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Welcome to our Math lesson on The Meaning of Decimal Number System. Writing a Number in Terms of Powers of Ten, this is the first lesson of our suite of math lessons covering the topic of Decimal Number System and Other Numbering Systems, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
We explained in tutorial 1.1 that we use a base 10 numbering system, in which we use 10 different digits to represent numbers (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9). All numbers are written by combining these 10 digits, similar to songs that are made by using the combination of 7 musical notes. We call this numbering system made of 10 basic digits the decimal system (decimal means relating to or denoting a system of numbers and arithmetic based on the number ten, tenth parts, and powers of ten). In simpler words, the units in the decimal system change 10 by 10.
Any number N in decimal system is written in the following form:
where n is the number of digits and a, b, c, d.... are the digits of the number.
For example, in the number 952,378, n = 6 because the number has 6 digits, a = 9, b = 5, c = 2 and so on. Hence, based on the above rule, this number is written as
Otherwise, we can write:
All the above forms represent decomposed forms of the original number. However, the formal one is the first form, that is the one written as a sum of powers of ten.
Decompose the number 42,100,038 in powers of ten.
Since the number has 8 digits (n = 8), powers range from 7 to 0 when viewed from left to right. This is because in general powers range from n - 1 to 0. Thus, we have
Likewise, we can also write the original number from its decomposed form. For example, if we have the number 4 × 104 + 9 × 103 + 1 × 102 + 5 × 101 + 7 × 100, we can write is as
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