Mental Math

Squaring Numbers

Instantly square numbers ending in 5, near 50, near 100, and difference of squares shortcuts.

Lessons included in this chapter:

• Square Numbers Ending in 5: Multiply the front digits by (front+1), then append 25.
• Square Numbers Ending in 1: 31² = 30² + 2×30 + 1 = 900 + 60 + 1 = 961.
• Square Numbers Ending in 9: 39² = 40² − 2×40 + 1 = 1600 − 80 + 1 = 1521.
• Square Numbers Near 50: (50±d)² = 2500 ± 100d + d².
• Square Numbers Near 100: (100±d)² = 10000 ± 200d + d².
• Difference of Squares: a² − b² = (a+b)(a−b). Find the midpoint, square it, subtract the gap squared.
• Same Tens, Units Sum to 10: tens × (tens+1) gives the left part; units₁ × units₂ (2 digits) gives the right.