P.3. Scientific Notation
Scientific notation uses powers of ten to express very large or very small numbers. An example of scientific notation is 2*103, which represents the number 2*10*10*10 = 2*1000 = 2000. Some other examples are -5.4*104, which is equal to -5.4*10,000 or -54,000 and 3.8*10-2 = 3.8*0.01 = 0.038. The number to the left is called the mantissa. It is a real number whose absolute value is greater than or equal to 1 and less than 10.
There is an easy way to write out the full form of any number given in scientific notation. If the exponent is positive, it is one less than the number of digits of the number to the left of the decimal point. We can see this in the first two examples given above. Thus, for instance, we know that 2*103 must have 3+1=4 digits to the left of the decimal point. To write it out, we start by writing the mantissa, namely 2, then add three zeros, which gives the number four digits. For the second example, to write out -5.4*104, we start by writing the mantissa without the decimal point, namely -54. Then we add enough zeros to give the number 4+1=5 digits to the left of the decimal point. Thus we see that the full form of -5.4*104 is -54,000.
What if the exponent is negative? In this case, the corresponding number has absolute value less that 1, so we start by writing 0. if the number is positive and -0. if the number is negative. Now we must add zeros to the right of the decimal point. How many? The total number of zeros, including the one to the left of the decimal point, is equal to the absolute value of the exponent, so the number to the right of the decimal point is one less than this. Thus, for instance, the expression 3.8*10-2 starts with two zeros, the first to the left and the second to the right of the decimal point. Finally, we write down the absolute value of the mantissa without the decimal point. Thus we see that the full form of 3.8*10-2 is 0.038.
Example 1: Express the following numbers, given in scientific notation, in full form.
- (a) 2.5*103
- (b) -1.7*102
- (c) 4*108
- (d) -3.14*10-6
- (e) 2.718*100
- (a) The exponent is 3, so the number has four digits to the left of the decimal point and starts off as 25. The result is 2500.
- (b) The exponent is 2, so the number has three digits to the left of the decimal point and starts off as -17. The result is -170.
- (c) The exponent is 8, so the number has nine digits to the left of the decimal point and starts with 4. The result is 400,000,000.
- (d) The exponent is -6, so the number has five zeros to the right of the decimal point, is negative, and starts with -314. The result is -0.00000314.
- (e) The exponent is 0, so the number is equal to the mantissa, which in this case is 2.718.
Given the full form of a number, it is also easy to represent the number in scientific notation. If the number has an absolute value greater than 1, we first count the number of digits of the number and set the exponent equal to one less than this number. The mantissa is the number with the same sign as the original number and absolute value greater than or equal to 1 and less than 10, which starts out the same way as the original number. Thus, for instance, the mantissa of 2000 is 2 and the exponent is 3, so the scientific notation of this number is 2*103. As a second example, consider the current (as of January 2008) estimated world population of 6,640,000,000. The mantissa is 6.64 and the exponent is 9, so the scientific notation for this number is 6.64 *109.
Example 2: Express the following numbers in scientific notation.
- (a) 65,000
- (b) -0.024
- (c) -180,000,000,000,000
- (d) 0.000000000529
- (e) 0
- (a) 65,000 has five digits, so the exponent is 4 and the mantissa is 6.5. The result is 6.5*104.
- (b) -0.024 has one zero to the right of the decimal point, so the exponent is -2 and the mantissa is -2.4. The result is -2.4*10-2.
- (c) -180,000,000,000,000 has 15 digits to the left of the decimal point, so the exponent is 14 and the mantissa is -1.8. The result is -1.8*1014.
- (d) 0.0000000000529 has ten zeros to the right of the decimal point, so the exponent is -11 and the mantissa is 5.29. The result is 5.29*10-11. (This is the Bohr radius of a hydrogen atom in meters).
- (e) The usual rules for converting a number to scientific notation do not apply to zero, so the scientific notation of 0 is simply 0.