1.3 - Decimal and Exponential Notation

Decimal notation

The word decimal means ten. The decimal number system is the familiar system that uses just ten symbols (numerals) to create any whole number, no matter how big. Those symbols are of course 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. (In contrast the binary number system uses just the two symbols 0 and 1 and the hexadecimal number system uses the sixteen symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F to create any number. Binary and hexadecimal are the number systems used in computers.)

To create numbers bigger than 9 the decimal system uses place-values. For example the place-value chart on the right shows that 3528 means

   3528 = 3 · 1000 + 5 · 100 + 2 · 10 + 8 · 1

This is because the 3 is in the thousands place, the 5 is in the hundreds place, the 2 is in the tens place and the 8 is in the ones place.

Notice that as we move from right to left in the place-value table, the value of each place is ten times the value of the place to its right.

If we continue this pattern to the right then we get the expanded place-value chart shown here:
A decimal point is used to separate the digit in the ones place from the digits to the right of it. The decimal number 3528.74 means:

Converting numbers from decimal to fraction notation:

We have seen before that whole numbers can be converted to fraction notation by simply putting them over 1.

Decimal numbers with digits to the right of the decimal point can be converted to fraction notation by multiplying them by a UFOO. First identify the place-value of the right-most digit. If the place-value is tenths then multiply by 10/10, if it is hundredths then multiply by 100/100, etc. Then simplify the numerator. This turns the numerator into a whole number. Here are some examples:

Exponential notation

Exponential notation is a convenient shorthand for repeated multiplication. The exponential b n means multiply b times itself n times:
b is called the base, n is called the exponent, and we say that we are “raising b to the n th power” (except when n is 2 we say that we are “squaring b” and when n is 3 we say that we are “cubing b”).

Here are two examples:
3 4 = 3 · 3 · 3 · 3 = 81

4 3 = 4 · 4 · 4 = 64
Important: To avoid confusion make sure you write the exponential neatly, with the exponent smaller and higher than the base. For example, did that person write the exponential 3 4 or just the number 34 ???

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