Numbers - Session 4

Did you know there are numbers that never end and never form a pattern, no matter how far you keep counting their digits? Irrational numbers are numbers that we cannot write as a simple fraction, like two over five or seven over eight. In other words, you cannot write them as p over q, where p and q are whole numbers and q is not zero. Their decimal form goes on forever without repeating any fixed pattern. For example, the square root of two is about one point four one four, but the digits keep going without ever repeating in a cycle. This makes irrational numbers different from rational numbers, which can either stop or repeat in their decimals.

An easy way to understand irrational numbers is to compare them with rational numbers. Rational numbers include fractions like one over two, which is zero point five, or one over three, which is zero point three three three repeating forever. Irrational numbers are different because their decimal part never ends and never repeats. For example, the square root of three begins with one point seven three two zero five and keeps going endlessly.

Another very famous irrational number is pi, which is used in almost every calculation involving circles. Pi starts with three point one four one five, but its digits never stop or repeat. Mathematicians have calculated billions of digits of pi, yet it keeps going without any pattern.

There is also the number e, which starts with two point seven one eight. This number appears in many areas of mathematics and science, especially in problems about growth and decay, like population growth or compound interest. In the mathematical field-of-study trigonometry, some values give irrational numbers when written as decimals. For example, the sine of forty-five degrees is irrational and cannot be expressed as a simple fraction.

A reciprocal is the flipped version of a number. To find the reciprocal of a number, you swap its numerator and denominator. For example, the reciprocal of two over three is three over two. For whole numbers, you can think of them as having a denominator of one. So, the reciprocal of five is one over five. Can you tell the reciprocal of six over seven?.

Here are some more examples to understand reciprocals better. The reciprocal of one over four is four over one, which is simply four. The reciprocal of eight is one over eight. The reciprocal of negative three over seven is negative seven over three. The reciprocal of zero point five, which is one over two, is two.

Decimals can also have reciprocals once we turn them into fractions. For example, the reciprocal of zero point two five is four because zero point two five is one over four and flipping it gives four over one. The reciprocal of one point five is two over three because one point five is three over two and flipping gives two over three. Also, epeating decimals after the dot can have reciprocals once you turn them into fractions first.

Reciprocals are closely linked to multiplication. When you multiply a number by its reciprocal, the result is always one. For example, three over five multiplied by five over three equals one. This works for positive and negative numbers, but it never works for zero because zero has no reciprocal.

Fractions make it easy to see what a reciprocal looks like. If you start with three over four, its reciprocal is four over three. If you start with seven over two, its reciprocal is two over seven. The process is the same for all proper fractions, improper fractions, and mixed numbers once you change mixed numbers into improper fractions first.

Negative numbers also have reciprocals, and the process stays the same. For example, the reciprocal of negative two over five is negative five over two. The negative-sign or positive-sign of the number does not change when you flip it. Multiplying a negative number by its reciprocal also gives one, because a negative multiplied by a negative gives a positive.

Reciprocals are very important in division. When we divide by a fraction, we multiply by its reciprocal instead. For example, dividing by two over three is the same as multiplying by three over two. This is a key rule for working with fractions in both arithmetic and algebra.

To change a number into words, you should start by breaking the number into groups of three digits, beginning from the right-side. Each group of three digits represents a level of value. The first group is the ones, tens, and hundreds, the second group is the thousands, ten thousands, and hundred thousands. the next group is the millions, then billions, and so on. This method helps you read-out numbers in an organized way and prevents confusion when the number has many digits.The illustration shows how the number represented by three-four-two-three-six-five is broken-down into one’s, ten’s, hundred’s, thousand’s, ten’s of thousand’s and hundred’s of thousand’s.

Let us take the number 4,572 as an example. First, group the digits from the right-hand-side as four and five hundred seventy two. The group five hundred seventy two is read-out as it is. The group four is in the thousands place, so you say four thousand. Put them together and you get four thousand five hundred seventy two. Can you change the number 87,664 into words?.

Another example is the number 235,814. First, group it as two hundred thirty five and eight hundred fourteen. The eight hundred fourteen is read-out exactly like that. The two hundred thirty five is in the thousands place, so you say two hundred thirty five thousand. Put them together and you have two hundred thirty five thousand eight hundred fourteen.

If the number has a decimal point, say the whole number first, then say point and read-out each decimal digit separately. For example, 45.67 becomes forty five point six seven.To change words into numbers, first check if it is in tens, hundreds, thousands, ten thousands, or hundred thousands and count digits according to it. Then, listen for the largest value first and write it down. Then add the smaller parts in the correct places. For example, two thousand three hundred nineteen is in thousands so it has 4 digits. Start with the two thousand, then add the three, then nineteen at the end, making 2,319.