Comparing Very Large and Very Small Numbers
Utilizing standard form representations to compare magnitudes of extreme values.
Definition
Comparing very large and very small numbers involves analyzing their scale using exponents or standard form rather than standard counting. By converting numbers to standard form with powers of 10, we can easily determine which number is greater by looking at the exponent values. If the exponents are equal, we compare the decimal coefficients.
Key formula
Compare a \times 10^m and b \times 10^n by comparing exponents m and n
Worked examples
Compare 3 \times 10^7 and 4 \times 10^5; since 7 > 5, the first number is much larger.
Compare 1.5 \times 10^{-4} and 2.5 \times 10^{-4}; since exponents are equal, 2.5 is greater than 1.5, so the second number is larger.
Determine which is larger between 500,000 (5 \times 10^5) and 0.00005 (5 \times 10^{-5}) by noting that a positive exponent is always larger than a negative one.
Common mistakes
- • Looking only at the decimal coefficient while ignoring the power of 10, incorrectly concluding that 9 \times 10^2 is larger than 1 \times 10^6.
- • Misinterpreting negative exponents, thinking that a more negative exponent like -10 represents a larger value than a less negative exponent like -3.
Introduction to Rational Numbers
Rational Numbers
Properties of Rational Numbers
Rational Numbers
Representation on Number Line
Rational Numbers
Rational Numbers Between Two Rational Numbers
Rational Numbers
Algebraic Expressions and Equations
Linear Equations in One Variable
Solving Equations with Variables on Both Sides
Linear Equations in One Variable