Experimental measurements in science may include very large multiples of units and very small submultiples of them. Since measurements are expressed as a product of a number and a unit, this means that scientists have had to express very large and very small numbers. For example, one copper Canadian penny contains 564,240,000,000,000,000,000,000 atoms of copper. This is a clumsy notation, and it implies that we actually know that the number is not 564,240,000,000,000,000,000,005 atoms. We do not, and cannot, know this. It is therefore both more convenient, and by implication more honest, to express the number of atoms of copper in a copper penny as 56.424 x 10+23 atoms in this exponential notation.
A number is written in exponential notation as the product of a real number (with a decimal point) multiplied by ten to some integral power (the exponent). Alternatively, but less conveniently, we could explicitly give the precision of the measurement. On an accurate analytical balance, the number of copper atoms in a copper penny might be 56.424 +/- 0.005 x 10+23 atoms. Scientists usually write numbers in a form of exponential notation called scientific notation, which means that the number is written with one non-zero digit to the left of the decimal point and an integer exponent or power of ten. The number of atoms of copper in a copper penny would be written as 5.6424 x 10+24 atoms in scientific notation.
Another form of exponential notation called engineering notation is also convenient. In engineering notation the number is written with one, two, or three digits to the left of the decimal point and the integer exponent is always expressed as a number divisible by three. For example, 5.6424 x 10+24 atoms is in both scientific notation and engineering notation while 56.424 x 10+23 atoms is in neither, although both represent exactly the same number in exponential notation. Engineering notation is particularly convenient in the International System of Units (SI) because many powers of ten which are evenly divisible by three have a named prefix with an easily identifiable symbol.
The expression of a number as a power of ten is convenient because our number system is base on decimal pattern. We call the power to which ten is raised an exponent of ten, and exponents are normally written as superscripts. Thus 10+2 = 10 x 10 = 100 and 10+3 = 10 x 10 x 10 = 1000. A number which is raised to a power is called, in mathematics, a base. Numbers other than ten can be used as bases. For example, 2+3 is base two to exponent three, more often described as two to the third power, and equals 2 x 2 x 2 = 8; 2+4 = 16, and so on. Use of base ten is convenient because shifting the decimal point one place to the right and increasing the exponent by one are equivalent operations, as are shifting the decimal point one place to the left and decreasing the exponent by one. If either of these operations is repeated until the exponent is zero, the exponential part disappears because 100 = 1, and the number is again in ordinary non-exponential notation.
All forms of exponential notation are particularly convenient when products, quotients, powers, and roots must be calculated. Multiplications of two numbers in exponential form involve addition of their exponents, while division of two numbers in exponential form involves subtraction of their exponents. For powers, the exponent is multiplied, and for roots, the exponent is divided. Taking of a root is easier if the exponent of the number whose root is to be taken is evenly divisible by the desired root.
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