The Alphabet of Math: Decimal numbers

Rational numbers are actually the division of whole numbers into smaller parts. A fraction represents a division problem. The numerator is the dividend and the denominator is the divisor. So 7/4 could also be written as 7÷4. 4 divides 7 once with a remainder of 3. We can represent this as a mixed number, a whole number for the whole number of divisions and a rational number for the remaining parts of the whole left over. So 7/4 is the same as 1 ¾. But, we won't represent rational numbers with mixed numbers. We are going to use a new type of number that looks more like whole number called a decimal number.

A decimal number has two parts, a whole number part and a fractional part separated by a decimal. To get a decimal representation of a rational number we use long division and instead of stopping when we get to the remainder, we keep dividing until we get to a zero remainder. 7/4 would be divided like this

		1
4	(	7
	-	4
		3

This is where we have been stopping. But we can continue the division process to get a decimal representation by placing a decimal after the 1 and using zeros to continue dividing until we get a zero remainder.

		1.	7	5
4	(	7
	-	4
		3	0
	-	2	8
			2	0
		-	2	0
				0

So 7/4 = 1.75

This is a special decimal number called a non-repeating, terminating decimal. This decimal number doesn;t repeat and it comes out to a zero remainder. What happens if the number doesn't come out to a zero remainder?

		2.	3	3
3	(	7
	-	6
		1	0
	-		9
			1	0
		-		9
				1

This decimal number repeats 3 over and over again. This is called a repeating decimal number. We usually only show the repating part once and draw a bar opver the top of the repeating part.

All rational numbers can be represented as either non-repeating, terminating decimals or repeating decimals.

Adding decimal numbers is done just like whole numbers. Line up the decimals then add the numbers just like whole numbers.

	1	.	5	4	3
+	2	.	3	4
	3	.	8	8	3

Multipying decimal numbers is similar to multiplying whole numbers. Multiply the numbers just like whole numbers disregarding the positions of the decimals. After you've finished multiplying, count the number of digits to the right of the decimal in each number. This is the total number of decimals that will be in the product. Starting from the right, you count positions to the left for as many decimals as you have in the multiplicand and multiplier and place the deicmal in that position.

	1	.	5	4	3
×		2	.	3	4
		6	1	7	2
	4	6	2	9
3	0	8	6
3	6	1	0	6	2

There are 3 decimals in the multiplicand and 2 in the multiplier so our answer needs 5 decimals. Starting from the right we count back 5 positions and place a decimal between 3 and 6. So 1.543×2.34=3.61062

Dividing decimals works just like dividing we have seen above but we start by making the divisor into a whole number by miving the decimal to the right until there are no more numbers. We also have to move the deicmal in the dividend the same number of places to the right. The decimal in the quotient will be at the same position as the new position in the dividend.

2.

3

4

(

3.

6

1

0

6

2

There are 2 decimal positions in the divisor. So move the decimal 2 places to the right in the divisor and the dividend. Now divide as usual.

						1.	5	4	3
2	3	4	(	3	6	1.	0	6	2
			-	2	3	4
				1	2	7	0
			-	1	1	7	0
					1	0	0	6
					-	9	3	6
							7	0	2
						-	7	0	2
									0

What happens if the quotient doesn't repeat and doesn't terminate? This is the last of our number types for our alphabet. They are called irrational numbers because they can't be represented as rational numbers. There are lots of irrational numbers and we'll discuss them in another post.