-2.5)\}.
Inside braces, 1.7+7.5=9.2. Hence -4.2-9.2=-13.4.
2. Find |-12.6|-|4.9-11.3|.
|-12.6|=12.6 and |-6.4|=6.4. Difference =6.2.
3. State whether each number is rational or irrational: \sqrt{49},\sqrt{50},\frac{\pi}{2},0.1010010001\ldots.
\sqrt{49}=7 is rational. \sqrt{50}=5\sqrt2, \pi/2 and the non-repeating non-terminating decimal are irrational.
4. A submarine is at -135 m and rises 48 m, descends 27.5 m and rises 66.5 m. Find its final depth.
-135+48-27.5+66.5=-48. It is 48 m below sea level.
