1.4 Scientific notation
It is an inescapable consequence of the immense scale of the Universe that astronomy deals with huge numbers. Our Sun has a mass of approx-imately 2,000,000,000,000,000,000,000,000,000,000 kilograms, there are about 300,000,000,000 stars in our Milky Way galaxy, and there are between 50,000,000,000 and 1,000,000,000,000 galaxies in the observable Universe.
You can express huge numbers such as these using words, such as two thousand
1.4 Scientific notation 29
billion billion billion kilograms, and three hundred billion stars, but this is still unwieldy. And you can’t do calculations with numbers that are written out in words. The most succinct and flexible way to write and manipulate very large (and very small) numbers is to use scientific notation.
Of course, the subject of scientific notation is covered fairly early in the curriculum of most schools, so you may be entirely comfortable with numbers expressed as 2× 1030or 6.67 × 10−11. If so, feel free to skip this section. But if it’s been a few years since you’ve encountered scientific notation, or if you have any doubt at all about the difference between 6× 10−3and−6 × 103, or how to calculate(8×107)/(2×1012) in your head, this section may be helpful for you.
1.4.1 Coefficient, base, and exponent
In scientific notation, the very large number 300,000,000 (which is the number of meters light travels in one second) is written as 3× 108, and the very small number 0.0000000000667 (which is the Universal Gravitational Constant in standard units) is written as 6.67 × 10−11. As shown in Figure 1.3, each of these expressions consists of three numbers called the coefficient, the base, and the exponent. The standard base for scientific notation is 10. Exponents are usually integers and can be positive or negative. The coefficient can be any number at all. If you see a number in scientific notation in which the coefficient is missing, such as 106, it is important to remember that a coefficient of 1.0 is implicit. That is, 106= 1 × 106.
Many astronomy texts use “normalized” scientific notation in which the decimal point in the coefficient always appears immediately to the right of the leftmost non-zero digit. So although 3.5 × 104 and 35. × 103 represent exactly the same number, astronomy texts are more likely to use the first ver-sion of this number. In normalized scientific notation, the coefficient is always between one and ten, and the exponent is called the “order of magnitude” of the number.
3 ×10 8
Coefficient Base Exponent
6.67 ×10 –11
Coefficient Base Exponent
Figure 1.3 The elements of a number in scientific notation.
30 Fundamentals
If you think about the mathematical operations represented in scientific notation, you can understand why these numbers are written this way. First, consider the number 300,000,000 or 3× 108, and recall that 108 is simply 10× 10 × 10 × 10 × 10 × 10 × 10 × 10, which is 100,000,000. So 3 × 108is just 3 times 100,000,000, which is 300,000,000.
The same logic applies to the number 0.0000000000667 or 6.67×10−11, but in this case the number 10−11is the very small number 10111, or 100,000,000,0001 , or 0.00000000001. So 6.67 × 10−11means 6.67 times 0.00000000001, which is 0.0000000000667.
One thing to remember when you’re dealing with numbers written in sci-entific notation is that a negative sign in front of the coefficient (such as
−6 × 103) means that the number is negative, but a negative sign in the exponent (such as 6× 10−3) does not have any effect on whether the num-ber is positive or negative. So what does a negative exponent mean? Simply this: the more negative the exponent, the closer the value of the number is to zero. So 6× 10−3is a small number, and 6.3 × 10−11 is a very very small number. In astronomy, you are unlikely to encounter many negative numbers, but you are very likely to see negative exponents. For example, the values of some of the physical constants, wavelengths of light, and masses of atoms are all very small and are often written using scientific notation with negative exponents.
Example: Identify the base, coefficient, and exponent in the numbers (a) 150× 106and (b) 1.6 × 10−19.
The coefficient is the number in front including any negative signs (though in this case, both numbers are positive), so the coefficients for (a) and (b) are 150 and 1.6, respectively. The base is 10 for both, the standard for scientific notation. The exponent is the power that 10 is raised to, including any negative signs, so the exponents are 6 and−19, respectively.
Note that the number in (a) above is not in normalized scientific notation because the coefficient (150) is not between 1 and 10. Sometimes you may wish to move the decimal point in the coefficient – perhaps to put it in nor-malized notation, or to facilitate comparing with other numbers in scientific notation, or to allow you to do a calculation in your head. This comes up a lot, so it’s a good idea to be comfortable with this procedure. The key to keep in mind is that you are not changing the value of the number; you are only changing the way it looks. So if you move the decimal point in the coefficient, the value of the number will change unless you adjust the exponent to com-pensate. For example, if you move the decimal point in the coefficient to the
1.4 Scientific notation 31
left some number of places, then you are making the coefficient smaller by that many powers of 10, so you must increase the exponent by the same number of powers of 10 to compensate. This ensures that the overall value of the entire number is not changed.
Example: Express 150× 106in normalized scientific notation.
The first step is to change “150.” (the decimal point after the zero was implicit) to “1.50,” which requires moving the decimal point two places to the left. This decreases the value of the coefficient by a factor of 100 (two powers of 10). You must then increase the value of the rest of the number (106) by two powers of 10, to 108. Thus the remodeled number is 1.5 × 108, which has exactly the same value as 150× 106.
On the other hand, if you move the decimal point in the coefficient to the right some number of places, you are making the coefficient larger by that many powers of 10, so you must decrease the exponent by the same number of powers to compensate. For example, if you have the number 0.026 × 103, you can put this into normalized scientific notation by moving the decimal point of the coefficient two places to the right (so 0.026 becomes 2.6). This is equivalent to multplying the coefficient by a factor of 100 (two powers of 10), so to compensate you need to reduce the rest of the number by a factor of 100. To do that, you can reduce the exponent by two, which turns 103into 101. Thus 0.026 × 103= 2.6 × 101.
Here are some equivalent expressions (not in normalized notation) for the numbers used in the preceding examples:
150× 106= (150 × 10) × 10(6−1)= 1500 × 105 and
1.6 × 10−19= (1.6 × 1000) × 10(−19−3)= 1600 × 10−22.
Exercise 1.18. Write the following numbers in scientific notation with the