Question 1
Juergen says pi may be called an irrational number, but it should also be called a rational number since 3.14159 can be written as a fraction: 314159/100000 . Is he correct?
No, Juergen is incorrect. The definition of a rational number is a number that can be expressed as the quotient or fraction of two integers. While 3.14159 can be approximated by the fraction 314159/100000, it is not an exact representation. The decimal representation of pi goes on infinitely without repeating, making it an irrational number.
shorter answer
No, Juergen is incorrect. Pi is an irrational number because it cannot be expressed exactly as a fraction of two integers.
Negative Numbers Integers include whole numbers and the opposites of whole numbers. This brings us to negative numbers, although not all negative numbers are integers. Negative numbers were used to indicate debts around 700 BCE in India. Yet as late as the seventeenth century CE, mathematicians in Europe spoke of negative numbers as nonsense. Now we use them for overdrawn balances, temperatures below zero, and elevations below sea level. Fractions Fractions, which are part of the rational numbers set, were already in use in ancient Egypt. But in Europe they were not used as we use them now until the seventeenth century CE. Now we use fractions whenever we need to deal with pieces of something, such as a half cup of flour. Decimals Islamic mathematicians were already using decimals in the ninth century CE. European mathematicians only started using them in the sixteenth century. Now we use decimals to express money in dollars and cents and for many types of measurements, particularly in the metric system. Irrational Numbers In ancient Greece, the followers of Pythagoras believed all numbers were rational numbers. There is a legend that they murdered the person who first proved some numbers were irrational. Today, we use irrational numbers in many situations. We need the square root of 2 to calculate the diagonal of a square. We need pi (π) to calculate the circumference and area of a circle.
shorten this
Negative numbers were initially used to indicate debts in ancient India, but were considered nonsense in Europe until the 17th century CE. Now negative numbers are used for things like overdrawn balances and temperatures below zero.
Fractions, part of the rational numbers set, were used in ancient Egypt but not widely used in Europe until the 17th century CE. Fractions are now used for measurements like a half cup of flour.
Decimals were used by Islamic mathematicians in the 9th century CE, but European mathematicians didn't widely adopt them until the 16th century. Decimals are now used for expressing money and measurements in the metric system.
Irrational numbers were not recognized by the followers of Pythagoras in ancient Greece, but they are now used in calculations involving the diagonal of a square (square root of 2) and the circumference and area of a circle (pi).