Integers
Integers encapsulate the universe of whole numbers, spanning from negative infinity to positive infinity.
Definition
An integer is a number without any fractional part. This includes all whole numbers and their negatives.
Properties
- Integers include both positive and negative numbers.
- They do not have fractional or decimal parts.
- Zero, neither positive nor negative, is an integer.
Examples
- The numbers -5, -2, 0, 1, 4, and 100 are all integers.
- Numbers like 2.5, -0.77, and 1/3 are not integers.
Real-World Application
Imagine owing someone £5. This debt can be represented as -5, while having £5 can be portrayed as +5.
Question:
Which of the following are integers: -3, 5.8, 7, 0, -11.1, 90?
Answer: -3, 7, 0, and 90 are integers.
Rationals
Rational numbers are incredibly versatile, found almost everywhere in mathematics.
Definition
Rationals are numbers that can be expressed as the quotient of two integers, where the denominator is not zero.
Properties
- All integers are, by definition, rational numbers.
- Rationals can appear in fractional or decimal form.
- Terminating decimals or those with a repeating pattern are rational.
Examples
- 2/3 is a rational number.
- 0.125 (as it can be expressed as 1/8).
- 0.666... (a repeating decimal, equivalent to 2/3).
Real-World Application
If you were to divide a chocolate bar among 3 friends, each would get 1/3 of it. This fraction is a rational number.
Question:
Is 0.777... rational?
Answer: Yes, 0.777... is equivalent to 7/9, which is rational.
Irrationals
Irrational numbers might seem strange, but they hold great significance in the world of mathematics.
Definition
Irrationals are numbers that cannot be expressed as a simple fraction.
Properties
- They never settle into a repetitive decimal pattern nor terminate.
- Square roots of numbers that aren't perfect squares are irrational.
- Constants like pi (π) and the golden ratio (φ) are irrational.
Examples
- The value of π is approximately 3.14159265..., it goes on forever without repeating.
- √3 is irrational and is roughly 1.732.
Real-World Application
The value of pi (π) is used to calculate the circumference of circles. So, every time you measure anything round, from a pizza to a planet, you are venturing into the realm of irrational numbers.
Question:
Is the square root of 7 rational or irrational?
Answer: The square root of 7 doesn't result in a whole number and cannot be expressed as a fraction, hence it's irrational.
Distinguishing Rationals from Irrationals
Confusion often arises when students attempt to classify numbers as either rational or irrational. Here are some strategies to help:
Tips
- Observe the decimal. If it's repeating or terminates, it's likely rational.
- Attempt to express the number as a fraction.
- Familiarise yourself with common irrationals, such as roots of non-perfect squares.
Question:
How would you classify 0.12112111211112...?
Answer: This doesn't show a consistent repeating pattern, making it irrational.
FAQ
Rational numbers are used frequently in our daily lives, often without us realising it. When we budget our expenses and divide our monthly income to allocate it to various needs, we're using rational numbers. If you were to slice a pizza into 8 equal parts and take 3 slices, you've consumed 3/8 of the pizza, a rational number. When fueling your car, if you put in half a tank or three-quarters of a tank, you are working with rational numbers. Measurements in cooking recipes, discounts in shopping, and calculating grades in school are all real-life applications of rational numbers.
Yes, the product of two irrational numbers can be rational. A classic example of this is the multiplication of √2 by itself. Both numbers are irrational, but their product is 2, which is a rational number. This showcases that irrational numbers, despite their non-repeating and non-terminating characteristics, can sometimes yield rational results when multiplied with other irrationals, depending on the specific numbers in question.
The nature of square roots as rational or irrational depends on the number being rooted. If the square root of a number results in a whole number or a fraction that can be expressed with a terminating or repeating decimal, it's rational. Essentially, if the number is a perfect square (like 4, 9, 16, 25, etc.), its square root will be rational. However, if the number isn't a perfect square, its square root won't neatly fit into a whole number or a simple fraction, making it irrational. For instance, the square root of 2 or 3 doesn't yield a whole number and can't be written as a precise fraction, so they are irrational.
Natural numbers are the set of positive counting numbers. They begin from 1 and go on indefinitely (1, 2, 3, 4, …). They don't include zero or any negative numbers. Integers, on the other hand, encompass a broader set of numbers. They include all whole numbers, both positive and negative, as well as zero. So, integers can be represented as (…, -3, -2, -1, 0, 1, 2, 3, …). In essence, while natural numbers are a subset of integers, integers offer a more comprehensive set that spans from negative infinity to positive infinity without any fractional or decimal components.
Irrational numbers can't be written as exact fractions because their decimal expansions are neither terminating nor repeating. This means they don't settle into a predictable pattern, and they don't stop, which are the two characteristics that would make a decimal representable as a fraction. When you attempt to write an irrational number as a fraction, you'd find that it's impossible to pinpoint two integers (a numerator and a denominator) that would represent that number precisely. The inherent nature of irrationals defies the conventional fractional representation.
Practice Questions
The number √3 cannot be expressed as a simple fraction and its decimal expansion does not terminate or repeat, so it's an irrational number. The number 7/2 can be expressed as a fraction (and is equivalent to 3.5 when expressed as a decimal), so it's a rational number. The number -8 is a whole number without any decimal or fractional part, so it's an integer. The number 0.444... has a repeating decimal, thus it is a rational number. Lastly, √121 is equal to 11, which is an integer.
The sum of two rational numbers is always a rational number. Rational numbers can be expressed as a fraction where the denominator isn’t zero. When we add two fractions (with like or unlike denominators), the result will also be a fraction. For example, consider the two rational numbers 1/3 and 2/5. Their sum is 5/15 + 6/15 = 11/15. This resultant number can still be expressed as a fraction, confirming that it is a rational number.
